Euler's sum of powers conjecture

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Revision as of 14:13, 17 February 2024 by imported>Pigeon768 (→‎Second version)
Task
Euler's sum of powers conjecture
You are encouraged to solve this task according to the task description, using any language you may know.

There is a conjecture in mathematics that held for over two hundred years before it was disproved by the finding of a counterexample in 1966 by Lander and Parkin.

This conjecture is called Euler's sum of powers conjecture and can be stated as such:

At least k positive kth powers are required to sum to a kth power, except for the trivial case of one kth power: yk = yk.

In 1966, Leon J. Lander and Thomas R. Parkin used a brute-force search on a CDC 6600 computer restricting numbers to those less than 250.

The task consists in writing a program to search for an integer solution of where all and are distinct integers between 0 and 250 (exclusive). Show an answer here.

Related tasks are:


11l

Translation of: Python
F eulers_sum_of_powers()
   V max_n = 250
   V pow_5 = (0 .< max_n).map(n -> Int64(n) ^ 5)
   V pow5_to_n = Dict(0 .< max_n, n -> (Int64(n) ^ 5, n))

   L(x0) 1 .< max_n
      L(x1) 1 .< x0
         L(x2) 1 .< x1
            L(x3) 1 .< x2
               V pow_5_sum = pow_5[x0] + pow_5[x1] + pow_5[x2] + pow_5[x3]
               I pow_5_sum C pow5_to_n
                  V y = pow5_to_n[pow_5_sum]
                  R (x0, x1, x2, x3, y)

V r = eulers_sum_of_powers()
print(‘#.^5 + #.^5 + #.^5 + #.^5 = #.^5’.format(r[0], r[1], r[2], r[3], r[4]))
Output:
133^5 + 110^5 + 84^5 + 27^5 = 144^5

360 Assembly

In the program we do not user System/360 integers (31 bits) unable to handle the problem, but System/360 packed decimal (15 digits). 250^5 needs 12 digits.
This program could have been run in 1964. Here, for maximum compatibility, we use only the basic 360 instruction set. Macro instruction XPRNT can be replaced by a WTO.

 EULERCO  CSECT
         USING  EULERCO,R13
         B      80(R15)
         DC     17F'0'
         DC     CL8'EULERCO'
         STM    R14,R12,12(R13)
         ST     R13,4(R15)
         ST     R15,8(R13)
         LR     R13,R15
         ZAP    X1,=P'1'
LOOPX1   ZAP    PT,MAXN            do x1=1 to maxn-4
         SP     PT,=P'4'
         CP     X1,PT
         BH     ELOOPX1
         ZAP    PT,X1
         AP     PT,=P'1'
         ZAP    X2,PT
LOOPX2   ZAP    PT,MAXN            do x2=x1+1 to maxn-3
         SP     PT,=P'3'
         CP     X2,PT
         BH     ELOOPX2
         ZAP    PT,X2
         AP     PT,=P'1'
         ZAP    X3,PT
LOOPX3   ZAP    PT,MAXN            do x3=x2+1 to maxn-2
         SP     PT,=P'2'
         CP     X3,PT
         BH     ELOOPX3
         ZAP    PT,X3
         AP     PT,=P'1'
         ZAP    X4,PT
LOOPX4   ZAP    PT,MAXN            do x4=x3+1 to maxn-1
         SP     PT,=P'1'
         CP     X4,PT
         BH     ELOOPX4
         ZAP    PT,X4
         AP     PT,=P'1'
         ZAP    X5,PT              x5=x4+1
         ZAP    SUMX,=P'0'         sumx=0
         ZAP    PT,X1              x1
         BAL    R14,POWER5
         AP     SUMX,PT
         ZAP    PT,X2              x2
         BAL    R14,POWER5
         AP     SUMX,PT
         ZAP    PT,X3              x3
         BAL    R14,POWER5
         AP     SUMX,PT
         ZAP    PT,X4              x4
         BAL    R14,POWER5
         AP     SUMX,PT            sumx=x1**5+x2**5+x3**5+x4**5
         ZAP    PT,X5              x5
         BAL    R14,POWER5
         ZAP    VALX,PT            valx=x5**5
LOOPX5   CP     X5,MAXN            while x5<=maxn & valx<=sumx
         BH     ELOOPX5
         CP     VALX,SUMX
         BH     ELOOPX5
         CP     VALX,SUMX          if valx=sumx 
         BNE    NOTEQUAL
         MVI    BUF,C' '
         MVC    BUF+1(79),BUF      clear buffer
         MVC    WC,MASK
         ED     WC,X1              x1
         MVC    BUF+0(8),WC+8     
         MVC    WC,MASK
         ED     WC,X2              x2
         MVC    BUF+8(8),WC+8    
         MVC    WC,MASK
         ED     WC,X3              x3
         MVC    BUF+16(8),WC+8    
         MVC    WC,MASK
         ED     WC,X4              x4
         MVC    BUF+24(8),WC+8     
         MVC    WC,MASK
         ED     WC,X5              x5
         MVC    BUF+32(8),WC+8     
         XPRNT  BUF,80             output x1,x2,x3,x4,x5
         B      ELOOPX1
NOTEQUAL ZAP    PT,X5
         AP     PT,=P'1'
         ZAP    X5,PT              x5=x5+1
         ZAP    PT,X5
         BAL    R14,POWER5
         ZAP    VALX,PT            valx=x5**5
         B      LOOPX5
ELOOPX5  AP     X4,=P'1'
         B      LOOPX4
ELOOPX4  AP     X3,=P'1'
         B      LOOPX3
ELOOPX3  AP     X2,=P'1'
         B      LOOPX2
ELOOPX2  AP     X1,=P'1'
         B      LOOPX1
ELOOPX1  L      R13,4(0,R13)
         LM     R14,R12,12(R13)
         XR     R15,R15
         BR     R14
POWER5   ZAP    PQ,PT              ^1
         MP     PQ,PT              ^2
         MP     PQ,PT              ^3
         MP     PQ,PT              ^4
         MP     PQ,PT              ^5
         ZAP    PT,PQ
         BR     R14
MAXN     DC     PL8'249'
X1       DS     PL8
X2       DS     PL8
X3       DS     PL8
X4       DS     PL8
X5       DS     PL8
SUMX     DS     PL8
VALX     DS     PL8
PT       DS     PL8
PQ       DS     PL8
WC       DS     CL17            
MASK     DC     X'40',13X'20',X'212060'  CL17
BUF      DS     CL80
         YREGS  
         END
Output:
      27      84     110     133     144

6502 Assembly

This is long enough as it is, so the code here is just the main task body. Details like the BASIC loader (for a C-64, which is what I ran this on) and the output routines (including the very tedious conversion of a 40-bit integer to decimal) were moved into external include files. Also in an external file is the prebuilt table of the first 250 fifth powers ... actually, just for ease of referencing, it's the first 256, 0...255, in five tables each holding one byte of the value.

; Prove Euler's sum of powers conjecture false by finding
; positive a,b,c,d,e such that a⁵+b⁵+c⁵+d⁵=e⁵.

; we're only looking for the first counterexample, which occurs with all
; integers less than this value
max_value      = $fa  ; decimal 250

; this header turns our code into a LOADable and RUNnable BASIC program
.include "basic_header.s"

; this contains the first 256 integers to the power of 5 broken up into
; 5 tables of one-byte values (power5byte0 with the LSBs through
; power5byte4 with the MSBs)
.include "power5table.s"

; this defines subroutines and macros for printing messages to
; the console, including `puts` for printing out a NUL-terminated string,
; puthex to display a one-byte value in hexadecimal, and putdec through
; putdec5 to display an N-byte value in decimal
.include "output.s"

; label strings for the result output
.feature string_escapes
success:   .asciiz "\r\rFOUND EXAMPLE:\r\r"
between:   .asciiz "^5 + "
penult:    .asciiz "^5 = "
eqend:     .asciiz "^5\r\r(SUM IS "

; the BASIC loader program prints the elapsed time at the end, so we include a
; label for that, too
tilabel:   .asciiz ")\r\rTIME:"

; ZP locations to store the integers to try
x0 = $f7
x1 = x0 + 1
x2 = x0 + 2
x3 = x0 + 3

; we use binary search to find integer roots; current bounds go here
low = x0 + 4
hi  = x0 + 5

; sum of powers of current candidate integers
sum: .res 5

; when we find a sum with an integer 5th root, we put it here
x4: .res 1

main: ; loop for x0 from 1 to max_value
       ldx #01
       stx x0

loop0: ; loop for x1 from x0+1 to max_value
       ldx x0
       inx
       stx x1

loop1: ; loop for x2 from x1+1 to max_value
       ldx x1
       inx
       stx x2

loop2: ; loop for x3 from x2+1 to max_value
       ldx x2
       inx
       stx x3

loop3: ; add up the fifth powers of the four numbers
       ; initialize to 0
       lda #00
       sta sum
       sta sum+1
       sta sum+2
       sta sum+3
       sta sum+4

       ; we use indexed addressing, taking advantage of the fact that the xn's
       ; are consecutive, so x0,1 = x1, etc.
       ldy #0
addloop:
       ldx x0,y
       lda sum
       clc
       adc power5byte0,x
       sta sum
       lda sum+1
       adc power5byte1,x
       sta sum+1
       lda sum+2
       adc power5byte2,x
       sta sum+2
       lda sum+3
       adc power5byte3,x
       sta sum+3
       lda sum+4
       adc power5byte4,x
       sta sum+4
       iny
       cpy #4
       bcc addloop
       ; now sum := x₀⁵+x₁⁵+x₂⁵+x₃⁵
       ; set initial bounds for binary search
       ldx x3
       inx
       stx low
       ldx #max_value
       dex
       stx hi

binsearch:
       ; compute midpoint
       lda low
       cmp hi
       beq notdone
       bcs done_search
notdone:
       ldx #0
       clc
       adc hi

       ; now a + carry bit = low+hi; rotating right will get the midpoint
       ror
       ; compare square of midpoint to sum
       tax
       lda sum+4
       cmp power5byte4,x
       bne notyet
       lda sum+3
       cmp power5byte3,x
       bne notyet
       lda sum+2
       cmp power5byte2,x
       bne notyet
       lda sum+1
       cmp power5byte1,x
       bne notyet
       lda sum
       cmp power5byte0,x
       beq found
notyet:
       bcc sum_lt_guess
       inx
       stx low
       bne endbin
       beq endbin
sum_lt_guess:
       dex
       stx hi
endbin:
       bne binsearch
       beq binsearch

done_search:
       inc x3
       lda x3
       cmp #max_value
       bcs end_loop3
       jmp loop3

end_loop3:
       inc x2
       lda x2
       cmp #max_value
       bcs end_loop2
       jmp loop2

end_loop2:
       inc x1
       lda x1
       cmp #max_value
       bcs end_loop1
       jmp loop1

end_loop1:
       inc x0
       lda x0
       cmp #max_value
       bcs end_loop0
       jmp loop0

end_loop0:
       ; should never get here, means we didn't find an example.
       brk

found: stx x4
       puts success
       putdec x0
       ldy #1

ploop: puts between
       putdec {x0,y}
       iny
       cpy #4
       bcc ploop
       puts penult
       putdec {x0,y}
       puts eqend
       putdec5 sum
       puts tilabel
       rts
Output:
    **** COMMODORE 64 BASIC V2 ****

 64K RAM SYSTEM  38911 BASIC BYTES FREE

READY.
LOAD"ESOP",8,1

SEARCHING FOR ESOP
LOADING
READY.
RUN:


FOUND EXAMPLE:

27^5 + 84^5 + 110^5 + 133^5 = 144^5

(SUM IS 61917364224)

TIME:095617

READY.

... almost ten hours, but it did find it!

Ada

with Ada.Text_IO;

procedure Sum_Of_Powers is
   
   type Base is range 0 .. 250; -- A, B, C, D and Y are in that range
   type Num is range 0 .. 4*(250**5); -- (A**5 + ... + D**5) is in that range
   subtype Fit is Num range 0 .. 250**5; -- Y**5 is in that range
   
   Modulus: constant Num := 254;
   type Modular is mod Modulus;
 
   type Result_Type is array(1..5) of Base; -- this will hold A,B,C,D and Y
  
   type Y_Type is array(Modular) of Base;
   type Y_Sum_Type is array(Modular) of Fit;

   Y_Sum: Y_Sum_Type := (others => 0);  
   Y: Y_Type := (others => 0);
      -- for I in 0 .. 250, we set Y_Sum(I**5 mod Modulus) := I**5
      --                       and Y(I**5 mod Modulus) := I
      -- Modulus has been chosen to avoid collisions on (I**5 mod Modulus)
      -- later we will compute Sum_ABCD := A**5 + B**5 + C**5 + D**5
      -- and check if Y_Sum(Sum_ABCD mod modulus) = Sum_ABCD
   
   function Compute_Coefficients return Result_Type is  
   
      Sum_A: Fit;
      Sum_AB, Sum_ABC, Sum_ABCD: Num;
      Short: Modular;
      
   begin
      for A in Base(0) .. 246 loop
         Sum_A := Num(A) ** 5;
         for B in A .. 247 loop
            Sum_AB := Sum_A + (Num(B) ** 5);
            for C in Base'Max(B,1) .. 248 loop -- if A=B=0 then skip C=0
               Sum_ABC := Sum_AB + (Num(C) ** 5);
               for D in C .. 249 loop
                  Sum_ABCD := Sum_ABC + (Num(D) ** 5);
                  Short    := Modular(Sum_ABCD mod Modulus);
                  if Y_Sum(Short) = Sum_ABCD then
                     return A & B & C & D & Y(Short);
                  end if;
               end loop;
            end loop;
         end loop;
      end loop;
      return 0 & 0 & 0 & 0 & 0;
   end Compute_Coefficients;

   Tmp: Fit;
   ABCD_Y: Result_Type;

begin -- main program

   -- initialize Y_Sum and Y
   for I in Base(0) .. 250 loop
      Tmp := Num(I)**5;
      if Y_Sum(Modular(Tmp mod Modulus)) /= 0 then 
         raise Program_Error with "Collision: Change Modulus and recompile!";
      else
         Y_Sum(Modular(Tmp mod Modulus)) := Tmp;
         Y(Modular(Tmp mod Modulus)) := I;
      end if;
   end loop;
   
   -- search for a solution (A, B, C, D, Y)
   ABCD_Y := Compute_Coefficients;

   -- output result
   for Number of ABCD_Y loop
      Ada.Text_IO.Put(Base'Image(Number));
   end loop;
   Ada.Text_IO.New_Line;
   
end Sum_Of_Powers;
Output:
 27 84 110 133 144

ALGOL 68

Works with: ALGOL 68G version Any - tested with release 2.8.win32
# max number will be the highest integer we will consider                    #
INT max number = 250;

# Construct a table of the fifth powers of 1 : max number                    #
[ max number ]LONG INT fifth;
FOR i TO max number DO
    LONG INT i2 =  i * i;
    fifth[ i ] := i2 * i2 * i
OD;

# find the first a, b, c, d, e such that a^5 + b^5 + c^5 + d^5 = e^5         #
# as the fifth powers are in order, we can use a binary search to determine  #
# whether the value is in the table                                          #
BOOL found := FALSE;
FOR a TO max number WHILE NOT found DO
    FOR b FROM a TO max number WHILE NOT found DO
        FOR c FROM b TO max number WHILE NOT found DO
            FOR d FROM c TO max number WHILE NOT found DO
                LONG INT sum   = fifth[a] + fifth[b] + fifth[c] + fifth[d];
                INT      low  := d;
                INT      high := max number;
                WHILE low < high
                  AND NOT found
                DO
                    INT e := ( low + high ) OVER 2;
                    IF fifth[ e ] = sum
                    THEN
                        # the value at e is a fifth power                    #
                        found := TRUE;
                        print( ( ( whole( a, 0 ) + "^5 + " + whole( b, 0 ) + "^5 + "
                                 + whole( c, 0 ) + "^5 + " + whole( d, 0 ) + "^5 = "
                                 + whole( e, 0 ) + "^5"
                                 )
                               , newline
                               )
                             )
                    ELIF sum < fifth[ e ]
                    THEN high := e - 1
                    ELSE low  := e + 1
                    FI
                OD
            OD
        OD
    OD
OD

Output:

27^5 + 84^5 + 110^5 + 133^5 = 144^5

ALGOL W

As suggested by the REXX solution, we find a solution to a^5 + b^5 + c^5 = e^5 - d^5 which results in a significant reduction in run time.
Algol W integers are 32-bit only, so we simulate the necessary 12 digit arithmetic with pairs of integers.

begin
    % find a, b, c, d, e such that a^5 + b^5 + c^5 + d^5 = e^5                              %
    %                        where 1 <= a <= b <= c <= d <= e <= 250                        %
    % we solve this using the equivalent equation a^5 + b^5 + c^5 = e^5 - d^5               %
    % 250^5 is 976 562 500 000 - too large for a 32 bit number so we will use pairs of      %
    % integers and constrain their values to be in 0..1 000 000                             %
    % Note only positive numbers are needed                                                 %
    integer MAX_NUMBER, MAX_V;
    MAX_NUMBER := 250;
    MAX_V      := 1000000;
    begin
        % quick sorts the fifth power differences table                                     %
        procedure quickSort5 ( integer value lb, ub ) ;
            if ub > lb then begin
                % more than one element, so must sort                                       %
                integer left, right, pivot, pivotLo, pivotHi;
                left    := lb;
                right   := ub;
                % choosing the middle element of the array as the pivot %
                pivot   := left + ( ( ( right + 1 ) - left ) div 2 );
                pivotLo := loD( pivot );
                pivotHi := hiD( pivot );
                while begin
                    while left  <= ub
                      and begin integer cmp;
                                cmp := hiD( left ) - pivotHi;
                                if cmp = 0 then cmp := loD( left ) - pivotLo;
                                cmp < 0
                          end
                    do left := left + 1;
                    while right >= lb
                      and begin integer cmp;
                                cmp := hiD( right ) - pivotHi;
                                if cmp = 0 then cmp := loD( right ) - pivotLo;
                                cmp > 0
                          end
                    do right := right - 1;
                    left <= right
                end do begin
                    integer swapLo, swapHi, swapD, swapE;
                    swapLo       := loD( left  );
                    swapHi       := hiD( left  );
                    swapD        := Dd(  left  );
                    swapE        := De(  left  );
                    loD( left  ) := loD( right );
                    hiD( left  ) := hiD( right );
                    Dd(  left  ) := Dd(  right );
                    De(  left  ) := De(  right );
                    loD( right ) := swapLo;
                    hiD( right ) := swapHi;
                    Dd(  right ) := swapD;
                    De(  right ) := swapE;
                    left         := left  + 1;
                    right        := right - 1
                end while_left_le_right ;
                quickSort5( lb,   right );
                quickSort5( left, ub    )
            end quickSort5 ;
        % table of fifth powers                                                             %
        integer array lo5, hi5         ( 1 :: MAX_NUMBER );
        % table if differences between fifth powers                                         %
        integer array loD, hiD, De, Dd ( 1 :: MAX_NUMBER * MAX_NUMBER );
        integer dUsed, dPos;
        % compute fifth powers                                                              %
        for i := 1 until MAX_NUMBER do begin
            lo5( i ) := i * i; hi5( i ) := 0;
            for p := 3 until 5 do begin
                integer carry;
                lo5( i ) := lo5( i ) * i;
                carry    := lo5( i ) div MAX_V;
                lo5( i ) := lo5( i ) rem MAX_V;
                hi5( i ) := hi5( i ) * i;
                hi5( i ) := hi5( i ) + carry
            end for_p
        end for_i ;
        % compute the differences between fifth powers e^5 - d^5, 1 <= d < e <= MAX_NUMBER  %
        dUsed := 0;
        for e := 2 until MAX_NUMBER do begin
            for d := 1 until e - 1  do begin
                dUsed := dUsed + 1;
                De(  dUsed ) := e;
                Dd(  dUsed ) := d;
                loD( dUsed ) := lo5( e ) - lo5( d );
                hiD( dUsed ) := hi5( e ) - hi5( d );
                if loD( dUsed ) < 0 then begin
                    loD( dUsed ) := loD( dUsed ) + MAX_V;
                    hiD( dUsed ) := hiD( dUsed ) - 1
                end if_need_to_borrow
            end for_d
        end for_e;
        % sort the fifth power differences                                                  %
        quickSort5( 1, dUsed );
        % attempt to find a^5 + b^5 + c^5 = e^5 - d^5                                       %
        for a := 1 until MAX_NUMBER do begin
            integer loA, hiA;
            loA := lo5( a ); hiA := hi5( a );
            for b := a until MAX_NUMBER do begin
                integer loB, hiB;
                loB := lo5( b ); hiB := hi5( b );
                for c := b until MAX_NUMBER do begin
                    integer low, high, loSum, hiSum;
                    loSum :=                       loA + loB + lo5( c );
                    hiSum := ( loSum div MAX_V ) + hiA + hiB + hi5( c );
                    loSum :=   loSum rem MAX_V;
                    % look for hiSum,loSum in hiD,loD                                       %
                    low   := 1;
                    high  := dUsed;
                    while low < high do begin
                        integer mid, cmp;
                        mid := ( low + high ) div 2;
                        cmp := hiD( mid ) - hiSum;
                        if cmp = 0 then cmp := loD( mid ) - loSum;
                        if cmp = 0 then begin
                            % the value at mid is the difference of two fifth powers        %
                            write( i_w := 1, s_w := 0
                                 , a, "^5 + ", b, "^5 + ", c, "^5 + "
                                 , Dd( mid ), "^5 = ", De( mid ), "^5"
                                 );
                            go to found
                            end
                        else if cmp > 0 then high := mid - 1
                        else                 low  := mid + 1
                    end while_low_lt_high
                end for_c
            end for_b
        end for_a ;
found :
    end
end.
Output:
27^5 + 84^5 + 110^5 + 133^5 = 144^5

Arturo

Translation of: Nim
eulerSumOfPowers: function [top][
    p5: map 0..top => [& ^ 5]

    loop 4..top 'a [
        loop 3..a-1 'b [
            loop 2..b-1 'c [
                loop 1..c-1 'd [
                    s: (get p5 a) + (get p5 b) + (get p5 c) + (get p5 d)
                    if integer? index p5 s ->
                        return ~"|a|^5 + |b|^5 + |c|^5 + |d|^5 = |index p5 s|^5"
                ]
            ]
        ]
    ]

    return "not found" ; shouldn't reach here
]

print eulerSumOfPowers 249
Output:
133^5 + 110^5 + 84^5 + 27^5 = 144^5

AWK

# syntax: GAWK -f EULERS_SUM_OF_POWERS_CONJECTURE.AWK
BEGIN {
    start_int = systime()
    main()
    printf("%d seconds\n",systime()-start_int)
    exit(0)
}
function main(  sum,s1,x0,x1,x2,x3) {
    for (x0=1; x0<=250; x0++) {
      for (x1=1; x1<=x0; x1++) {
        for (x2=1; x2<=x1; x2++) {
          for (x3=1; x3<=x2; x3++) {
            sum = (x0^5) + (x1^5) + (x2^5) + (x3^5)
            s1 = int(sum ^ 0.2)
            if (sum == s1^5) {
              printf("%d^5 + %d^5 + %d^5 + %d^5 = %d^5\n",x0,x1,x2,x3,s1)
              return
            }
          }
        }
      }
    }
}
Output:
133^5 + 110^5 + 84^5 + 27^5 = 144^5
15 seconds

BASIC

BASIC256

Translation of: FreeBASIC
arraybase 1
max = 250
pow5_max = max * max * max * max * max
limit_x1 = (pow5_max / 4) ^ 0.2
limit_x2 = (pow5_max / 3) ^ 0.2
limit_x3 = (pow5_max / 2) ^ 0.2

dim pow5(max)
for x1 = 1 to max
	pow5[x1] = x1 * x1 * x1 * x1 * x1
next x1

for x1 = 1 to limit_x1
	for x2 = x1 +1 to limit_x2
		m1 = x1 + x2
		ans1 = pow5[x1] + pow5[x2]
		if ans1 > pow5_max then exit for
		for x3 = x2 +1 to limit_x3
			ans2 = ans1 + pow5[x3]
			if ans2 > pow5_max then exit for
			m2 = (m1 + x3) % 30
			if m2 = 0 then m2 = 30
			for x4 = x3 +1 to max -1
				ans3 = ans2 + pow5[x4]
				if ans3 > pow5_max then exit for
				for x5 = x4 + m2 to max step 30
					if ans3 < pow5[x5] then exit for
					if ans3 = pow5[x5] then
						print x1; "^5 + "; x2; "^5 + "; x3; "^5 + "; x4; "^5 = "; x5; "^5"
						end
                    end if
                next x5
            next x4
        next x3
    next x2
next x1
Output:
27^5 + 84^5 + 110^5 + 133^5 = 144^5

Chipmunk Basic

Works with: Chipmunk Basic version 3.6.4
Translation of: Run BASIC
100 max = 250
110 for w = 1 to max
120  for x = 1 to w
130   for y = 1 to x
140    for z = 1 to y
150     sum = w^5+x^5+y^5+z^5
160     sol = int(sum^0.2)
170     if sum = sol^5 then
180      print w;"^5 + ";x;"^5 + ";y;"^5 + ";z;"^5 = ";sol;"^5"
190      end
200     endif
210    next z
220   next y
230  next x
240 next w
Output:
133 ^5 + 110 ^5 + 84 ^5 + 27 ^5 = 144 ^5

FreeBASIC

' version 14-09-2015
' compile with: fbc -s console

' some constants calculated when the program is compiled

Const As UInteger max = 250
Const As ULongInt pow5_max = CULngInt(max) * max * max * max * max
' limit x1, x2, x3
Const As UInteger limit_x1 = (pow5_max / 4) ^ 0.2
Const As UInteger limit_x2 = (pow5_max / 3) ^ 0.2
Const As UInteger limit_x3 = (pow5_max / 2) ^ 0.2

' ------=< MAIN >=------

Dim As ULongInt pow5(max), ans1, ans2, ans3
Dim As UInteger x1, x2, x3, x4, x5 , m1, m2

Cls : Print

For x1 = 1 To max
    pow5(x1) = CULngInt(x1) * x1 * x1 * x1 * x1
Next x1

For x1 = 1 To limit_x1
    For x2 = x1 +1 To limit_x2
        m1 = x1 + x2
        ans1 = pow5(x1) + pow5(x2)
        If ans1 > pow5_max Then Exit For
        For x3 = x2 +1 To limit_x3
            ans2 = ans1 + pow5(x3)
            If ans2 > pow5_max Then Exit For
            m2 = (m1 + x3) Mod 30
            If m2 = 0 Then m2 = 30
            For x4 = x3 +1 To max -1
                ans3 = ans2 + pow5(x4)
                If ans3 > pow5_max Then Exit For
                For x5 = x4 + m2 To max Step 30
                    If ans3 < pow5(x5) Then Exit For
                    If ans3 = pow5(x5) Then
                        Print x1; "^5 + "; x2; "^5 + "; x3; "^5 + "; _
                              x4; "^5 = "; x5; "^5"
                        Exit For, For
                    EndIf
                Next x5
            Next x4
        Next x3
    Next x2
Next x1

Print
Print "done"

' empty keyboard buffer
While Inkey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End
Output:
27^5 + 84^5 + 110^5 + 133^5 = 144^5

Microsoft Small Basic

' Euler sum of powers conjecture - 03/07/2015
  'find: x1^5+x2^5+x3^5+x4^5=x5^5
  '-> x1=27 x2=84 x3=110 x4=133 x5=144
  maxn=250
  For i=1 to maxn
	p5[i]=Math.Power(i,5)
  EndFor
  For x1=1 to maxn-4
    For x2=x1+1 to maxn-3
      'TextWindow.WriteLine("x1="+x1+", x2="+x2)
      For x3=x2+1 to maxn-2
        'TextWindow.WriteLine("x1="+x1+", x2="+x2+", x3="+x3)
        For x4=x3+1 to maxn-1
          'TextWindow.WriteLine("x1="+x1+", x2="+x2+", x3="+x3+", x4="+x4)
          x5=x4+1
          valx=p5[x5]
          sumx=p5[x1]+p5[x2]+p5[x3]+p5[x4]
          While x5<=maxn and valx<=sumx
            If valx=sumx Then
              TextWindow.WriteLine("Found!")
              TextWindow.WriteLine("-> "+x1+"^5+"+x2+"^5+"+x3+"^5+"+x4+"^5="+x5+"^5")
              TextWindow.WriteLine("x5^5="+sumx)
              Goto EndPgrm
            EndIf
            x5=x5+1
            valx=p5[x5]
          EndWhile 'x5
        EndFor 'x4
      EndFor 'x3
    EndFor 'x2
  EndFor 'x1
 EndPgrm:
Output:
Found!
-> 27^5+84^5+110^5+133^5=144^5
x5^5=61917364224 

Minimal BASIC

Translation of: Run BASIC
100 LET m = 250
110 FOR w = 1 TO m
120  FOR x = 1 TO w
130   FOR y = 1 TO x
140    FOR z = 1 TO y
150     LET s = w^5+x^5+y^5+z^5
160     LET r = INT(s^0.2)
170     IF s = r^5 THEN 220
180    NEXT z
190   NEXT y
200  NEXT x
210 NEXT w
220 PRINT w;"^5 +";x;"^5 +";y;"^5+ ";z;"^5 =";r;"^5"
230 END
Output:
133 ^5 + 110 ^5 + 84 ^5 + 27 ^5 = 144 ^5

PureBasic

EnableExplicit

; assumes an array of non-decreasing positive integers
Procedure.q BinarySearch(Array a.q(1), Target.q)
  Protected l = 0, r = ArraySize(a()), m
  Repeat
    If l > r : ProcedureReturn 0 : EndIf; no match found
    m = (l + r) / 2
    If a(m) < target
      l = m + 1
    ElseIf a(m) > target
      r = m - 1
    Else
      ProcedureReturn m ; match found
    EndIf  
  ForEver
EndProcedure
  
Define i, x0, x1, x2, x3, y
Define.q sum
Define Dim p5.q(249)

For i = 1 To 249
  p5(i) = i * i * i * i * i
Next

If OpenConsole()
  For x0 = 1 To 249
    For x1 = 1 To x0 - 1
      For x2 = 1 To x1 - 1 
        For x3 = 1 To x2 - 1 
          sum = p5(x0) + p5(x1) + p5(x2) + p5(x3)
          y = BinarySearch(p5(), sum)
          If y > 0          
            PrintN(Str(x0) + "^5 + " + Str(x1) + "^5 + " + Str(x2) + "^5 + " + Str(x3) + "^5 = " + Str(y) + "^5")
            Goto finish
          EndIf
        Next x3
      Next x2
    Next x1
  Next x0
  
  PrintN("No solution was found")
  finish:
  PrintN("")
  PrintN("Press any key to close the console")
  Repeat: Delay(10) : Until Inkey() <> ""
  CloseConsole()
EndIf
Output:
133^5 + 110^5 + 84^5 + 27^5 = 144^5

QL SuperBASIC

This program enhances a modular brute-force search, posted on fidonet in the 1980s, via number theoretic enhancements as used by the program for ZX Spectrum Basic, but without the early decrement of the RHS in the control framework due to being backward compatible with the lack of floating point in Sinclair ZX80 BASIC (whereby the latter is truly 'zeroeth' generation). To emulate running on a ZX80 (needing a 16K RAM pack & MODifications, some given below) that completes the task sooner than the program for the OEM Spectrum, it relies entirely on integer functions, their upper limit being 2^15- 1 in ZX80 BASIC as well. Thus, the "slide rule" calculation of each percentage on the Spectrum is replaced by that of ones' digits across "abaci" of relatively prime bases Pi. Given that each Pi is to be <= 2^7 for said limit's sake, it takes six prime numbers or powers thereof to serve as bases of such a mixed-base number system, since it is necessary that ΠPi > 249^5 for unambiguous representation (as character strings). On ZX80s there are at most 64 consecutive printable characters (in the inverse video block: t%=48 thus becomes T=128). Just seven bases Pi <= 2^6 will be needed when the difference between 64 & a base is expressible as a four-bit offset, by which one must 'multiply' (since Z80s lack MUL) in the reduction step of the optimal assembly algorithm for MOD Pi. Such bases are: 49, 53, 55, 57, 59, 61, 64. In disproving Euler's conjecture, the program demonstrates that using 60 bits of integer precision in 1966 was 2-fold overkill, or even more so in terms of overhead cost vis-a-vis contemporaneous computers less sophisticated than a CDC 6600.

1 CLS
2 DIM i%(255,6) : DIM a%(6) : DIM c%(6)
3 DIM v%(255,6) : DIM b%(6) : DIM d%(29)
4 RESTORE 137
6      FOR m=0 TO 6
7    READ t% 
8    FOR j=1 TO 255
11 LET i%(j,m)=j MOD t%
12 LET v%(j,m)=(i%(j,m) * i%(j,m))MOD t%
14 LET v%(j,m)=(v%(j,m) * v%(j,m))MOD t%
15 LET v%(j,m)=(v%(j,m) * i%(j,m))MOD t%
17  END FOR j : END FOR m
19 PRINT "Abaci ready"
21 FOR j=10 TO 29: d%(j)=210+ j
24 FOR j=0 TO 9: d%(j)=240+ j
25 LET t%=48 
30      FOR w=6 TO 246 STEP 3
33     LET o=w
42     FOR x=4 TO 248 STEP 2
44    IF o<x THEN o=x
46    FOR m=1 TO 6: a%(m)=i%((v%(w,m)+v%(x,m)),m)
50    FOR y=10 TO 245 STEP 5
54   IF o<y THEN o=y
56   FOR m=1 TO 6: b%(m)=i%((a%(m)+v%(y,m)),m)
57   FOR z=14 TO 245 STEP 7
59  IF o<z THEN o=z
60  FOR m=1 TO 6: c%(m)=i%((b%(m)+v%(z,m)),m)
65  LET s$="" : FOR m=1 TO 6: s$=s$&CHR$(c%(m)+t%)
70  LET o=o+1 : j=d%(i%((i%(w,0)+i%(x,0)+i%(y,0)+i%(z,0)),0))
75  IF j<o THEN NEXT z
80  FOR k=j TO o STEP -30
85 LET e$="" : FOR m=1 TO 6: e$=e$&CHR$(v%(k,m)+t%)
90 IF s$=e$ THEN PRINT w,x,y,z,k,s$,e$: STOP
95  END FOR k : END FOR z : END FOR y : END FOR x : END FOR w
137 DATA 30,97,113,121,125,127,128
Output:
Abaci ready
  27      84     110     133     144   bT`íα0   bT`íα0

Run BASIC

max = 250
FOR w = 1 TO max
  FOR x = 1 TO w
    FOR y = 1 TO x
      FOR z = 1 TO y
      sum = w^5 + x^5 + y^5 + z^5
      s1  = INT(sum^0.2)
      IF sum = s1^5 THEN 
        PRINT w;"^5 + ";x;"^5 + ";y;"^5 + ";z;"^5 = ";s1;"^5" 
        end
      end if
      NEXT z
    NEXT y
  NEXT x
NEXT w
133^5 + 110^5 + 84^5 + 27^5 = 144^5

True BASIC

Translation of: Run BASIC
LET max = 250
FOR w = 1 TO max
    FOR x = 1 TO w
        FOR y = 1 TO x
            FOR z = 1 TO y
                LET sum = w^5 + x^5 + y^5 + z^5
                LET s1  = INT(sum^0.2)
                IF sum = s1^5 THEN
                   PRINT w;"^5 + ";x;"^5 + ";y;"^5 + ";z;"^5 = ";s1;"^5"
                   EXIT FOR
                END IF
            NEXT z
        NEXT y
    NEXT x
NEXT w
END
Output:
Same as Run BASIC entry.

VBA

Translation of: AWK
Public Declare Function GetTickCount Lib "kernel32.dll" () As Long
Public Sub begin()
    start_int = GetTickCount()
    main
    Debug.Print (GetTickCount() - start_int) / 1000 & " seconds"
End Sub
Private Function pow(x, y) As Variant
    pow = CDec(Application.WorksheetFunction.Power(x, y))
End Function
Private Sub main()
    For x0 = 1 To 250
        For x1 = 1 To x0
            For x2 = 1 To x1
                For x3 = 1 To x2
                    sum = CDec(pow(x0, 5) + pow(x1, 5) + pow(x2, 5) + pow(x3, 5))
                    s1 = Int(pow(sum, 0.2))
                    If sum = pow(s1, 5) Then
                        Debug.Print x0 & "^5 + " & x1 & "^5 + " & x2 & "^5 + " & x3 & "^5 = " & s1
                        Exit Sub
                    End If
                Next x3
            Next x2
        Next x1
    Next x0
End Sub
Output:
33^5 + 110^5 + 84^5 + 27^5 = 144
160,187 seconds

Visual Basic .NET

Paired Powers Algorithm

Translation of: Python
Module Module1

    Structure Pair
        Dim a, b As Integer
        Sub New(x as integer, y as integer)
            a = x : b = y
        End Sub
    End Structure

    Dim max As Integer = 250
    Dim p5() As Long,
        sum2 As SortedDictionary(Of Long, Pair) = New SortedDictionary(Of Long, Pair)

    Function Fmt(p As Pair) As String
        Return String.Format("{0}^5 + {1}^5", p.a, p.b)
    End Function

    Sub Init()
        p5(0) = 0 : p5(1) = 1 : For i As Integer = 1 To max - 1
            For j As Integer = i + 1 To max
                p5(j) = CLng(j) * j : p5(j) *= p5(j) * j
                sum2.Add(p5(i) + p5(j), New Pair(i, j))
            Next
        Next
    End Sub

    Sub Calc(Optional findLowest As Boolean = True)
        For i As Integer = 1 To max : Dim p As Long = p5(i)
            For Each s In sum2.Keys
                Dim t As Long = p - s : If t <= 0 Then Exit For
                If sum2.Keys.Contains(t) AndAlso sum2.Item(t).a > sum2.Item(s).b Then
                    Console.WriteLine("  {1} + {2} = {0}^5", i, Fmt(sum2.Item(s)), Fmt(sum2.Item(t)))
                    If findLowest Then Exit Sub
                End If
            Next : Next
    End Sub

    Sub Main(args As String())
        If args.Count > 0 Then
            Dim t As Integer = 0 : Integer.TryParse(args(0), t)
            If t > 0 AndAlso t < 5405 Then max = t
        End If
        Console.WriteLine("Checking from 1 to {0}...", max)
        For i As Integer = 0 To 1
            ReDim p5(max) : sum2.Clear()
            Dim st As DateTime = DateTime.Now
            Init() : Calc(i = 0)
            Console.WriteLine("{0}  Computation time to {2} was {1} seconds{0}", vbLf,
                (DateTime.Now - st).TotalSeconds, If(i = 0, "find lowest one", "check entire space"))
        Next
        If Diagnostics.Debugger.IsAttached Then Console.ReadKey()
    End Sub
End Module
Output:

(No command line arguments)

Checking from 1 to 250...
  27^5 + 84^5 + 110^5 + 133^5 = 144^5
  
  Computation time to find lowest one was 0.0807819 seconds
  
  27^5 + 84^5 + 110^5 + 133^5 = 144^5
  
  Computation time to check entire space was 0.3830103 seconds

Command line argument = "1000"

Checking from 1 to 1000...
  27^5 + 84^5 + 110^5 + 133^5 = 144^5

  Computation time to find lowest one was 0.3112007 seconds

  27^5 + 84^5 + 110^5 + 133^5 = 144^5
  54^5 + 168^5 + 220^5 + 266^5 = 288^5
  81^5 + 252^5 + 330^5 + 399^5 = 432^5
  108^5 + 336^5 + 440^5 + 532^5 = 576^5
  135^5 + 420^5 + 550^5 + 665^5 = 720^5
  162^5 + 504^5 + 660^5 + 798^5 = 864^5

  Computation time to check entire space was 28.8847393 seconds

Paired Powers w/ Mod 30 Shortcut and Threading

If one divides the searched array of powers (sum2m()) into 30 pieces, the search time can be reduced by only searching the appropriate one (determined by the Mod 30 value of the value being sought). Once broken down by this, it is now easier to use threading to further reduce the computation time.
The following compares the plain paired powers algorithm to the plain powers plus the mod 30 shortcut algorithm, without and with threading.

Module Module1

    Structure Pair
        Dim a, b As Integer
        Sub New(x As Integer, y As Integer)
            a = x : b = y
        End Sub
    End Structure

    Dim min As Integer = 1, max As Integer = 250
    Dim p5() As Long,
        sum2 As SortedDictionary(Of Long, Pair) = New SortedDictionary(Of Long, Pair),
        sum2m(29) As SortedDictionary(Of Long, Pair)

    Function Fmt(p As Pair) As String
        Return String.Format("{0}^5 + {1}^5", p.a, p.b)
    End Function

    Sub Init()
        p5(0) = 0 : p5(min) = CLng(min) * min : p5(min) *= p5(min) * min
        For i As Integer = min To max - 1
            For j As Integer = i + 1 To max
                p5(j) = CLng(j) * j : p5(j) *= p5(j) * j
                If j = max Then Continue For
                sum2.Add(p5(i) + p5(j), New Pair(i, j))
            Next
        Next
    End Sub

    Sub InitM()
        For i As Integer = 0 To 29 : sum2m(i) = New SortedDictionary(Of Long, Pair) : Next
        p5(0) = 0 : p5(min) = CLng(min) * min : p5(min) *= p5(min) * min
        For i As Integer = min To max - 1
            For j As Integer = i + 1 To max
                p5(j) = CLng(j) * j : p5(j) *= p5(j) * j
                If j = max Then Continue For
                Dim x As Long = p5(i) + p5(j)
                sum2m(x Mod 30).Add(x, New Pair(i, j))
            Next
        Next
    End Sub

    Sub Calc(Optional findLowest As Boolean = True)
        For i As Integer = min To max : Dim p As Long = p5(i)
            For Each s In sum2.Keys
                Dim t As Long = p - s : If t <= 0 Then Exit For
                If sum2.Keys.Contains(t) AndAlso sum2.Item(t).a > sum2.Item(s).b Then
                    Console.WriteLine("  {1} + {2} = {0}^5", i, 
                        Fmt(sum2.Item(s)), Fmt(sum2.Item(t)))
                    If findLowest Then Exit Sub
                End If
            Next : Next
    End Sub

    Function CalcM(m As Integer) As List(Of String)
        Dim res As New List(Of String)
        For i As Integer = min To max
            Dim pm As Integer = i Mod 30,
                mp As Integer = (pm - m + 30) Mod 30
            For Each s In sum2m(m).Keys
                Dim t As Long = p5(i) - s : If t <= 0 Then Exit For
                If sum2m(mp).Keys.Contains(t) AndAlso
                  sum2m(mp).Item(t).a > sum2m(m).Item(s).b Then
                    res.Add(String.Format("  {1} + {2} = {0}^5",
                        i, Fmt(sum2m(m).Item(s)), Fmt(sum2m(mp).Item(t))))
                End If
            Next : Next
        Return res
    End Function

    Function Snip(s As String) As Integer
        Dim p As Integer = s.IndexOf("=") + 1
        Return s.Substring(p, s.IndexOf("^", p) - p)
    End Function

    Function CompareRes(ByVal x As String, ByVal y As String) As Integer
        CompareRes = Snip(x).CompareTo(Snip(y))
        If CompareRes = 0 Then CompareRes = x.CompareTo(y)
    End Function

    Function Validify(def As Integer, s As String) As Integer
        Validify = def : Dim t As Integer = 0 : Integer.TryParse(s, t)
        If t >= 1 AndAlso Math.Pow(t, 5) < (Long.MaxValue >> 1) Then Validify = t
    End Function

    Sub Switch(ByRef a As Integer, ByRef b As Integer)
        Dim t As Integer = a : a = b : b = t
    End Sub

    Sub Main(args As String())
        Select Case args.Count
            Case 1 : max = Validify(max, args(0))
            Case > 1
                min = Validify(min, args(0))
                max = Validify(max, args(1))
                If max < min Then Switch(max, min)
        End Select
        Console.WriteLine("Paired powers, checking from {0} to {1}...", min, max)
        For i As Integer = 0 To 1
            ReDim p5(max) : sum2.Clear()
            Dim st As DateTime = DateTime.Now
            Init() : Calc(i = 0)
            Console.WriteLine("{0}  Computation time to {2} was {1} seconds{0}", vbLf,
                (DateTime.Now - st).TotalSeconds, If(i = 0, "find lowest one", "check entire space"))
        Next
        For i As Integer = 0 To 1
            Console.WriteLine("Paired powers with Mod 30 shortcut (entire space) {2}, checking from {0} to {1}...",
                min, max, If(i = 0, "sequential", "parallel"))
            ReDim p5(max)
            Dim res As List(Of String) = New List(Of String)
            Dim st As DateTime = DateTime.Now
            Dim taskList As New List(Of Task(Of List(Of String)))
            InitM()
            Select Case i
                Case 0
                    For j As Integer = 0 To 29
                        res.AddRange(CalcM(j))
                    Next
                Case 1
                    For j As Integer = 0 To 29 : Dim jj = j
                        taskList.Add(Task.Run(Function() CalcM(jj)))
                    Next
                    Task.WhenAll(taskList)
                    For Each item In taskList.Select(Function(t) t.Result)
                        res.AddRange(item) : Next
            End Select
            res.Sort(AddressOf CompareRes)
            For Each item In res
                Console.WriteLine(item) : Next
            Console.WriteLine("{0}  Computation time was {1} seconds{0}", vbLf, (DateTime.Now - st).TotalSeconds)
        Next
        If Diagnostics.Debugger.IsAttached Then Console.ReadKey()
    End Sub
End Module
Output:

(No command line arguments)

Paired powers, checking from 1 to 250...

27^5 + 84^5 + 110^5 + 133^5 = 144^5

Computation time to find lowest one was 0.0781252 seconds

27^5 + 84^5 + 110^5 + 133^5 = 144^5

Computation time to check entire space was 0.3280574 seconds

Paired powers with Mod 30 shortcut (entire space) sequential, checking from 1 to 250... 27^5 + 84^5 + 110^5 + 133^5 = 144^5

Computation time was 0.2655529 seconds

Paired powers with Mod 30 shortcut (entire space) parallel, checking from 1 to 250... 27^5 + 84^5 + 110^5 + 133^5 = 144^5

Computation time was 0.0624651 seconds

(command line argument = "1000")

Paired powers, checking from 1 to 1000...
  27^5 + 84^5 + 110^5 + 133^5 = 144^5

  Computation time to find lowest one was 0.2499343 seconds

  27^5 + 84^5 + 110^5 + 133^5 = 144^5
  54^5 + 168^5 + 220^5 + 266^5 = 288^5
  81^5 + 252^5 + 330^5 + 399^5 = 432^5
  108^5 + 336^5 + 440^5 + 532^5 = 576^5
  135^5 + 420^5 + 550^5 + 665^5 = 720^5
  162^5 + 504^5 + 660^5 + 798^5 = 864^5

  Computation time to check entire space was 27.805961 seconds

Paired powers with Mod 30 shortcut (entire space) sequential, checking from 1 to 1000...
  27^5 + 84^5 + 110^5 + 133^5 = 144^5
  54^5 + 168^5 + 220^5 + 266^5 = 288^5
  81^5 + 252^5 + 330^5 + 399^5 = 432^5
  108^5 + 336^5 + 440^5 + 532^5 = 576^5
  135^5 + 420^5 + 550^5 + 665^5 = 720^5
  162^5 + 504^5 + 660^5 + 798^5 = 864^5

  Computation time was 23.8068928 seconds

Paired powers with Mod 30 shortcut (entire space) parallel, checking from 1 to 1000...
  27^5 + 84^5 + 110^5 + 133^5 = 144^5
  54^5 + 168^5 + 220^5 + 266^5 = 288^5
  81^5 + 252^5 + 330^5 + 399^5 = 432^5
  108^5 + 336^5 + 440^5 + 532^5 = 576^5
  135^5 + 420^5 + 550^5 + 665^5 = 720^5
  162^5 + 504^5 + 660^5 + 798^5 = 864^5

  Computation time was 5.4205943 seconds

(command line arguments = "27 864")

Paired powers, checking from 27 to 864...
  27^5 + 84^5 + 110^5 + 133^5 = 144^5

  Computation time to find lowest one was 0.1562309 seconds

  27^5 + 84^5 + 110^5 + 133^5 = 144^5
  54^5 + 168^5 + 220^5 + 266^5 = 288^5
  81^5 + 252^5 + 330^5 + 399^5 = 432^5
  108^5 + 336^5 + 440^5 + 532^5 = 576^5
  135^5 + 420^5 + 550^5 + 665^5 = 720^5
  162^5 + 504^5 + 660^5 + 798^5 = 864^5

  Computation time to check entire space was 15.8243802 seconds

Paired powers with Mod 30 shortcut (entire space) sequential, checking from 27 to 864...
  27^5 + 84^5 + 110^5 + 133^5 = 144^5
  54^5 + 168^5 + 220^5 + 266^5 = 288^5
  81^5 + 252^5 + 330^5 + 399^5 = 432^5
  108^5 + 336^5 + 440^5 + 532^5 = 576^5
  135^5 + 420^5 + 550^5 + 665^5 = 720^5
  162^5 + 504^5 + 660^5 + 798^5 = 864^5

  Computation time was 13.0438215 seconds

Paired powers with Mod 30 shortcut (entire space) parallel, checking from 27 to 864...
  27^5 + 84^5 + 110^5 + 133^5 = 144^5
  54^5 + 168^5 + 220^5 + 266^5 = 288^5
  81^5 + 252^5 + 330^5 + 399^5 = 432^5
  108^5 + 336^5 + 440^5 + 532^5 = 576^5
  135^5 + 420^5 + 550^5 + 665^5 = 720^5
  162^5 + 504^5 + 660^5 + 798^5 = 864^5

  Computation time was 3.0305365 seconds

(command line arguments = "189 1008")

Paired powers, checking from 189 to 1008...
  189^5 + 588^5 + 770^5 + 931^5 = 1008^5
  
  Computation time to find lowest one was 14.6840411 seconds
  
  189^5 + 588^5 + 770^5 + 931^5 = 1008^5
  
  Computation time to check entire space was 14.7777685 seconds
  
Paired powers with Mod 30 shortcut (entire space) sequential, checking from 189 to 1008...
  189^5 + 588^5 + 770^5 + 931^5 = 1008^5
  
  Computation time was 12.4814705 seconds
  
Paired powers with Mod 30 shortcut (entire space) parallel, checking from 189 to 1008...
  189^5 + 588^5 + 770^5 + 931^5 = 1008^5
  
  Computation time was 2.7180777 seconds

Yabasic

Translation of: FreeBASIC
limit = 250
pow5_limit = limit * limit * limit * limit * limit
limit_x1 = (pow5_limit / 4) ^ 0.2
limit_x2 = (pow5_limit / 3) ^ 0.2
limit_x3 = (pow5_limit / 2) ^ 0.2

dim pow5(limit)
for x1 = 1 to limit
	pow5(x1) = x1 * x1 * x1 * x1 * x1
next x1

for x1 = 1 to limit_x1
	for x2 = x1 +1 to limit_x2
		m1 = x1 + x2
		ans1 = pow5(x1) + pow5(x2)
		if ans1 > pow5_limit  break
		for x3 = x2 +1 to limit_x3
			ans2 = ans1 + pow5(x3)
			if ans2 > pow5_limit  break
			m2 = mod((m1 + x3), 30)
			if m2 = 0  m2 = 30
			for x4 = x3 +1 to limit -1
				ans3 = ans2 + pow5(x4)
				if ans3 > pow5_limit  break
				for x5 = x4 + m2 to limit step 30
					if ans3 < pow5(x5)  break
					if ans3 = pow5(x5) then
						print x1, "^5 + ", x2, "^5 + ", x3, "^5 + ", x4, "^5 = ", x5, "^5"
						break
                    fi
                next x5
            next x4
        next x3
    next x2
next x1
end
Output:
27^5 + 84^5 + 110^5 + 133^5 = 144^5

ZX Spectrum Basic

This "abacus revision" reverts back to an earlier one, i.e. the "slide rule" one calculating the logarithmic 'percentage' for each of 4 summands. It also calculates their ones' digits as if on a base m abacus, that complements the other base m digits embedded in their percentages via a check sum of the ones' digits when the percentages add up to 1. Because any other argument is less than m, there is sufficient additional precision so as to not find false solutions while seeking the first primitive solution. 1 is excluded a priori for (e.g.) w, since it is well-known that the only integral points on 1=m^5-x^5-y^5-z^5 are the obvious ones (that aren't distinct\all >0). Nonetheless, 1 (as the zeroeth 'prime' Po) can be the first of 3 factors (one of the others being a power of the first 4 primes) for specifying a LHS argument. Given 4 LHS arguments each raised to a 5th power as is the RHS for which m<=ΣΠPi<2^(3+5), i=0 to 4, the LHS solution (if one exists) will consist of 4 elements of a factorial domain that are each a triplet (a prime\Po * a prime^1st\2nd power * a prime^1st\4th power) multiple having a distinct pair of the first 4 primes present & absent. Such potential solutions are generated by aiming for simplicity rather than efficiency (e.g. not generating potential solutions where each multiple is even). Two theorems' consequences intended for an even earlier revision are also applied: Fermat's Little Theorem (as proven later by Euler) whereby Xi^5 mod P = Xi mod P for the first 3 primes; and the Chinese Remainder Theorem in reverse whereby m= k- i*2*3*5, such that k is chosen to be the highest allowed m congruent mod 30 to the sum of the LHS arguments of a potential solution. Although still slow, this revision performs the given task on any ZX Spectrum, despite being limited to 32-bit precision.

1 CLS
2 DIM k(29): DIM q(249)
5 FOR i=4 TO 249: LET q(i)=LN i : NEXT i
6 REM enhancements for the much expanded Spectrum Next:  DIM p(248,249)
7 REM FOR j=4TO 248:FOR i=j TO 249:LET p(j,i)=EXP (q(j)-q(i))*5:NEXT i:NEXT j
9 PRINT "slide rule ready"
15 FOR i=0 TO 9: LET k(i)=240+ i : NEXT i
17 FOR i=10 TO 29: LET k(i)=210+ i : NEXT i
20 FOR w=6 TO 246 STEP 3
21 LET o=w
22 FOR x=4 TO 248 STEP 2
23 IF o<x THEN LET o=x
24 FOR y=10 TO 245 STEP 5
25 IF o<y THEN LET o=y
26 FOR z=14 TO 245 STEP 7
27 IF o<z THEN LET o=z
30 LET o=o+1 : LET m=k(FN f((w+x+y+z),30))
34 IF m<o THEN GO TO 90
40 REM LET s=p(w,m)+p(x,m)+p(y,m)+p(z,m) instead of:
42 LET s=EXP((q(w)-q(m))*5)
43 LET s=EXP((q(x)-q(m))*5)+ s
45 LET s=EXP((q(y)-q(m))*5)+ s
47 LET s=EXP((q(z)-q(m))*5)+ s
50 IF s<>1 THEN GO TO 80
52 LET a=FN f(w*w,m) : LET a=FN f(a*a*w,m)
53 LET b=FN f(x*x,m) : LET b=FN f(b*b*x,m)
55 LET c=FN f(y*y,m) : LET c=FN f(c*c*y,m)
57 LET d=FN f(z*z,m) : LET d=FN f(d*d*z,m)
60 LET u=FN f((a+b+c+d),m)
65 IF u THEN GO TO 80
73 PRINT w;"^5+";x;"^5+";y;"^5+";z;"^5=";m;"^5": STOP
80 IF s<1 THEN m=m-30 : GO TO 34
90 NEXT z: NEXT y: NEXT x: NEXT w
100 DEF FN f(e,n)=e- INT(e/n)*n
Output:
slide rule ready
27^5+84^5+110^5+133^5=144^5

BCPL

Translation of: Go
GET "libhdr"

LET solve() BE {
    LET pow5 = VEC 249
    LET sum = ?

    FOR i = 1 TO 249
        pow5!i := i * i * i * i * i

    FOR w = 4 TO 249
        FOR x = 3 TO w - 1
            FOR y = 2 TO x - 1
                FOR z = 1 TO y - 1 {
                    sum := pow5!w + pow5!x + pow5!y + pow5!z
                    FOR a = w + 1 TO 249
                        IF pow5!a = sum {
                            writef("solution found: %d  %d  %d  %d  %d *n", w, x, y, z, a)
                            RETURN
                        }
                }
    writef("Sorry, no solution found.*n")
}    

LET start() = VALOF {
    solve()
    RESULTIS 0
}
Output:
solution found: 133 110 84 27 144

Bracmat

  0:?x0
&   whl
  ' ( 1+!x0:<250:?x0
    & out$(x0 !x0)
    & 0:?x1
    &   whl
      ' ( 1+!x1:~>!x0:?x1
        & out$(x0 !x0 x1 !x1)
        & 0:?x2
        &   whl
          ' ( 1+!x2:~>!x1:?x2
            & 0:?x3
            &   whl
              ' ( 1+!x3:~>!x2:?x3
                &   (!x0^5+!x1^5+!x2^5+!x3^5)^1/5
                  : (   #?y
                      & out$(x0 !x0 x1 !x1 x2 !x2 x3 !x3 y !y)
                      & 250:?x0:?x1:?x2:?x3
                    | ?
                    )
                )
            )
        )
    )

Output

x0 133 x1 110 x2 84 x3 27 y 144

C

The trick to speed up was the observation that for any x we have x^5=x modulo 2, 3, and 5, according to the Fermat's little theorem. Thus, based on the Chinese Remainder Theorem we have x^5==x modulo 30 for any x. Therefore, when we have computed the left sum s=a^5+b^5+c^5+d^5, then we know that the right side e^5 must be such that s==e modulo 30. Thus, we do not have to consider all values of e, but only values in the form e=e0+30k, for some starting value e0, and any k. Also, we follow the constraints 1<=a<b<c<d<e<N in the main loop.

// Alexander Maximov, July 2nd, 2015
#include <stdio.h>
#include <time.h>
typedef long long mylong;

void compute(int N, char find_only_one_solution)
{	const int M = 30;   /* x^5 == x modulo M=2*3*5 */
	int a, b, c, d, e;
	mylong s, t, max, *p5 = (mylong*)malloc(sizeof(mylong)*(N+M));
 
	for(s=0; s < N; ++s)
		p5[s] = s * s, p5[s] *= p5[s] * s;
	for(max = p5[N - 1]; s < (N + M); p5[s++] = max + 1);
 
	for(a = 1; a < N; ++a)
	for(b = a + 1; b < N; ++b)
	for(c = b + 1; c < N; ++c)
	for(d = c + 1, e = d + ((t = p5[a] + p5[b] + p5[c]) % M); ((s = t + p5[d]) <= max); ++d, ++e)
	{	for(e -= M; p5[e + M] <= s; e += M); /* jump over M=30 values for e>d */
		if(p5[e] == s)
		{	printf("%d %d %d %d %d\r\n", a, b, c, d, e);
			if(find_only_one_solution) goto onexit;
		}
	}
onexit:
	free(p5);
}

int main(void)
{
	int tm = clock();
	compute(250, 0);
	printf("time=%d milliseconds\r\n", (int)((clock() - tm) * 1000 / CLOCKS_PER_SEC));
	return 0;
}
Output:

The fair way to measure the speed of the code above is to measure it's run time to find all possible solutions to the problem, given N (and not just a single solution, since then the time may depend on the way and the order we organize for-loops).

27 84 110 133 144
time=235 milliseconds

Another test with N=1000 produces the following results:

27 84 110 133 144
54 168 220 266 288
81 252 330 399 432
108 336 440 532 576
135 420 550 665 720
162 504 660 798 864
time=65743 milliseconds

PS: The solution for C++ provided below is actually quite good in its design idea behind. However, with all proposed tricks to speed up, the measurements for C++ solution for N=1000 showed the execution time 81447ms (+23%) on the same environment as above for C solution (same machine, same compiler, 64-bit platform). The reason that C++ solution is a bit slower is, perhaps, the fact that the inner loops over rs have complexity ~N/2 steps in average, while with the modulo 30 trick that complexity can be reduced down to ~N/60 steps, although one "expensive" extra %-operation is still needed.

C#

Loops

Translation of: Java
using System;

namespace EulerSumOfPowers {
    class Program {
        const int MAX_NUMBER = 250;

        static void Main(string[] args) {
            bool found = false;
            long[] fifth = new long[MAX_NUMBER];

            for (int i = 1; i <= MAX_NUMBER; i++) {
                long i2 = i * i;
                fifth[i - 1] = i2 * i2 * i;
            }

            for (int a = 0; a < MAX_NUMBER && !found; a++) {
                for (int b = a; b < MAX_NUMBER && !found; b++) {
                    for (int c = b; c < MAX_NUMBER && !found; c++) {
                        for (int d = c; d < MAX_NUMBER && !found; d++) {
                            long sum = fifth[a] + fifth[b] + fifth[c] + fifth[d];
                            int e = Array.BinarySearch(fifth, sum);
                            found = e >= 0;
                            if (found) {
                                Console.WriteLine("{0}^5 + {1}^5 + {2}^5 + {3}^5 = {4}^5", a + 1, b + 1, c + 1, d + 1, e + 1);
                            }
                        }
                    }
                }
            }
        }
    }
}

Paired Powers, Mod 30, etc...

Translation of: vbnet
using System;
using System.Collections.Generic;
using System.Linq;
using System.Threading.Tasks;

namespace Euler_cs
{
    class Program
    {
        struct Pair
        {
            public int a, b;
            public Pair(int x, int y)
            {
                a = x; b = y;
            }
        }

        static int min = 1, max = 250;
        static ulong[] p5;
        static SortedDictionary<ulong, Pair>[] sum2 =
                   new SortedDictionary<ulong, Pair>[30];

        static string Fmt(Pair p)
        {
            return string.Format("{0}^5 + {1}^5", p.a, p.b);
        }

        public static void InitM()
        {
            for (int i = 0; i <= 29; i++)
                sum2[i] = new SortedDictionary<ulong, Pair>();
            p5 = new ulong[max + 1];
            p5[min] = Convert.ToUInt64(min) * Convert.ToUInt64(min);
            p5[min] *= p5[min] * Convert.ToUInt64(min);
            for (int i = min; i <= max - 1; i++)
            {
                for (int j = i + 1; j <= max; j++)
                {
                    p5[j] = Convert.ToUInt64(j) * Convert.ToUInt64(j);
                    p5[j] *= p5[j] * Convert.ToUInt64(j);
                    if (j == max) continue;
                    ulong x = p5[i] + p5[j];
                    sum2[x % 30].Add(x, new Pair(i, j));
                }
            }
        }

        static List<string> CalcM(int m)
        {
            List<string> res = new List<string>();
            for (int i = max; i >= min; i--)
            {
                ulong p = p5[i]; int pm = i % 30, mp = (pm - m + 30) % 30;
                foreach (var s in sum2[m].Keys)
                {
                    if (p <= s) break;
                    ulong t = p - s;
                    if (sum2[mp].Keys.Contains(t) && sum2[mp][t].a > sum2[m][s].b)
                        res.Add(string.Format("  {1} + {2} = {0}^5",
                            i, Fmt(sum2[m][s]), Fmt(sum2[mp][t])));
                }
            }
            return res;
        }

        static int Snip(string s)
        {
            int p = s.IndexOf("=") + 1;
            return Convert.ToInt32(s.Substring(p, s.IndexOf("^", p) - p));
        }

        static int CompareRes(string x, string y)
        {
            int res = Snip(x).CompareTo(Snip(y));
            if (res == 0) res = x.CompareTo(y);
            return res;
        }

        static int Validify(int def, string s)
        {
            int res = def, t = 0; int.TryParse(s, out t);
            if (t >= 1 && t < Math.Pow((double)(ulong.MaxValue >> 1), 0.2))
                res = t;
            return res;
        }

        static void Switch(ref int a, ref int b)
        {
            int t = a; a = b; b = t;
        }

        static void Main(string[] args)
        {
            if (args.Count() > 1)
            {
                min = Validify(min, args[0]);
                max = Validify(max, args[1]);
                if (max < min) Switch(ref max, ref min);
            }
            else if (args.Count() == 1)
                max = Validify(max, args[0]);
            Console.WriteLine("Mod 30 shortcut with threading, checking from {0} to {1}...", min, max);
            List<string> res = new List<string>();
            DateTime st = DateTime.Now;
            List<Task<List<string>>> taskList = new List<Task<List<string>>>();
            InitM();
            for (int j = 0; j <= 29; j++)
            {
                var jj = j;
                taskList.Add(Task.Run(() => CalcM(jj)));
            }
            Task.WhenAll(taskList);
            foreach (var item in taskList.Select(t => t.Result))
                res.AddRange(item);
            res.Sort(CompareRes);
            foreach (var item in res)
                Console.WriteLine(item);
            Console.WriteLine("  Computation time to check entire space was {0} seconds",
                (DateTime.Now - st).TotalSeconds);
            if (System.Diagnostics.Debugger.IsAttached)
                Console.ReadKey();
        }
    }
}
Output:

(no command line arguments)

Mod 30 shortcut with threading, checking from 1 to 250...

27^5 + 84^5 + 110^5 + 133^5 = 144^5

Computation time to check entire space was 0.0838058 seconds

(command line argument = "1000")

Mod 30 shortcut with threading, checking from 1 to 1000...
  27^5 + 84^5 + 110^5 + 133^5 = 144^5
  54^5 + 168^5 + 220^5 + 266^5 = 288^5
  81^5 + 252^5 + 330^5 + 399^5 = 432^5
  108^5 + 336^5 + 440^5 + 532^5 = 576^5
  135^5 + 420^5 + 550^5 + 665^5 = 720^5
  162^5 + 504^5 + 660^5 + 798^5 = 864^5
  Computation time to check entire space was 5.4109744 seconds

C++

First version

The simplest brute-force find is already reasonably quick:

#include <algorithm>
#include <iostream>
#include <cmath>
#include <set>
#include <vector>

using namespace std;

bool find()
{
    const auto MAX = 250;
    vector<double> pow5(MAX);
    for (auto i = 1; i < MAX; i++)
        pow5[i] = (double)i * i * i * i * i;
    for (auto x0 = 1; x0 < MAX; x0++) {
        for (auto x1 = 1; x1 < x0; x1++) {
            for (auto x2 = 1; x2 < x1; x2++) {
                for (auto x3 = 1; x3 < x2; x3++) {
                    auto sum = pow5[x0] + pow5[x1] + pow5[x2] + pow5[x3];
                    if (binary_search(pow5.begin(), pow5.end(), sum))
                    {
                        cout << x0 << " " << x1 << " " << x2 << " " << x3 << " " << pow(sum, 1.0 / 5.0) << endl;
                        return true;
                    }
                }
            }
        }
    }
    // not found
    return false;
}

int main(void)
{
    int tm = clock();
    if (!find())
        cout << "Nothing found!\n";
    cout << "time=" << (int)((clock() - tm) * 1000 / CLOCKS_PER_SEC) << " milliseconds\r\n";
    return 0;
}
Output:
133 110 84 27 144
time=234 milliseconds

We can accelerate this further by creating a parallel std::set<double> p5s containing the elements of the std::vector pow5, and using it to replace the call to std::binary_search:

	set<double> pow5s;
	for (auto i = 1; i < MAX; i++) 
	{
		pow5[i] = (double)i * i * i * i * i;
		pow5s.insert(pow5[i]);
	}
	//...
        if (pow5s.find(sum) != pow5s.end())

This reduces the timing to 125 ms on the same hardware.

A different, more effective optimization is to note that each successive sum is close to the previous one, and use a bidirectional linear search with memory. We also note that inside the innermost loop, we only need to search upward, so we hoist the downward linear search to the loop over x2.

bool find() 
{
	const auto MAX = 250;
	vector<double> pow5(MAX);
	for (auto i = 1; i < MAX; i++) 
		pow5[i] = (double)i * i * i * i * i;
	auto rs = 5;
	for (auto x0 = 1; x0 < MAX; x0++) {
		for (auto x1 = 1; x1 < x0; x1++) {
			for (auto x2 = 1; x2 < x1; x2++) {
				auto s2 = pow5[x0] + pow5[x1] + pow5[x2];
				while (rs > 0 && pow5[rs] > s2) --rs;
				for (auto x3 = 1; x3 < x2; x3++) {
					auto sum = s2 + pow5[x3];
					while (rs < MAX - 1 && pow5[rs] < sum) ++rs;
					if (pow5[rs] == sum)
					{
						cout << x0 << " " << x1 << " " << x2 << " " << x3 << " " << pow(sum, 1.0 / 5.0) << endl;
						return true;
					}
				}
			}
		}
	}
	// not found
	return false;
}

This reduces the timing to around 25 ms. We could equally well replace rs with an iterator inside pow5; the timing is unaffected.

For comparison with the C code, we also check the timing of an exhaustive search up to MAX=1000. (Don't try this in Python.) This takes 87.2 seconds on the same hardware, comparable to the results found by the C code authors, and supports their conclusion that the mod-30 trick used in the C solution leads to better scalability than the iterator optimizations.

Fortunately, we can incorporate the same trick into the above code, by inserting a forward jump to a feasible solution mod 30 into the loop over x3:

				for (auto x3 = 1; x3 < x2; x3++) 
				{
					// go straight to the first appropriate x3, mod 30
					if (int err30 = (x0 + x1 + x2 + x3 - rs) % 30)
						x3 += 30 - err30;
					if (x3 >= x2)
						break;
					auto sum = s2 + pow5[x3];

With this refinement, the exhaustive search up to MAX=1000 takes 16.9 seconds.

Thanks, C guys!

Second version

We can create a more efficient method by using the idea (taken from the EchoLisp listing below) of precomputing difference between pairs of fifth powers. If we combine this with the above idea of using linear search with memory, this still requires asymptotically O(N^4) time (because of the linear search within diffs), but is at least twice as fast as the solution above using the mod-30 trick. Exhaustive search up to MAX=1000 took 6.2 seconds for me (64-bit on 3.4GHz i7). It is not clear how it can be combined with the mod-30 trick.

The asymptotic behavior can be improved to O(N^3 ln N) by replacing the linear search with an increasing-increment "hunt" (and the outer linear search, which is also O(N^4), with a call to std::upper_bound). With this replacement, the first solution was found in 0.05 seconds; exhaustive search up to MAX=1000 took 2.80 seconds; and the second nontrivial solution (discarding multiples of the first solution), at y==2615, was found in 94.6 seconds. Note: there is no solution 2615, because 645^5 + 1523^5 + 1722^5 +2506^5 = 122 280 854 808 884 376, but 2615^5=122 280 854 808 884 375. This is an error due to limitation in mantissa of double type (52 bits). 128 bit type is required for the next solution 85359.


template<class C_, class LT_> C_ Unique(const C_& src, const LT_& less)
{
	C_ retval(src);
	std::sort(retval.begin(), retval.end(), less);
	retval.erase(unique(retval.begin(), retval.end()), retval.end());
	return retval;
}

template<class I_, class P_> I_ HuntFwd(const I_& hint, const I_& end, const P_& less)	// if less(x) is false, then less(x+1) must also be false
{
	I_ retval(hint);
	int step = 1;
	// expanding phase
	while (end - retval > step)
	{
		I_ test = retval + step;
		if (!less(test))
			break;
		retval = test;
		step <<= 1;
	}
	// contracting phase
	while (step > 1)
	{
		step >>= 1;
		if (end - retval <= step)
			continue;
		I_ test = retval + step;
		if (less(test))
			retval = test;
	}
	if (retval != end && less(retval))
		++retval;
	return retval;
}

bool DPFind(int how_many)
{
	const int MAX = 1000;
	vector<double> pow5(MAX);
	for (int i = 1; i < MAX; i++)
		pow5[i] = (double)i * i * i * i * i;
	vector<pair<double, int>> diffs;
	for (int i = 2; i < MAX; ++i)
	{
		for (int j = 1; j < i; ++j)
			diffs.emplace_back(pow5[i] - pow5[j], j);
	}
	auto firstLess = [](const pair<double, int>& lhs, const pair<double, int>& rhs) { return lhs.first < rhs.first; };
	diffs = Unique(diffs, firstLess);

	for (int x4 = 4; x4 < MAX - 1; ++x4)
	{
		for (int x3 = 3; x3 < x4; ++x3)
		{
			// if (133 * x3 == 110 * x4) continue;	// skip duplicates of first solution
			const auto s2 = pow5[x4] + pow5[x3];
			auto pd = upper_bound(diffs.begin() + 1, diffs.end(), make_pair(s2, 0), firstLess) - 1;
			for (int x2 = 2; x2 < x3; ++x2)
			{
				const auto sum = s2 + pow5[x2];
				pd = HuntFwd(pd, diffs.end(), [&](decltype(pd) it){ return it->first < sum; });
				if (pd != diffs.end() && pd->first == sum && pd->second < x3)	// find each solution only once
				{
					const double y = pow(pd->first + pow5[pd->second], 0.2);
					cout << x4 << " " << x3 << " " << x2 << " " << pd->second << " -> " << static_cast<int>(y + 0.5) << "\n";
					if (--how_many <= 0)
						return true;
				}
			}
		}
	}
	return false;
}

Thanks, EchoLisp guys!

Third version

We expand on the second version with two main improvements. First, we use a hash table instead of binary search to improve the runtime from O(n^3 log n) to O(n^3). Second, we adapt the fast inverse square root algorithm to quickly compute the fifth root. Combined this gives a 7.3x speedup over the Second Version.

#include <array>
#include <cstdint>
#include <iostream>
#include <memory>
#include <numeric>

template <size_t n, typename T> static constexpr T power(T base) {
  if constexpr (n == 0)
    return 1;
  else if constexpr (n == 1)
    return base;
  else if constexpr (n & 1)
    return power<n / 2, T>(base * base) * base;
  else
    return power<n / 2, T>(base * base);
}

static constexpr int count = 1024;
static constexpr uint64_t count_diff = []() {
  // finds something that looks kinda sorta prime.                                                                                           
  const uint64_t coprime_to_this = 2llu * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23 * 29 * 31 * 37 * 41 * 43 * 59;
  // we make this oversized to reduce collisions and just hope it fits in cache.                                                             
  uint64_t guess = 7 * count * count;
  for (; std::gcd(guess, coprime_to_this) > 1; ++guess)
    ;
  return guess;
}();

static constexpr int fast_integer_root5(const double x) {
  constexpr uint64_t magic = 3685583637919522816llu;
  uint64_t x_i = std::bit_cast<uint64_t>(x);
  x_i /= 5;
  x_i += magic;
  const double x_f = std::bit_cast<double>(x_i);
  const double x5 = power<5>(x_f);
  return x_f * ((x - x5) / (3 * x5 + 2 * x)) + (x_f + .5);
}

static constexpr uint64_t hash(uint64_t h) { return h % count_diff; }

void euler() {
  std::array<int64_t, count> pow5;
  for (int64_t i = 0; i < count; i++)
    pow5[i] = power<5>(i);

  // build hash table                                                                                                                        
  constexpr int oversize_fudge = 8;
  std::unique_ptr<int16_t[]> differences = std::make_unique<int16_t[]>(count_diff + oversize_fudge);
  std::fill(differences.get(), differences.get() + count_diff + oversize_fudge, 0);
  for (int64_t n = 4; n < count; n++)
    for (int64_t d = 3; d < n; d++) {
      uint64_t h = hash(pow5[n] - pow5[d]);
      for (; differences[h]; ++h)
        if (h >= count_diff + oversize_fudge - 2) {
          std::cerr << "too many collisions; increase fudge factor or hash table size\n";
          return;
        }
      differences[h] = d;
      if (h >= count_diff)
        differences[h - count_diff] = d;
    }

  // brute force a,b,c                                                                                                                       
  const int a_max = fast_integer_root5(.25 * pow5.back());
  for (int a = 0; a <= a_max; a++) {
    const int b_max = fast_integer_root5((1.0 / 3.0) * (pow5.back() - pow5[a]));
    for (int b = a; b <= b_max; b++) {
      const int64_t a5_p_b5 = pow5[a] + pow5[b];
      const int c_max = fast_integer_root5(.5 * (pow5.back() - a5_p_b5));
      for (int c = b; c <= c_max; c++) {
        // lookup d in hash table                                                                                                            
        const int64_t n5_minus_d5 = a5_p_b5 + pow5[c];
	    //this loop is O(1)                                                                                                                 
	    for (uint64_t h = hash(n5_minus_d5); differences[h]; ++h) {
          if (const int d = differences[h]; d >= c)
            // calculate n from d                                                                                                            
            if (const int n = fast_integer_root5(n5_minus_d5 + pow5[d]);
                // check whether this is a solution                                                                                       
                n < count && n5_minus_d5 == pow5[n] - pow5[d] && d != n)
              std::cout << a << "^5 + " << b << "^5 + " << c << "^5 + " << d << "^5 = " << n << "^5\t"
                        << pow5[a] + pow5[b] + pow5[c] + pow5[d] << " = " << pow5[n] << '\n';
        }
      }
    }
  }
}

int main() {
  std::ios::sync_with_stdio(false);
  euler();
  return 0;
}

Clojure

(ns test-p.core
  (:require [clojure.math.numeric-tower :as math])
  (:require [clojure.data.int-map :as i]))

(defn solve-power-sum [max-value max-sols]
  " Finds solutions by using method approach of EchoLisp
    Large difference is we store a dictionary of all combinations 
    of y^5 - x^5 with the x, y value so we can simply lookup rather than have to search "
  (let [pow5 (mapv #(math/expt % 5) (range 0 (* 4 max-value)))                  ; Pow5 = Generate Lookup table for x^5
        y5-x3 (into (i/int-map) (for [x (range 1 max-value)                     ; For x0^5 + x1^5 + x2^5 + x3^5  = y^5
                                      y (range (+ 1 x) (* 4 max-value))]        ; compute y5-x3 = set of all possible differnences
                                  [(- (get pow5 y) (get pow5 x)) [x y]])) ; note: (get pow5 y) is closure for: pow5[y]
        solutions-found (atom 0)]

    (for [x0 (range 1 max-value)                                    ; Search over x0, x1, x2 for sums equal y5-x3
          x1 (range 1 x0)
          x2 (range 1 x1)
          :when (< @solutions-found max-sols)
          :let [sum (apply + (map pow5 [x0 x1 x2]))]         ; compute sum of items to the 5th power
          :when (contains? y5-x3 sum)]                       ; check if sum is in set of differences
      (do
        (swap! solutions-found inc)                          ; increment counter for solutions found
        (concat [x0 x1 x2] (get y5-x3 sum))))))              ; create result (since in set of differences)

; Output results with numbers in ascending order placing results into a set (i.e. #{}) so duplicates are discarded
                                                            ; CPU i7 920 Quad Core @2.67 GHz clock Windows 10
(println (into #{} (map sort (solve-power-sum 250 1))))   ; MAX = 250, find only 1 value: Duration was 0.26 seconds
(println (into #{} (map sort (solve-power-sum 1000 1000))));MAX = 1000, high max-value so all solutions found: Time = 4.8 seconds

Output

1st Solution with MAX = 250 (Solution Time: 260 ms CPU i7 920 Quad Core)
#{(27 84 110 133 144))

All Solutions with MAX = 1000 (Solution Time: 4.8 seconds CPU i7 920 Quad Core)
#{(27 84 110 133 144) 
(162 504 660 798 864) 
(135 420 550 665 720) 
(108 336 440 532 576) 
(189 588 770 931 1008) 
(54 168 220 266 288) 
(81 252 330 399 432)}

COBOL

       IDENTIFICATION DIVISION.
       PROGRAM-ID. EULER.
       DATA DIVISION.
       FILE SECTION.
       WORKING-STORAGE SECTION.
       1   TABLE-LENGTH CONSTANT 250.
       1   SEARCHING-FLAG     PIC 9.
        88  FINISHED-SEARCHING VALUE IS 1
                               WHEN SET TO FALSE IS 0.
       1  CALC.
        3  A               PIC 999 USAGE COMPUTATIONAL-5.
        3  B               PIC 999 USAGE COMPUTATIONAL-5.
        3  C               PIC 999 USAGE COMPUTATIONAL-5.
        3  D               PIC 999 USAGE COMPUTATIONAL-5.
        3  ABCD            PIC 9(18) USAGE COMPUTATIONAL-5.
        3  FIFTH-ROOT-OFFS PIC 999 USAGE COMPUTATIONAL-5.
        3  POWER-COUNTER   PIC 999 USAGE COMPUTATIONAL-5.
        88 POWER-MAX       VALUE TABLE-LENGTH.
       
       1   PRETTY.
        3  A               PIC ZZ9.
        3  FILLER          VALUE "^5 + ".
        3  B               PIC ZZ9.
        3  FILLER          VALUE "^5 + ".
        3  C               PIC ZZ9.
        3  FILLER          VALUE "^5 + ".
        3  D               PIC ZZ9.
        3  FILLER          VALUE "^5 = ".
        3  FIFTH-ROOT-OFFS PIC ZZ9.
        3  FILLER          VALUE "^5.".

       1   FIFTH-POWER-TABLE   OCCURS TABLE-LENGTH TIMES
                               ASCENDING KEY IS FIFTH-POWER
                               INDEXED BY POWER-INDEX.
        3  FIFTH-POWER PIC 9(18) USAGE COMPUTATIONAL-5.


       PROCEDURE DIVISION.
       MAIN-PARAGRAPH.
           SET FINISHED-SEARCHING TO FALSE.
           PERFORM POWERS-OF-FIVE-TABLE-INIT.
           PERFORM VARYING
               A IN CALC
               FROM 1 BY 1 UNTIL A IN CALC = TABLE-LENGTH

               AFTER B IN CALC
               FROM 1 BY 1 UNTIL B IN CALC = A IN CALC

               AFTER C IN CALC
               FROM 1 BY 1 UNTIL C IN CALC = B IN CALC

               AFTER D IN CALC
               FROM 1 BY 1 UNTIL D IN CALC = C IN CALC

               IF FINISHED-SEARCHING
                   STOP RUN
               END-IF

               PERFORM POWER-COMPUTATIONS

           END-PERFORM.

       POWER-COMPUTATIONS.

           MOVE ZERO TO ABCD IN CALC.

           ADD FIFTH-POWER(A IN CALC)
               FIFTH-POWER(B IN CALC)
               FIFTH-POWER(C IN CALC)
               FIFTH-POWER(D IN CALC)
                   TO ABCD IN CALC.

           SET POWER-INDEX TO 1.

           SEARCH ALL FIFTH-POWER-TABLE
               WHEN FIFTH-POWER(POWER-INDEX) = ABCD IN CALC
                      MOVE POWER-INDEX TO FIFTH-ROOT-OFFS IN CALC
                      MOVE CORRESPONDING CALC TO PRETTY
                      DISPLAY PRETTY END-DISPLAY
                      SET FINISHED-SEARCHING TO TRUE
           END-SEARCH

           EXIT PARAGRAPH.

       POWERS-OF-FIVE-TABLE-INIT.
           PERFORM VARYING POWER-COUNTER FROM 1 BY 1 UNTIL POWER-MAX
               COMPUTE FIFTH-POWER(POWER-COUNTER) = 
                   POWER-COUNTER *
                   POWER-COUNTER *
                   POWER-COUNTER *
                   POWER-COUNTER *
                   POWER-COUNTER 
               END-COMPUTE
           END-PERFORM.
           EXIT PARAGRAPH.

       END PROGRAM EULER.

Output

133^5 + 110^5 +  84^5 +  27^5 = 144^5.

Common Lisp

(ql:quickload :alexandria)
(let ((fifth-powers (mapcar #'(lambda (x) (expt x 5))
                            (alexandria:iota 250))))
  (loop named outer for x0 from 1 to (length fifth-powers) do
    (loop for x1 from 1 below x0 do
      (loop for x2 from 1 below x1 do
        (loop for x3 from 1 below x2 do
          (let ((x-sum (+ (nth x0 fifth-powers)
                          (nth x1 fifth-powers)
                          (nth x2 fifth-powers)
                          (nth x3 fifth-powers))))
            (if (member x-sum fifth-powers)
                  (return-from outer (list x0 x1 x2 x3 (round (expt x-sum 0.2)))))))))))
Output:
(133 110 84 27 144)

D

First version

Translation of: Rust
import std.stdio, std.range, std.algorithm, std.typecons;

auto eulersSumOfPowers() {
    enum maxN = 250;
    auto pow5 = iota(size_t(maxN)).map!(i => ulong(i) ^^ 5).array.assumeSorted;

    foreach (immutable x0; 1 .. maxN)
        foreach (immutable x1; 1 .. x0)
            foreach (immutable x2; 1 .. x1)
                foreach (immutable x3; 1 .. x2) {
                    immutable powSum = pow5[x0] + pow5[x1] + pow5[x2] + pow5[x3];
                    if (pow5.contains(powSum))
                        return tuple(x0, x1, x2, x3, pow5.countUntil(powSum));
                }
    assert(false);
}

void main() {
    writefln("%d^5 + %d^5 + %d^5 + %d^5 == %d^5", eulersSumOfPowers[]);
}
Output:
133^5 + 110^5 + 84^5 + 27^5 == 144^5

Run-time about 0.64 seconds. A Range-based Haskell-like solution is missing because of Issue 14833.

Second version

Translation of: Python
void main() {
    import std.stdio, std.range, std.algorithm, std.typecons;

    enum uint MAX = 250;
    uint[ulong] p5;
    Tuple!(uint, uint)[ulong] sum2;

    foreach (immutable i; 1 .. MAX) {
        p5[ulong(i) ^^ 5] = i;
        foreach (immutable j; i .. MAX)
            sum2[ulong(i) ^^ 5 + ulong(j) ^^ 5] = tuple(i, j);
    }

    const sk = sum2.keys.sort().release;
    foreach (p; p5.keys.sort())
        foreach (immutable s; sk) {
            if (p <= s)
                break;
            if (p - s in sum2) {
                writeln(p5[p], " ", tuple(sum2[s][], sum2[p - s][]));
                return; // Finds first only.
            }
        }
}
Output:
144 Tuple!(uint, uint, uint, uint)(27, 84, 110, 133)

Run-time about 0.10 seconds.

Third version

Translation of: C++

This solution is a brutal translation of the iterator-based C++ version, and it should be improved to use more idiomatic D Ranges.

import core.stdc.stdio, std.typecons, std.math, std.algorithm, std.range;

alias Pair = Tuple!(double, int);
alias PairPtr = Pair*;

// If less(x) is false, then less(x + 1) must also be false.
PairPtr huntForward(Pred)(PairPtr hint, const PairPtr end, const Pred less) pure nothrow @nogc {
    PairPtr result = hint;
    int step = 1;

    // Expanding phase.
    while (end - result > step) {
        PairPtr test = result + step;
        if (!less(test))
            break;
        result = test;
        step <<= 1;
    }

    // Contracting phase.
    while (step > 1) {
        step >>= 1;
        if (end - result <= step)
            continue;
        PairPtr test = result + step;
        if (less(test))
            result = test;
    }
    if (result != end && less(result))
        ++result;
    return result;
}


bool dPFind(int how_many) nothrow {
    enum MAX = 1_000;

    double[MAX] pow5;
    foreach (immutable i; 1 .. MAX)
        pow5[i] = double(i) ^^ 5;

    Pair[] diffs0; // Will contain (MAX-1) * (MAX-2) / 2 pairs.
    foreach (immutable i; 2 .. MAX)
        foreach (immutable j; 1 .. i)
            diffs0 ~= Pair(pow5[i] - pow5[j], j);

    // Remove pairs with duplicate first items.
    diffs0.length -= diffs0.sort!q{ a[0] < b[0] }.uniq.copy(diffs0).length;
    auto diffs = diffs0.assumeSorted!q{ a[0] < b[0] };

    foreach (immutable x4; 4 .. MAX - 1) {
        foreach (immutable x3; 3 .. x4) {
            immutable s2 = pow5[x4] + pow5[x3];
            auto pd0 = diffs[1 .. $].upperBound(Pair(s2, 0));
            PairPtr pd = &pd0[0] - 1;
            foreach (immutable x2; 2 .. x3) {
                immutable sum = s2 + pow5[x2];
                const PairPtr endPtr = &diffs[$ - 1] + 1;
                // This lambda heap-allocates.
                pd = huntForward(pd, endPtr, (in PairPtr p) pure => (*p)[0] < sum);
                if (pd != endPtr && (*pd)[0] == sum && (*pd)[1] < x3) { // Find each solution only once.
                    immutable y = ((*pd)[0] + pow5[(*pd)[1]]) ^^ 0.2;
                    printf("%d %d %d %d : %d\n", x4, x3, x2, (*pd)[1], cast(int)(y + 0.5));
                    if (--how_many <= 0)
                        return true;
                }
            }
        }
    }

    return false;
}


void main() nothrow {
    if (!dPFind(100))
        printf("Search finished.\n");
}
Output:
133 110 27 84 : 144
133 110 84 27 : 144
266 220 54 168 : 288
266 220 168 54 : 288
399 330 81 252 : 432
399 330 252 81 : 432
532 440 108 336 : 576
532 440 336 108 : 576
665 550 135 420 : 720
665 550 420 135 : 720
798 660 162 504 : 864
798 660 504 162 : 864
Search finished.

Run-time about 7.1 seconds.

Delphi

See Pascal.

EasyLang

n = 250
len p5[] n
len h5[] 65537
for i = 0 to n - 1
   p5[i + 1] = i * i * i * i * i
   h5[p5[i + 1] mod 65537 + 1] = 1
.
func search a s .
   y = -1
   b = n
   while a + 1 < b
      i = (a + b) div 2
      if p5[i + 1] > s
         b = i
      elif p5[i + 1] < s
         a = i
      else
         a = b
         y = i
      .
   .
   return y
.
for x0 = 0 to n - 1
   for x1 = 0 to x0
      sum1 = p5[x0 + 1] + p5[x1 + 1]
      for x2 = 0 to x1
         sum2 = p5[x2 + 1] + sum1
         for x3 = 0 to x2
            sum = p5[x3 + 1] + sum2
            if h5[sum mod 65537 + 1] = 1
               y = search x0 sum
               if y >= 0
                  print x0 & " " & x1 & " " & x2 & " " & x3 & " " & y
                  break 4
               .
            .
         .
      .
   .
.

EchoLisp

To speed up things, we search for x0, x1, x2 such as x0^5 + x1^5 + x2^5 = a difference of 5-th powers.

(define dim 250)

;; speed up n^5
(define (p5 n) (* n n n n n)) 
(remember 'p5) ;; memoize

;; build vector of all  y^5 - x^5 diffs - length 30877
(define all-y^5-x^5 
	(for*/vector 
		[(x (in-range 1 dim))  (y (in-range (1+ x) dim))] 
		(- (p5 y) (p5 x))))
		
;; sort to use vector-search
(begin (vector-sort! <  all-y^5-x^5) 'sorted) 

 ;; find couple (x y) from y^5 - x^5
(define (x-y y^5-x^5)
	(for*/fold (x-y null) 
	[(x (in-range 1 dim)) (y (in-range (1+ x ) dim))]
		(when 
			(= (- (p5 y) (p5 x)) y^5-x^5) 
			(set! x-y (list x y)) 
			(break #t)))) ; stop on first

;; search
(for*/fold  (sol null)  
	[(x0 (in-range 1 dim)) (x1 (in-range (1+ x0) dim)) (x2 (in-range (1+ x1) dim))]
	(set! sol (+ (p5 x0) (p5 x1) (p5 x2)))
 	(when 
 		(vector-search sol all-y^5-x^5)  ;; x0^5 + x1^5 + x2^5 = y^5 - x3^5 ???
 		(set! sol (append (list x0 x1 x2) (x-y  sol))) ;; found
 		(break #t))) ;; stop on first

     (27 84 110 133 144) ;; time 2.8 sec

Elixir

Translation of: Ruby
defmodule Euler do
  def sum_of_power(max \\ 250) do
    {p5, sum2} = setup(max)
    sk = Enum.sort(Map.keys(sum2))
    Enum.reduce(Enum.sort(Map.keys(p5)), Map.new, fn p,map ->
      sum(sk, p5, sum2, p, map)
    end)
  end
  
  defp setup(max) do
    Enum.reduce(1..max, {%{}, %{}}, fn i,{p5,sum2} ->
      i5 = i*i*i*i*i
      add = for j <- i..max, into: sum2, do: {i5 + j*j*j*j*j, [i,j]}
      {Map.put(p5, i5, i), add}
    end)
  end
  
  defp sum([], _, _, _, map), do: map
  defp sum([s|_], _, _, p, map) when p<=s, do: map
  defp sum([s|t], p5, sum2, p, map) do
    if sum2[p - s],
      do:   sum(t, p5, sum2, p, Map.put(map, Enum.sort(sum2[s] ++ sum2[p-s]), p5[p])),
      else: sum(t, p5, sum2, p, map)
  end
end

Enum.each(Euler.sum_of_power, fn {k,v} ->
  IO.puts Enum.map_join(k, " + ", fn i -> "#{i}**5" end) <> " = #{v}**5"
end)
Output:
27**5 + 84**5 + 110**5 + 133**5 = 144**5

ERRE

PROGRAM EULERO

CONST MAX=250

!$DOUBLE

FUNCTION POW5(X)
    POW5=X*X*X*X*X
END FUNCTION

!$INCLUDE="PC.LIB"

BEGIN
   CLS
   FOR X0=1 TO MAX DO
     FOR X1=1 TO X0 DO
        FOR X2=1 TO X1 DO
           FOR X3=1 TO X2 DO
              LOCATE(3,1) PRINT(X0;X1;X2;X3)
              SUM=POW5(X0)+POW5(X1)+POW5(X2)+POW5(X3)
              S1=INT(SUM^0.2#+0.5#)
              IF SUM=POW5(S1) THEN PRINT(X0,X1,X2,X3,S1) END IF
           END FOR
        END FOR
     END FOR
   END FOR
END PROGRAM
Output:
133 110 84 27 144

F#

//Find 4 integers whose 5th powers sum to the fifth power of an integer (Quickly!) - Nigel Galloway: April 23rd., 2015
let G =
  let GN = Array.init<float> 250 (fun n -> (float n)**5.0)
  let rec gng (n, i, g, e) =
    match (n, i, g, e) with
    | (250,_,_,_) -> "No Solution Found"
    | (_,250,_,_) -> gng (n+1, n+1, n+1, n+1)
    | (_,_,250,_) -> gng (n, i+1, i+1, i+1)
    | (_,_,_,250) -> gng (n, i, g+1, g+1)
    | _ -> let l = GN.[n] + GN.[i] + GN.[g] + GN.[e]
           match l with
           | _ when l > GN.[249]           -> gng(n,i,g+1,g+1)
           | _ when l = round(l**0.2)**5.0 -> sprintf "%d**5 + %d**5 + %d**5 + %d**5 = %d**5" n i g e (int (l**0.2))
           | _                             -> gng(n,i,g,e+1)
  gng (1, 1, 1, 1)
Output:
"27**5 + 84**5 + 110**5 + 133**5 = 144**5"

Factor

This solution uses Factor's backtrack vocabulary (based on continuations) to simplify the reduction of the search space. Each time xn is called, a new summand is introduced which can only take on a value as high as the previous summand - 1. This also creates a checkpoint for the backtracker. fail causes the backtracking to occur.

USING: arrays backtrack kernel literals math.functions
math.ranges prettyprint sequences ;

CONSTANT: pow5 $[ 0 250 [a,b) [ 5 ^ ] map ]

: xn ( n1 -- n2 n2 ) [1,b) amb-lazy dup ;

250 xn xn xn xn drop 4array dup pow5 nths sum dup pow5
member? [ pow5 index suffix . ] [ 2drop fail ] if
Output:
{ 133 110 84 27 144 }

Forth

Translation of: Go
: sq  dup * ;
: 5^  dup sq sq * ;

create pow5 250 cells allot
:noname
   250 0 DO  i 5^  pow5 i cells + !  LOOP ; execute

: @5^  cells pow5 + @ ;

: solution? ( n -- n )
   pow5 250 cells bounds DO
      dup i @ = IF  drop i pow5 - cell / unloop EXIT  THEN
   cell +LOOP drop 0 ;
 
\ GFORTH only provides 2 index variables: i, j
\ so the code creates locals for two outer loop vars, k & l

: euler  ( -- )
   250 4 DO i { l }
      l 3 DO i { k }
         k 2 DO
            i 1 DO
               i @5^ j @5^ + k @5^ + l @5^ + solution?
               dup IF
                  l . k . j . i . . cr
                  unloop unloop unloop unloop EXIT
               ELSE
                  drop
               THEN
            LOOP
         LOOP
      LOOP
   LOOP ;

euler
bye
Output:
$ gforth-fast ./euler.fs
133 110 84 27 144 

Fortran

FORTRAN IV

To solve this problem, we must handle integers up 250**5 ~= 9.8*10**11 . So we need integers with at less 41 bits. In 1966 all Fortrans were not equal. On IBM360, INTEGER was a 32-bit integer; on CDC6600, INTEGER was a 60-bit integer. And Leon J. Lander and Thomas R. Parkin used the CDC6600.

C EULER SUM OF POWERS CONJECTURE - FORTRAN IV
C FIND I1,I2,I3,I4,I5 : I1**5+I2**5+I3**5+I4**5=I5**5
      INTEGER I,P5(250),SUMX
      MAXN=250
      DO 1 I=1,MAXN
   1  P5(I)=I**5
      DO 6 I1=1,MAXN
      DO 6 I2=1,MAXN
      DO 6 I3=1,MAXN
      DO 6 I4=1,MAXN	  
      SUMX=P5(I1)+P5(I2)+P5(I3)+P5(I4)
      I5=1
   2  IF(I5-MAXN) 3,3,6
   3  IF(P5(I5)-SUMX) 5,4,6
   4  WRITE(*,300) I1,I2,I3,I4,I5
      STOP
   5  I5=I5+1
      GOTO 2
   6  CONTINUE
 300  FORMAT(5(1X,I3))
      END
Output:
  27  84 110 133 144

Fortran 95

Works with: Fortran version 95 and later
program sum_of_powers
  implicit none

  integer, parameter :: maxn = 249      
  integer, parameter :: dprec = selected_real_kind(15)
  integer :: i, x0, x1, x2, x3, y
  real(dprec) :: n(maxn), sumx

  n = (/ (real(i, dprec)**5, i = 1, maxn) /)
 
outer: do x0 = 1, maxn
         do x1 = 1, maxn
           do x2 = 1, maxn
             do x3 = 1, maxn
               sumx = n(x0)+ n(x1)+ n(x2)+ n(x3)
               y = 1
               do while(y <= maxn .and. n(y) <= sumx)
                 if(n(y) == sumx) then
                   write(*,*) x0, x1, x2, x3, y
                   exit outer
                 end if
                 y = y + 1
               end do  
             end do
           end do
         end do
       end do outer
        
end program
Output:
          27          84         110         133         144


FutureBasic

void local fn EulersSumOfPower( max as int )
  long w, x, y, z, sum, s1
  
  for w = 1 to max
    for x = 1 to w
      for y = 1 to x
        for z = 1 to y
          sum = w^5 + x^5 + y^5 + z^5
          s1  = int(sum^0.2)
          if ( sum == s1 ^ 5 )
            print w;"^5 + ";x;"^5 + ";y;"^5 + ";z;"^5 = ";s1;"^5"
            exit fn
          end if
        next
      next
    next
  next
end fn

fn EulersSumOfPower( 250 )

HandleEvents
Output:
133^5 + 110^5 + 84^5 + 27^5 = 144^5


Go

Translation of: Python
package main

import (
	"fmt"
	"log"
)

func main() {
	fmt.Println(eulerSum())
}

func eulerSum() (x0, x1, x2, x3, y int) {
	var pow5 [250]int
	for i := range pow5 {
		pow5[i] = i * i * i * i * i
	}
	for x0 = 4; x0 < len(pow5); x0++ {
		for x1 = 3; x1 < x0; x1++ {
			for x2 = 2; x2 < x1; x2++ {
				for x3 = 1; x3 < x2; x3++ {
					sum := pow5[x0] +
						pow5[x1] +
						pow5[x2] +
						pow5[x3]
					for y = x0 + 1; y < len(pow5); y++ {
						if sum == pow5[y] {
							return
						}
					}
				}
			}
		}
	}
	log.Fatal("no solution")
	return
}
Output:
133 110 84 27 144

Groovy

Translation of: Java
class EulerSumOfPowers {
    static final int MAX_NUMBER = 250

    static void main(String[] args) {
        boolean found = false
        long[] fifth = new long[MAX_NUMBER]

        for (int i = 1; i <= MAX_NUMBER; i++) {
            long i2 = i * i
            fifth[i - 1] = i2 * i2 * i
        }

        for (int a = 0; a < MAX_NUMBER && !found; a++) {
            for (int b = a; b < MAX_NUMBER && !found; b++) {
                for (int c = b; c < MAX_NUMBER && !found; c++) {
                    for (int d = c; d < MAX_NUMBER && !found; d++) {
                        long sum = fifth[a] + fifth[b] + fifth[c] + fifth[d]
                        int e = Arrays.binarySearch(fifth, sum)
                        found = (e >= 0)
                        if (found) {
                            println("${a + 1}^5 + ${b + 1}^5 + ${c + 1}^5 + ${d + 1}^5 + ${e + 1}^5")
                        }
                    }
                }
            }
        }
    }
}
Output:
27^5 + 84^5 + 110^5 + 133^5 + 144^5

Haskell

import Data.List
import Data.List.Ordered

main :: IO ()
main = print $ head [(x0,x1,x2,x3,x4) | 
                                        -- choose x0, x1, x2, x3 
                                        -- so that 250 < x3 < x2 < x1 < x0
                                        x3 <- [1..250-1], 
                                        x2 <- [1..x3-1], 
                                        x1 <- [1..x2-1], 
                                        x0 <- [1..x1-1], 

                                        let p5Sum = x0^5 + x1^5 + x2^5 + x3^5,

                                        -- lazy evaluation of powers of 5
                                        let p5List = [i^5|i <- [1..]], 

                                        -- is sum a power of 5 ?
                                        member p5Sum p5List, 

                                        -- which power of 5 is sum ?
                                        let Just x4 = elemIndex p5Sum p5List ]
Output:
(27,84,110,133,144)

Or, using dictionaries of powers and sums, and thus rather faster:

Translation of: Python
import qualified Data.Map.Strict as M
import Data.List (find, intercalate)
import Data.Maybe (maybe)


------------- EULER'S SUM OF POWERS CONJECTURE -----------

counterExample :: (M.Map Int (Int, Int), M.Map Int Int) -> Maybe (Int, Int)
counterExample (sumMap, powerMap) =
  find
    (\(p, s) -> M.member (p - s) sumMap)
    (M.keys powerMap >>=
     (((>>=) . flip takeWhile (M.keys sumMap) . (>)) <*> \ p s -> [(p, s)]))

sumMapForRange :: [Int] -> M.Map Int (Int, Int)
sumMapForRange xs =
  M.fromList
    [ ((x ^ 5) + (y ^ 5), (x, y))
    | x <- xs 
    , y <- tail xs 
    , x > y ]

powerMapForRange :: [Int] -> M.Map Int Int
powerMapForRange = M.fromList . (zip =<< fmap (^ 5))


--------------------------- TEST -------------------------
main :: IO ()
main =
  putStrLn $
  "Euler's sum of powers conjecture – " <>
  maybe
    ("no counter-example found in the range " <> rangeString xs)
    (showExample sumsAndPowers xs)
    (counterExample sumsAndPowers)
  where
    xs = [1 .. 249]
    sumsAndPowers = ((,) . sumMapForRange <*> powerMapForRange) xs

showExample :: (M.Map Int (Int, Int), M.Map Int Int) -> [Int] -> (Int, Int) -> String
showExample (sumMap, powerMap) xs (p, s) =
  "a counter-example in range " <> rangeString xs <> ":\n\n" <>
  intercalate "^5 + " (show <$> [a, b, c, d]) <>
  "^5 = " <>
  show (powerMap M.! p) <>
  "^5"
  where
    (a, b) = sumMap M.! (p - s)
    (c, d) = sumMap M.! s

rangeString :: [Int] -> String
rangeString [] = "[]"
rangeString (x:xs) = '[' : show x <> " .. " <> show (last xs) <> "]"
Output:
Euler's sum of powers conjecture – a counter-example in range [1 .. 249]:

133^5 + 110^5 + 84^5 + 27^5 = 144^5

J

   require 'stats'
   (#~ (= <.)@((+/"1)&.:(^&5)))1+4 comb 248
27 84 110 133

Explanation:

1+4 comb 248

finds all the possibilities for our four arguments. Then,

(#~ (= <.)@((+/"1)&.:(^&5)))

discards the cases we are not interested in. (It only keeps the case(s) where the fifth root of the sum of the fifth powers is an integer.)

Only one possibility remains.

Here's a significantly faster approach (about 100 times faster), based on the echolisp implementation:

find5=:3 :0
  y=. 250
  n=. i.y
  p=. n^5
  a=. (#~ 0&<),-/~p
  s=. /:~a
  l=. (i.*:y)(#~ 0&<),-/~p
  c=. 3 comb <.5%:(y^5)%4
  t=. +/"1 c{p
  x=. (t e. s)#t
  |.,&<&~./|:(y,y)#:l#~a e. x
)

Use:

   find5''
┌─────────────┬───┐
27 84 110 133144
└─────────────┴───┘

Note that this particular implementation is a bit hackish, since it relies on the solution being unique for the range of numbers being considered. If there were more solutions it would take a little extra code (though not much time) to untangle them.

Java

Translation of: ALGOL 68

Tested with Java 6.

public class eulerSopConjecture
{

    static final int    MAX_NUMBER = 250;

    public static void main( String[] args )
    {
        boolean found = false;
        long[]  fifth = new long[ MAX_NUMBER ];

        for( int i = 1; i <= MAX_NUMBER; i ++ )
        {
            long i2 =  i * i;
            fifth[ i - 1 ] = i2 * i2 * i;
        } // for i

        for( int a = 0; a < MAX_NUMBER && ! found ; a ++ )
        {
            for( int b = a; b < MAX_NUMBER && ! found ; b ++ )
            {
                for( int c = b; c < MAX_NUMBER && ! found ; c ++ )
                {
                    for( int d = c; d < MAX_NUMBER && ! found ; d ++ )
                    {
                        long sum  = fifth[a] + fifth[b] + fifth[c] + fifth[d];
                        int  e = java.util.Arrays.binarySearch( fifth, sum );
                        found  = ( e >= 0 );
                        if( found )
                        {
                            // the value at e is a fifth power
                            System.out.print( (a+1) + "^5 + "
                                            + (b+1) + "^5 + "
                                            + (c+1) + "^5 + "
                                            + (d+1) + "^5 = "
                                            + (e+1) + "^5"
                                            );
                        } // if found;;
                    } // for d
                } // for c
            } // for b
        } // for a
    } // main

} // eulerSopConjecture

Output:

27^5 + 84^5 + 110^5 + 133^5 = 144^5

JavaScript

ES5

var eulers_sum_of_powers = function (iMaxN) {

    var aPow5 = [];
    var oPow5ToN = {};

    for (var iP = 0; iP <= iMaxN; iP++) {
        var iPow5 = Math.pow(iP, 5);
        aPow5.push(iPow5);
        oPow5ToN[iPow5] = iP;
    }

    for (var i0 = 1; i0 <= iMaxN; i0++) {
        for (var i1 = 1; i1 <= i0; i1++) {
            for (var i2 = 1; i2 <= i1; i2++) {
                for (var i3 = 1; i3 <= i2; i3++) {
                    var iPow5Sum = aPow5[i0] + aPow5[i1] + aPow5[i2] + aPow5[i3];
                    if (typeof oPow5ToN[iPow5Sum] != 'undefined') {
                        return {
                            i0: i0,
                            i1: i1,l
                            i2: i2,
                            i3: i3,
                            iSum: oPow5ToN[iPow5Sum]
                        };
                    }
                }
            }
        }
    }

};

var oResult = eulers_sum_of_powers(250);

console.log(oResult.i0 + '^5 + ' + oResult.i1 + '^5 + ' + oResult.i2 +
    '^5 + ' + oResult.i3 + '^5 = ' + oResult.iSum + '^5');
Output:
133^5 + 110^5 + 84^5 + 27^5 = 144^5

This

Translation of: D

that verify: a^5 + b^5 + c^5 + d^5 = x^5

var N=1000, first=false
var ns={}, npv=[]
for (var n=0; n<=N; n++) {
	var np=Math.pow(n,5); ns[np]=n; npv.push(np)
}
loop:
for (var a=1;   a<=N; a+=1) 
for (var b=a+1; b<=N; b+=1)
for (var c=b+1; c<=N; c+=1)
for (var d=c+1; d<=N; d+=1) {
	var x = ns[ npv[a]+npv[b]+npv[c]+npv[d] ]
	if (!x) continue
	print( [a, b, c, d, x] )
	if (first) break loop
}
function print(c) {
	var e='<sup>5</sup>', ep=e+' + '
	document.write(c[0], ep, c[1], ep, c[2], ep, c[3], e, ' = ', c[4], e, '<br>')
}

Or this

Translation of: C

that verify: a^5 + b^5 + c^5 + d^5 = x^5

var N=1000, first=false
var npv=[], M=30 // x^5 == x modulo M (=2*3*5) 
for (var n=0; n<=N; n+=1) npv[n]=Math.pow(n, 5)
var mx=1+npv[N]; while(n<=N+M) npv[n++]=mx

loop:
for (var a=1;   a<=N; a+=1)
for (var b=a+1; b<=N; b+=1)
for (var c=b+1; c<=N; c+=1)
for (var t=npv[a]+npv[b]+npv[c], d=c+1, x=t%M+d; (n=t+npv[d])<mx; d+=1, x+=1) {
	while (npv[x]<=n) x+=M; x-=M // jump over M=30 values for x>d
	if (npv[x] != n) continue
	print( [a, b, c, d, x] )
	if (first) break loop;
}
function print(c) {
	var e='<sup>5</sup>', ep=e+' + '
	document.write(c[0], ep, c[1], ep, c[2], ep, c[3], e, ' = ', c[4], e, '<br>')
}

Or this

Translation of: EchoLisp

that verify: a^5 + b^5 + c^5 = x^5 - d^5

var N=1000, first=false
var dxs={}, pow=Math.pow 
for (var d=1; d<=N; d+=1)
	for (var dp=pow(d,5), x=d+1; x<=N; x+=1)
		dxs[pow(x,5)-dp]=[d,x]
loop:
for (var a=1; a<N; a+=1)
for (var ap=pow(a,5), b=a+1; b<N; b+=1) 
for (var abp=ap+pow(b,5), c=b+1; c<N; c+=1) {
	var dx = dxs[ abp+pow(c,5) ]
	if (!dx || c >= dx[0]) continue
	print( [a, b, c].concat( dx ) )  
	if (first) break loop
}
function print(c) {
	var e='<sup>5</sup>', ep=e+' + '
	document.write(c[0], ep, c[1], ep, c[2], ep, c[3], e, ' = ', c[4], e, '<br>')
}

Or this

Translation of: Python

that verify: a^5 + b^5 = x^5 - (c^5 + d^5)

var N=1000, first=false
var is={}, ipv=[], ijs={}, ijpv=[], pow=Math.pow
for (var i=1; i<=N; i+=1) {
	var ip=pow(i,5); is[ip]=i; ipv.push(ip)
	for (var j=i+1; j<=N; j+=1) {
		var ijp=ip+pow(j,5); ijs[ijp]=[i,j]; ijpv.push(ijp)
	} 
}
ijpv.sort( function (a,b) {return a - b } )
loop:
for (var i=0, ei=ipv.length; i<ei; i+=1)
for (var xp=ipv[i], j=0, je=ijpv.length; j<je; j+=1) {
	var cdp = ijpv[j]
	if (cdp >= xp) break
	var cd = ijs[xp-cdp]
	if (!cd) continue
	var ab = ijs[cdp]
	if (ab[1] >= cd[0]) continue
	print( [].concat(ab, cd, is[xp]) )
	if (first) break loop
}
function print(c) {
	var e='<sup>5</sup>', ep=e+' + '
	document.write(c[0], ep, c[1], ep, c[2], ep, c[3], e, ' = ', c[4], e, '<br>')
}
Output:
 275 + 845 + 1105 + 1335 = 1445
 545 + 1685 + 2205 + 2665 = 2885
 815 + 2525 + 3305 + 3995 = 4325
 1085 + 3365 + 4405 + 5325 = 5765
 1355 + 4205 + 5505 + 6655 = 7205
 1625 + 5045 + 6605 + 7985 = 8645

ES6

Procedural

(() => {
    'use strict';

    const eulersSumOfPowers = intMax => {
        const
            pow = Math.pow,
            xs = range(0, intMax)
            .map(x => pow(x, 5)),
            dct = xs.reduce((a, x, i) =>
                (a[x] = i,
                    a
                ), {});

        for (let a = 1; a <= intMax; a++) {
            for (let b = 2; b <= a; b++) {
                for (let c = 3; c <= b; c++) {
                    for (let d = 4; d <= c; d++) {
                        const sumOfPower = dct[xs[a] + xs[b] + xs[c] + xs[d]];
                        if (sumOfPower !== undefined) {
                            return [a, b, c, d, sumOfPower];
                        }
                    }
                }
            }
        }
        return undefined;
    };

    // range :: Int -> Int -> [Int]
    const range = (m, n) =>
        Array.from({
            length: Math.floor(n - m) + 1
        }, (_, i) => m + i);

    // TEST
    const soln = eulersSumOfPowers(250);
    return soln ? soln.slice(0, 4)
        .map(x => `${x}^5`)
        .join(' + ') + ` = ${soln[4]}^5` : 'No solution found.'

})();
Output:
133^5 + 110^5 + 84^5 + 27^5 = 144^5

Functional

Using dictionaries of powers and sums, and a little faster than the procedural version above:

Translation of: Python
Translation of: Haskell
(() => {
    'use strict';

    const main = () => {

        const
            iFrom = 1,
            iTo = 249,
            xs = enumFromTo(1, 249),
            p5 = x => Math.pow(x, 5);

        const
            // powerMap :: Dict Int Int
            powerMap = mapFromList(
                zip(map(p5, xs), xs)
            ),
            // sumMap :: Dict Int (Int, Int)
            sumMap = mapFromList(
                bind(
                    xs,
                    x => bind(
                        tail(xs),
                        y => Tuple(
                            p5(x) + p5(y),
                            Tuple(x, y)
                        )
                    )
                )
            );

        // mbExample :: Maybe (Int, Int)
        const mbExample = find(
            tpl => member(fst(tpl) - snd(tpl), sumMap),
            bind(
                map(x => parseInt(x, 10),
                    keys(powerMap)
                ),
                p => bind(
                    takeWhile(
                        x => x < p,
                        map(x => parseInt(x, 10),
                            keys(sumMap)
                        )
                    ),
                    s => [Tuple(p, s)]
                )
            )
        );

        // showExample :: (Int, Int) -> String
        const showExample = tpl => {
            const [p, s] = Array.from(tpl);
            const [a, b] = Array.from(sumMap[p - s]);
            const [c, d] = Array.from(sumMap[s]);
            return 'Counter-example found:\n' + intercalate(
                '^5 + ',
                map(str, [a, b, c, d])
            ) + '^5 = ' + str(powerMap[p]) + '^5';
        };

        return maybe(
            'No counter-example found',
            showExample,
            mbExample
        );
    };

    // GENERIC FUNCTIONS ----------------------------------

    // Just :: a -> Maybe a
    const Just = x => ({
        type: 'Maybe',
        Nothing: false,
        Just: x
    });

    // Nothing :: Maybe a
    const Nothing = () => ({
        type: 'Maybe',
        Nothing: true,
    });

    // Tuple (,) :: a -> b -> (a, b)
    const Tuple = (a, b) => ({
        type: 'Tuple',
        '0': a,
        '1': b,
        length: 2
    });

    // bind (>>=) :: [a] -> (a -> [b]) -> [b]
    const bind = (xs, mf) => [].concat.apply([], xs.map(mf));

    // concat :: [[a]] -> [a]
    // concat :: [String] -> String
    const concat = xs =>
        0 < xs.length ? (() => {
            const unit = 'string' !== typeof xs[0] ? (
                []
            ) : '';
            return unit.concat.apply(unit, xs);
        })() : [];


    // enumFromTo :: (Int, Int) -> [Int]
    const enumFromTo = (m, n) =>
        Array.from({
            length: 1 + n - m
        }, (_, i) => m + i);

    // find :: (a -> Bool) -> [a] -> Maybe a
    const find = (p, xs) => {
        for (let i = 0, lng = xs.length; i < lng; i++) {
            if (p(xs[i])) return Just(xs[i]);
        }
        return Nothing();
    };

    // fst :: (a, b) -> a
    const fst = tpl => tpl[0];

    // intercalate :: [a] -> [[a]] -> [a]
    // intercalate :: String -> [String] -> String
    const intercalate = (sep, xs) =>
        0 < xs.length && 'string' === typeof sep &&
        'string' === typeof xs[0] ? (
            xs.join(sep)
        ) : concat(intersperse(sep, xs));

    // intersperse(0, [1,2,3]) -> [1, 0, 2, 0, 3]

    // intersperse :: a -> [a] -> [a]
    // intersperse :: Char -> String -> String
    const intersperse = (sep, xs) => {
        const bln = 'string' === typeof xs;
        return xs.length > 1 ? (
            (bln ? concat : x => x)(
                (bln ? (
                    xs.split('')
                ) : xs)
                .slice(1)
                .reduce((a, x) => a.concat([sep, x]), [xs[0]])
            )) : xs;
    };

    // keys :: Dict -> [String]
    const keys = Object.keys;

    // Returns Infinity over objects without finite length.
    // This enables zip and zipWith to choose the shorter
    // argument when one is non-finite, like cycle, repeat etc

    // length :: [a] -> Int
    const length = xs =>
        (Array.isArray(xs) || 'string' === typeof xs) ? (
            xs.length
        ) : Infinity;

    // map :: (a -> b) -> [a] -> [b]
    const map = (f, xs) =>
        (Array.isArray(xs) ? (
            xs
        ) : xs.split('')).map(f);

    // mapFromList :: [(k, v)] -> Dict
    const mapFromList = kvs =>
        kvs.reduce(
            (a, kv) => {
                const k = kv[0];
                return Object.assign(a, {
                    [
                        (('string' === typeof k) && k) || JSON.stringify(k)
                    ]: kv[1]
                });
            }, {}
        );

    // Default value (v) if m.Nothing, or f(m.Just)

    // maybe :: b -> (a -> b) -> Maybe a -> b
    const maybe = (v, f, m) =>
        m.Nothing ? v : f(m.Just);

    // member :: Key -> Dict -> Bool
    const member = (k, dct) => k in dct;

    // snd :: (a, b) -> b
    const snd = tpl => tpl[1];

    // str :: a -> String
    const str = x => x.toString();

    // tail :: [a] -> [a]
    const tail = xs => 0 < xs.length ? xs.slice(1) : [];

    // take :: Int -> [a] -> [a]
    // take :: Int -> String -> String
    const take = (n, xs) =>
        'GeneratorFunction' !== xs.constructor.constructor.name ? (
            xs.slice(0, n)
        ) : [].concat.apply([], Array.from({
            length: n
        }, () => {
            const x = xs.next();
            return x.done ? [] : [x.value];
        }));

    // takeWhile :: (a -> Bool) -> [a] -> [a]
    // takeWhile :: (Char -> Bool) -> String -> String
    const takeWhile = (p, xs) =>
        xs.constructor.constructor.name !==
        'GeneratorFunction' ? (() => {
            const lng = xs.length;
            return 0 < lng ? xs.slice(
                0,
                until(
                    i => lng === i || !p(xs[i]),
                    i => 1 + i,
                    0
                )
            ) : [];
        })() : takeWhileGen(p, xs);


    // until :: (a -> Bool) -> (a -> a) -> a -> a
    const until = (p, f, x) => {
        let v = x;
        while (!p(v)) v = f(v);
        return v;
    };

    // Use of `take` and `length` here allows for zipping with non-finite
    // lists - i.e. generators like cycle, repeat, iterate.

    // zip :: [a] -> [b] -> [(a, b)]
    const zip = (xs, ys) => {
        const lng = Math.min(length(xs), length(ys));
        return Infinity !== lng ? (() => {
            const bs = take(lng, ys);
            return take(lng, xs).map((x, i) => Tuple(x, bs[i]));
        })() : zipGen(xs, ys);
    };

    // MAIN ---
    return main();
})();
Output:
Counter-example found:
133^5 + 110^5 + 84^5 + 27^5 = 144^5

jq

Works with: jq version 1.4

This version finds all non-decreasing solutions within the specified bounds, using a brute-force but not entirely blind approach.

# Search for y in 1 .. maxn (inclusive) for a solution to SIGMA (xi ^ 5) = y^5
# and for each solution with x0<=x1<=...<x3, print [x0, x1, x3, x3, y]
#
def sum_of_powers_conjecture(maxn):
  def p5: . as $in | (.*.) | ((.*.) * $in);
  def fifth: log / 5 | exp;

  # return the fifth root if . is a power of 5
  def integral_fifth_root: fifth | if . == floor then . else false end;

  (maxn | p5) as $uber
  | range(1; maxn) as $x0
  | ($x0 | p5) as $s0
  | if $s0 < $uber then range($x0; ($uber - $s0 | fifth) + 1) as $x1
    | ($s0 + ($x1 | p5)) as $s1
    | if $s1 < $uber then range($x1; ($uber - $s1 | fifth) + 1) as $x2
      | ($s1 + ($x2 | p5)) as $s2
        | if $s2 < $uber then range($x2; ($uber - $s2 | fifth) + 1) as $x3
          | ($s2 + ($x3 | p5)) as $sumx
	  | ($sumx | integral_fifth_root)
	  | if . then [$x0,$x1,$x2,$x3,.] else empty end
	  else empty
	  end
      else empty
      end
    else empty
    end ;

The task:

sum_of_powers_conjecture(249)
Output:
$ jq -c -n -f Euler_sum_of_powers_conjecture_fifth_root.jq
[27,84,110,133,144]

Julia

const lim = 250
const pwr = 5
const p = [i^pwr for i in 1:lim]

x = zeros(Int, pwr-1)
y = 0

for a in combinations(1:lim, pwr-1)
    b = searchsorted(p, sum(p[a]))
    0 < length(b) || continue
    x = a
    y = b[1]
    break
end

if y == 0
    println("No solution found for power = ", pwr, " and limit = ", lim, ".")
else
    s = [@sprintf("%d^%d", i, pwr) for i in x]
    s = join(s, " + ")
    println("A solution is ", s, " = ", @sprintf("%d^%d", y, pwr), ".")
end
Output:
A solution is 27^5 + 84^5 + 110^5 + 133^5 = 144^5.

Kotlin

fun main(args: Array<String>) {
    val p5 = LongArray(250){ it.toLong() * it * it * it * it }
    var sum: Long
    var y: Int
    var found = false
    loop@ for (x0 in 0 .. 249)
        for (x1 in 0 .. x0 - 1)
            for (x2 in 0 .. x1 - 1)
                for (x3 in 0 .. x2 - 1) {
                    sum = p5[x0] + p5[x1] + p5[x2] + p5[x3]
                    y = p5.binarySearch(sum)
                    if (y >= 0) {
                        println("$x0^5 + $x1^5 + $x2^5 + $x3^5 = $y^5")
                        found = true
                        break@loop
                    }
                }
    if (!found) println("No solution was found")
}
Output:
133^5 + 110^5 + 84^5 + 27^5 = 144^5

Lua

Brute force but still takes under two seconds with LuaJIT.

-- Fast table search (only works if table values are in order)
function binarySearch (t, n)
    local start, stop, mid = 1, #t
    while start < stop do
        mid = math.floor((start + stop) / 2)
        if n == t[mid] then
            return mid
        elseif n < t[mid] then
            stop = mid - 1
        else
            start = mid + 1
        end
    end
    return nil
end

-- Test Euler's sum of powers conjecture
function euler (limit)
    local pow5, sum = {}
    for i = 1, limit do pow5[i] = i^5 end
    for x0 = 1, limit do
        for x1 = 1, x0 do
            for x2 = 1, x1 do
                for x3 = 1, x2 do
                    sum = pow5[x0] + pow5[x1] + pow5[x2] + pow5[x3]
                    if binarySearch(pow5, sum) then
                        print(x0 .. "^5 + " .. x1 .. "^5 + " .. x2 .. "^5 + " .. x3 .. "^5 = " .. sum^(1/5) .. "^5")
                        return true
                    end
                end
            end
        end
    end
    return false
end

-- Main procedure
if euler(249) then
    print("Time taken: " .. os.clock() .. " seconds")
else
    print("Looks like he was right after all...")
end
Output:
133^5 + 110^5 + 84^5 + 27^5 = 144^5
Time taken: 1.247 seconds

Mathematica /Wolfram Language

Sort[FindInstance[
   x0^5 + x1^5 + x2^5 + x3^5 == y^5 && x0 > 0 && x1 > 0 && x2 > 0 && 
    x3 > 0, {x0, x1, x2, x3, y}, Integers][[1, All, -1]]]
Output:
{27,84,110,133,144}

Modula-2

This example does not show the output mentioned in the task description on this page (or a page linked to from here). Please ensure that it meets all task requirements and remove this message.
Note that phrases in task descriptions such as "print and display" and "print and show" for example, indicate that (reasonable length) output be a part of a language's solution.


MODULE EulerConjecture;
FROM FormatString IMPORT FormatString;
FROM Terminal IMPORT WriteString,WriteLn,ReadChar;

PROCEDURE Pow5(a : LONGINT) : LONGINT;
BEGIN
    RETURN a * a * a * a * a
END Pow5;

VAR
    buf : ARRAY[0..63] OF CHAR;
    a,b,c,d,e,sum,curr : LONGINT;
BEGIN
    FOR a:=0 TO 250 DO
        FOR b:=a TO 250 DO
            IF b=a THEN CONTINUE END;
            FOR c:=b TO 250 DO
                IF (c=a) OR (c=b) THEN CONTINUE END;
                FOR d:=c TO 250 DO
                    IF (d=a) OR (d=b) OR (d=c) THEN CONTINUE END;
                    sum := Pow5(a) + Pow5(b) + Pow5(c) + Pow5(d);
                    FOR e:=d TO 250 DO
                        IF (e=a) OR (e=b) OR (e=c) OR (e=d) THEN CONTINUE END;
                        curr := Pow5(e);
                        IF (sum#0) AND (sum=curr) THEN
                            FormatString("%l^5 + %l^5 + %l^5 + %l^5 = %l^5\n", buf, a, b, c, d, e);
                            WriteString(buf)
                        ELSIF curr > sum THEN
                            BREAK
                        END
                    END;
                END;
            END;
        END;
    END;

    WriteString("Done");
    WriteLn;
    ReadChar
END EulerConjecture.

Alternative

Works with: XDS Modula-2

Uses the same idea as the QL SuperBASIC solution, i.e. if the desired equation holds modulo several positive integers M0, M1, ..., and if the LCM of M0, M1, ... is greater than 4*(249^5), then the equation holds absolutely. This makes it unnecessary to have variables with enough bits to store the fifth powers up to 249^5. It isn't clear why the fifth powers in a counterexample must be all different, as the task description seems to assume, so the demo program doesn't enforce this.

MODULE EulerFifthPowers;
FROM InOut IMPORT WriteString, WriteLn, WriteCard;

TYPE
  (* CARDINAL in XDS Modula-2 is 32-bit unsigned integer *)
  TModArray = ARRAY [0..4] OF CARDINAL;
CONST
  Modulus = TModArray {327, 329, 334, 335, 337};
  MaxTerm = 249;
VAR
  res : ARRAY [0..4] OF ARRAY [0..Modulus[4]-1] OF CARDINAL;
  inv : ARRAY [0..Modulus[0]-1] OF CARDINAL;
  m, s0, s1, s2, s3, x, x0, x1, x2, x3, y : CARDINAL;
BEGIN
  (* Set up arrays of 5th powers w.r.t. each modulus *)
  FOR m := 0 TO 4 DO
    FOR x := 0 TO Modulus[m] - 1 DO
      y := (x*x) MOD Modulus[m];
      y := (y*y) MOD Modulus[m];
      res[m][x] := (x*y) MOD Modulus[m];
    END;
  END;

  (* For Modulus[0] only, set up an inverse array (having checked
     elsewhere that 5th powers MOD Modulus[0] are all different) *)
  FOR x := 0 TO Modulus[0] - 1 DO
    y := res[0][x];
    inv[y] := x;
  END;

  (* Test sums of four 5th powers *)
  FOR x0 := 1 TO MaxTerm DO
    s0 := res[0][x0];
    FOR x1 := x0 TO MaxTerm DO
      s1 := (s0 + res[0][x1]) MOD Modulus[0];
      FOR x2 := x1 TO MaxTerm DO
        s2 := (s1 + res[0][x2]) MOD Modulus[0];
        FOR x3 := x2 TO MaxTerm DO
          s3 := (s2 + res[0][x3]) MOD Modulus[0];
          y := inv[s3];
          IF y <= MaxTerm THEN

            (* Here, by definition of y, we have
                 x0^5 + x1^5 + x2^5 + x3^5 = y^5 MOD Modulus[0].
               Now test the congruence for the other moduli. *)
            m := 1;
            WHILE (m <= 4) AND
              ((res[m][x0] + res[m][x1] + res[m][x2] + res[m][x3])
                MOD Modulus[m] = res[m][y])
            DO INC(m); END;
            IF (m = 5) THEN
              WriteString('Counterexample: ');
              WriteCard( x0, 1); WriteString('^5 + ');
              WriteCard( x1, 1); WriteString('^5 + ');
              WriteCard( x2, 1); WriteString('^5 + ');
              WriteCard( x3, 1); WriteString('^5 = ');
              WriteCard( y, 1); WriteString('^5');
              WriteLn;
            END; (* IF m... *)
          END; (* IF y... *)
        END; (* FOR x3 *)
      END; (* FOR x2 *)
    END; (* FOR x1 *)
  END; (* FOR x0 *)
END EulerFifthPowers.
Output:
Counterexample: 27^5 + 84^5 + 110^5 + 133^5 = 144^5

Nim

Translation of: PureBasic
# Brute force approach

import times

# assumes an array of non-decreasing positive integers
proc binarySearch(a : openArray[int], target : int) : int = 
  var left, right, mid : int
  left = 0
  right = len(a) - 1
  while true :
    if left > right : return 0  # no match found
    mid = (left + right) div 2
    if a[mid] < target :
      left = mid + 1
    elif a[mid] > target :
      right = mid - 1
    else :
      return mid  # match found

var 
  p5 : array[250, int]
  sum = 0
  y, t1 : int

let t0 = cpuTime()

for i in 1 .. 249 :
  p5[i] = i * i * i * i * i
 
for x0 in 1 .. 249 :
  for x1 in 1 .. x0 - 1 :
    for x2 in 1 .. x1 - 1 :
      for x3 in 1 .. x2 - 1 :
        sum = p5[x0] + p5[x1] + p5[x2] + p5[x3]
        y = binarySearch(p5, sum)
        if y > 0 :
          t1 = int((cputime() - t0) * 1000.0) 
          echo "Time : ", t1, " milliseconds"
          echo  $x0 & "^5 + " & $x1 & "^5 + " & $x2 & "^5 + " & $x3 & "^5 = " & $y & "^5"
          quit()

if y == 0 :
  echo "No solution was found"
Output:
Time : 156 milliseconds
133^5 + 110^5 + 84^5 + 27^5 = 144^5

Oforth

: eulerSum
| i j k l ip jp kp |
   250 loop: i [
      i 5 pow ->ip
      i 1 + 250 for: j [
         j 5 pow ip + ->jp
         j 1 + 250 for: k [
            k 5 pow jp + ->kp
            k 1 + 250 for: l [
               kp l 5 pow + 0.2 powf dup asInteger == ifTrue: [ [ i, j, k, l ] println ]
              ]
            ]
         ]
      ] ;
Output:
>eulerSum
[27, 84, 110, 133]

PARI/GP

Naive script:

forvec(v=vector(4,i,[0,250]), if(ispower(v[1]^5+v[2]^5+v[3]^5+v[4]^5,5,&n), print(n" "v)), 2)
Output:
144 [27, 84, 110, 133]

Naive + caching (setbinop):

{
v2=setbinop((x,y)->[min(x,y),max(x,y),x^5+y^5],[0..250]); \\ sums of two fifth powers
for(i=2,#v2,
  for(j=1,i-1,
    if(v2[i][2]<v2[j][2] && ispower(v2[i][3]+v2[j][3],5,&n) && #(v=Set([v2[i][1],v2[i][2],v2[j][1],v2[j][2]]))==4,
      print(n" "v)
    )
  )
)
}
Output:
144 [27, 84, 110, 133]

Pascal

Works with: Free Pascal

slightly improved.Reducing calculation time by temporary sum and early break.

program Pot5Test;
{$IFDEF FPC} {$MODE DELPHI}{$ELSE]{$APPTYPE CONSOLE}{$ENDIF}
type
  tTest = double;//UInt64;{ On linux 32Bit double is faster than  Uint64 } 
var
  Pot5 : array[0..255] of tTest;
  res,tmpSum : tTest;
  x0,x1,x2,x3, y : NativeUint;//= Uint32 or 64 depending on OS xx-Bit
  i : byte;
BEGIN
  For i := 1 to 255 do
    Pot5[i] := (i*i*i*i)*Uint64(i);

  For x0 := 1 to 250-3 do
    For x1 := x0+1 to 250-2 do
      For x2 := x1+1 to 250-1 do
      Begin
        //set y here only, because pot5 is strong monoton growing,
        //therefor the sum is strong monoton growing too.
        y := x2+2;// aka x3+1
        tmpSum := Pot5[x0]+Pot5[x1]+Pot5[x2];
        For x3 := x2+1 to 250 do
        Begin
          res := tmpSum+Pot5[x3];
          while (y< 250) AND (res > Pot5[y]) do
            inc(y);
          IF y > 250 then BREAK;
          if res = Pot5[y] then
            writeln(x0,'^5+',x1,'^5+',x2,'^5+',x3,'^5 = ',y,'^5');
        end;
      end;
END.
output
27^5+84^5+110^5+133^5 = 144^5
real  0m1.091s {Uint64; Linux 32}real  0m0.761s {double; Linux 32}real  0m0.511s{Uint64; Linux 64}

Perl

Brute Force:

use constant MAX => 250;
my @p5 = (0,map { $_**5 } 1 .. MAX-1);
my $s = 0;
my %p5 = map { $_ => $s++ } @p5;
for my $x0 (1..MAX-1) {
  for my $x1 (1..$x0-1) {
    for my $x2 (1..$x1-1) {
      for my $x3 (1..$x2-1) {
        my $sum = $p5[$x0] + $p5[$x1] + $p5[$x2] + $p5[$x3];
        die "$x3 $x2 $x1 $x0 $p5{$sum}\n" if exists $p5{$sum};
      }
    }
  }
}
Output:
27 84 110 133 144

Adding some optimizations makes it 5x faster with similar output, but obfuscates things.

Translation of: C++
use constant MAX => 250;
my @p5 = (0,map { $_**5 } 1 .. MAX-1);
my $rs = 5;
for my $x0 (1..MAX-1) {
  for my $x1 (1..$x0-1) {
    for my $x2 (1..$x1-1) {
      my $s2 = $p5[$x0] + $p5[$x1] + $p5[$x2];
      $rs-- while $rs > 0 && $p5[$rs] > $s2;
      for (my $x3 = 1;  $x3 < $x2;  $x3++) {
        my $e30 = ($x0 + $x1 + $x2 + $x3 - $rs) % 30;
        $x3 += (30-$e30) if $e30;
        last if $x3 >= $x2;
        my $sum = $s2 + $p5[$x3];
        $rs++ while $rs < MAX-1 && $p5[$rs] < $sum;
        die "$x3 $x2 $x1 $x0 $rs\n" if $p5[$rs] == $sum;
      }
    }
  }
}

Phix

Translation of: Python

Around four seconds, not spectacularly fast. My naive brute force was over a minute. This is not where Phix shines.
Quitting when the first is found drops the main loop to 0.7s, so 1.1s in all, vs 4.3s for the full search.
Without the return 0, you just get six permutes (of ordered pairs) for 144.

with javascript_semantics
constant MAX = 250
 
constant p5 = new_dict(),
         sum2 = new_dict()
 
atom t0 = time()
for i=1 to MAX do
    atom i5 = power(i,5)
    setd(i5,i,p5)
    for j=1 to i-1 do
        atom j5 = power(j,5)
        setd(j5+i5,{j,i},sum2)
    end for
end for
 
?time()-t0
 
function forsum2(object s, object data, object p)
    if p<=s then return 0 end if
    integer k = getd_index(p-s,sum2)
    if k!=NULL then
        ?getd(p,p5)&data&getd_by_index(k,sum2)
        return 0 -- (show one solution per p)
    end if
    return 1
end function
 
function forp5(object key, object /*data*/, object /*user_data*/)
    traverse_dict(forsum2,key,sum2)
    return 1
end function
 
traverse_dict(forp5,0,p5)
 
?time()-t0
Output:
0.421
{144,27,84,110,133}
4.312

PHP

Translation of: Python
<?php

function eulers_sum_of_powers () {
	$max_n = 250;
	$pow_5 = array();
	$pow_5_to_n = array();
	for ($p = 1; $p <= $max_n; $p ++) {
		$pow5 = pow($p, 5);
		$pow_5 [$p] = $pow5;
		$pow_5_to_n[$pow5] = $p;
	}
	foreach ($pow_5 as $n_0 => $p_0) {
		foreach ($pow_5 as $n_1 => $p_1) {
			if ($n_0 < $n_1) continue;
			foreach ($pow_5 as $n_2 => $p_2) {
				if ($n_1 < $n_2) continue;
				foreach ($pow_5 as $n_3 => $p_3) {
					if ($n_2 < $n_3) continue;
					$pow_5_sum = $p_0 + $p_1 + $p_2 + $p_3;
					if (isset($pow_5_to_n[$pow_5_sum])) {
						return array($n_0, $n_1, $n_2, $n_3, $pow_5_to_n[$pow_5_sum]);
					}
				}
			}
		}
	}
}

list($n_0, $n_1, $n_2, $n_3, $y) = eulers_sum_of_powers();

echo "$n_0^5 + $n_1^5 + $n_2^5 + $n_3^5 = $y^5";

?>
Output:
133^5 + 110^5 + 84^5 + 27^5 = 144^5

Picat

import sat. 
main =>	
    X = new_list(5), X :: 1..150, decreasing_strict(X),
    X[1]**5 #= sum([X[I]**5 : I in 2..5]),
    solve(X), printf("%d**5 = %d**5 + %d**5 + %d**5 + %d**5", X[1], X[2], X[3], X[4], X[5]).
Output:
144**5 = 133**5 + 110**5 + 84**5 + 27**5
CPU time 6.626 seconds

PicoLisp

(off P)
(off S)

(for I 250
   (idx
      'P
      (list (setq @@ (** I 5)) I)
      T )
   (for (J I (>= 250 J) (inc J))
      (idx
         'S
         (list (+ @@ (** J 5)) (list I J))
         T ) ) )
(println
   (catch 'found
      (for A (idx 'P)
         (for B (idx 'S)
            (T (<= (car A) (car B)))
            (and
               (lup S (- (car A) (car B)))
               (throw 'found
                  (conc
                     (cadr (lup S (car B)))
                     (cadr (lup S (- (car A) (car B))))
                     (cdr (lup P (car A))) ) ) ) ) ) ) )
Output:
(27 84 110 133 144)

PowerShell

Brute Force Search
This is a slow algorithm, so attempts have been made to speed it up, including pre-computing the powers, using an ArrayList for them, and using [int] to cast the 5th root rather than use truncate.

# EULER.PS1
$max = 250

$powers =  New-Object System.Collections.ArrayList
for ($i = 0; $i -lt $max; $i++) {
  $tmp = $powers.Add([Math]::Pow($i, 5)) 
}

for ($x0 = 1; $x0 -lt $max; $x0++) {
  for ($x1 = 1; $x1 -lt $x0; $x1++) {
    for ($x2 = 1; $x2 -lt $x1; $x2++) {
      for ($x3 = 1; $x3 -lt $x2; $x3++) {
        $sum = $powers[$x0] + $powers[$x1] + $powers[$x2] + $powers[$x3]
        $S1 = [int][Math]::pow($sum,0.2)

        if ($sum -eq $powers[$S1]) {
          Write-host "$x0^5 + $x1^5 + $x2^5 + $x3^5 = $S1^5"
          return
        }
      }
    }
  }
}
Output:
PS > measure-command { .\euler.ps1 | out-default }
133^5 + 110^5 + 84^5 + 27^5 = 144^5


Days              : 0
Hours             : 0
Minutes           : 0
Seconds           : 31
Milliseconds      : 608
Ticks             : 316082251
TotalDays         : 0.000365835938657407

Prolog

makepowers :-
    retractall(pow5(_, _)),
    between(1, 249, X),
    Y is X * X * X * X * X,
    assert(pow5(X, Y)),
    fail.
makepowers.

within(A, Bx, N) :-  % like between but with an exclusive upper bound
   succ(B, Bx),
   between(A, B, N).

solution(X0, X1, X2, X3, Y) :-
    makepowers,
    within(4, 250, X0), pow5(X0, X0_5th),
    within(3, X0,  X1), pow5(X1, X1_5th),
    within(2, X1,  X2), pow5(X2, X2_5th),
    within(1, X2,  X3), pow5(X3, X3_5th),
    Y_5th is X0_5th + X1_5th + X2_5th + X3_5th,
    pow5(Y, Y_5th).
Output:
?- solution(X0,X1,X2,X3,Y).
X0 = 133,
X1 = 110,
X2 = 84,
X3 = 27,
Y = 144 .

Python

Procedural

def eulers_sum_of_powers():
    max_n = 250
    pow_5 = [n**5 for n in range(max_n)]
    pow5_to_n = {n**5: n for n in range(max_n)}
    for x0 in range(1, max_n):
        for x1 in range(1, x0):
            for x2 in range(1, x1):
                for x3 in range(1, x2):
                    pow_5_sum = sum(pow_5[i] for i in (x0, x1, x2, x3))
                    if pow_5_sum in pow5_to_n:
                        y = pow5_to_n[pow_5_sum]
                        return (x0, x1, x2, x3, y)

print("%i**5 + %i**5 + %i**5 + %i**5 == %i**5" % eulers_sum_of_powers())
Output:
133**5 + 110**5 + 84**5 + 27**5 == 144**5

The above can be written as:

Works with: Python version 2.6+
from itertools import combinations

def eulers_sum_of_powers():
    max_n = 250
    pow_5 = [n**5 for n in range(max_n)]
    pow5_to_n = {n**5: n for n in range(max_n)}
    for x0, x1, x2, x3 in combinations(range(1, max_n), 4):
        pow_5_sum = sum(pow_5[i] for i in (x0, x1, x2, x3))
        if pow_5_sum in pow5_to_n:
            y = pow5_to_n[pow_5_sum]
            return (x0, x1, x2, x3, y)

print("%i**5 + %i**5 + %i**5 + %i**5 == %i**5" % eulers_sum_of_powers())
Output:
27**5 + 84**5 + 110**5 + 133**5 == 144**5

It's much faster to cache and look up sums of two fifth powers, due to the small allowed range:

MAX = 250
p5, sum2 = {}, {}

for i in range(1, MAX):
	p5[i**5] = i
	for j in range(i, MAX):
		sum2[i**5 + j**5] = (i, j)

sk = sorted(sum2.keys())
for p in sorted(p5.keys()):
	for s in sk:
		if p <= s: break
		if p - s in sum2:
			print(p5[p], sum2[s] + sum2[p-s])
			exit()
Output:
144 (27, 84, 110, 133)

Composition of pure functions

Works with: Python version 3.7
'''Euler's sum of powers conjecture'''

from itertools import (chain, takewhile)


# main :: IO ()
def main():
    '''Search for counter-example'''

    xs = enumFromTo(1)(249)

    powerMap = {x**5: x for x in xs}
    sumMap = {
        x**5 + y**5: (x, y)
        for x in xs[1:]
        for y in xs if x > y
    }

    # isExample :: (Int, Int) -> Bool
    def isExample(ps):
        p, s = ps
        return p - s in sumMap

    # display :: (Int, Int) -> String
    def display(ps):
        p, s = ps
        a, b = sumMap[p - s]
        c, d = sumMap[s]
        return '^5 + '.join([str(n) for n in [a, b, c, d]]) + (
            '^5 = ' + str(powerMap[p]) + '^5'
        )

    print(__doc__ + ' – counter-example:\n')
    print(
        maybe('No counter-example found.')(display)(
            find(isExample)(
                bind(powerMap.keys())(
                    lambda p: bind(
                        takewhile(
                            lambda x: p > x,
                            sumMap.keys()
                        )
                    )(lambda s: [(p, s)])
                )
            )
        )
    )


# ----------------------- GENERIC ------------------------

# Just :: a -> Maybe a
def Just(x):
    '''Constructor for an inhabited Maybe (option type) value.
       Wrapper containing the result of a computation.
    '''
    return {'type': 'Maybe', 'Nothing': False, 'Just': x}


# Nothing :: () -> Maybe a
def Nothing():
    '''Constructor for an empty Maybe (option type) value.
       Empty wrapper returned where a computation is not possible.
    '''
    return {'type': 'Maybe', 'Nothing': True}


# bind (>>=) :: [a] -> (a -> [b]) -> [b]
def bind(xs):
    '''List monad injection operator.
       Two computations sequentially composed,
       with any value produced by the first
       passed as an argument to the second.
    '''
    def go(f):
        return chain.from_iterable(map(f, xs))
    return go


# enumFromTo :: (Int, Int) -> [Int]
def enumFromTo(m):
    '''Integer enumeration from m to n.'''
    return lambda n: range(m, 1 + n)


# find :: (a -> Bool) -> [a] -> Maybe a
def find(p):
    '''Just the first element in the list that matches p,
       or Nothing if no elements match.
    '''
    def go(xs):
        try:
            return Just(next(x for x in xs if p(x)))
        except StopIteration:
            return Nothing()
    return go


# maybe :: b -> (a -> b) -> Maybe a -> b
def maybe(v):
    '''Either the default value v, if m is Nothing,
       or the application of f to x,
       where m is Just(x).
    '''
    return lambda f: lambda m: v if (
        None is m or m.get('Nothing')
    ) else f(m.get('Just'))


# MAIN ---
if __name__ == '__main__':
    main()
Output:
Euler's sum of powers conjecture – counter-example:

133^5 + 110^5 + 84^5 + 27^5 = 144^5

Racket

Translation of: C++
#lang racket
(define MAX 250)
(define pow5 (make-vector MAX))
(for ([i (in-range 1 MAX)])
  (vector-set! pow5 i (expt i 5)))  
(define pow5s (list->set (vector->list pow5)))
(let/ec break
  (for* ([x0 (in-range 1 MAX)]
         [x1 (in-range 1 x0)]
         [x2 (in-range 1 x1)]
         [x3 (in-range 1 x2)])
    (define sum (+ (vector-ref pow5 x0)
                   (vector-ref pow5 x1)
                   (vector-ref pow5 x2)
                   (vector-ref pow5 x3)))
    (when (set-member? pow5s sum)
      (displayln (list x0 x1 x2 x3 (inexact->exact (round (expt sum 1/5)))))
      (break))))
Output:
(133 110 84 27 144)

Raku

(formerly Perl 6)

Translation of: Python
constant MAX = 250;

my  %po5{Int};
my %sum2{Int};

for 1..MAX -> \i {
    %po5{i⁵} = i;
    for 1..MAX -> \j {
        %sum2{i⁵ + j⁵} = i, j;
    }
}

%po5.keys.sort.race.map: -> \p {
    for %sum2.keys.sort -> \s {
        if p > s and %sum2{p - s} {
            say ((sort |%sum2{s},|%sum2{p - s}) X~ '⁵').join(' + '), " = %po5{p}", "⁵" and exit
        }
    }
}
Output:
27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵

REXX

fast computation

Programming note:   the 3rd argument can be specified which causes an attempt to find   N   solutions.  
The starting and ending (low and high) values can also be specified   (to limit or expand the search range).
If any of the arguments are omitted, they default to the Rosetta Code task's specifications.

The method used is:

  •   precompute all powers of five   (within the confines of allowed integers)
  •   precompute all (positive) differences between two applicable 5th powers
  •   see if any of the sums of any three   5th   powers are equal to any of those (above) differences
  •           {thanks to the real nifty idea   (↑↑↑)   from user ID   G. Brougnard}
  •   see if the sum of any four   5th   powers is equal to   any   5th power
  •           (this is needed as the fourth number   d   isn't known yet).
  •   {all of the above utilizes REXX's   sparse stemmed array hashing   which eliminates the need for sorting.}

By implementing (user ID)   G. Brougnard's   idea of   differences of two 5th powers,  
the time used for computation was reduced by over a factor of   seventy.


In essence, the new formula being solved is:       a5   +   b5   +   c5     ==     x5   ─   d5

which lends itself to algorithm optimization by (only) having to:

  •   [the right side of the above equation]   pre-compute all possible differences between any two applicable
            integer powers of five   (there are   30,135   unique differences)
  •   [the   left side of the above equation]   sum any applicable three integer powers of five  
  •   [the   ==   part of the above equation]   see if any of the above left─side sums match any of the   ≈30k   right─side differences
/*REXX program finds unique positive integers for ────────── aⁿ+bⁿ+cⁿ+dⁿ==xⁿ  where n=5 */
parse arg L H N .                                /*get optional  LOW, HIGH,  #solutions.*/
if L=='' | L==","  then L=   0  + 1              /*Not specified?  Then use the default.*/
if H=='' | H==","  then H= 250  - 1              /* "      "         "   "   "     "    */
if N=='' | N==","  then N=   1                   /* "      "         "   "   "     "    */
w= length(H)                                     /*W:  used for display aligned numbers.*/
say center(' 'subword(sourceLine(1), 9, 3)" ", 70 +5*w, '─')  /*show title from 1st line*/
numeric digits 1000                              /*be able to handle the next expression*/
numeric digits max(9, length(3*H**5) )           /* "   "   "    "   3* [H to 5th power]*/
bH= H - 2;                 cH= H - 1             /*calculate the upper  DO  loop limits.*/
!.= 0                                            /* [↓]  define values of  5th  powers. */
       do pow=1  for H;    @.pow= pow**5;     _= @.pow;        !._= 1;          $._= pow
       end   /*pow*/
?.= !.
       do    j=4   for H-3                       /*use the range of:   four  to   cH.   */
          do k=j+1  to H;  _= @.k - @.j;  ?._= 1 /*compute the   xⁿ - dⁿ    differences.*/
          end   /*k*/                            /* [↑]  diff. is always positive as k>j*/
       end      /*j*/                            /*define [↑]    5th  power differences.*/
#= 0                                             /*#:  is the number of solutions found.*/   /* [↓]  for N=∞ solutions.*/
    do       a=L    to H-3                       /*traipse through possible  A  values. */   /*◄──done       246 times.*/
      do     b=a+1  to bH;      s1= @.a + @.b    /*   "       "        "     B    "     */   /*◄──done    30,381 times.*/
        do   c=b+1  to cH;      s2= s1  + @.c    /*   "       "        "     C    "     */   /*◄──done 2,511,496 times.*/
        if ?.s2  then do d=c+1  to H;  s= s2+@.d /*find the appropriate solution.       */
                      if !.s  then call show     /*Is it a solution?   Then display it. */
                      end   /*d*/                /* [↑]    !.S  is a boolean.           */
        end                 /*c*/
      end                   /*b*/
    end                     /*a*/

if #==0  then say "Didn't find a solution.";           exit 0
/*──────────────────────────────────────────────────────────────────────────────────────*/
show: _= left('', 5);     #= # + 1               /*_:  used as a spacer; bump # counter.*/
      say _  'solution'   right(#, length(N))":"  _  'a='right(a, w)   _  "b="right(b, w),
          _  'c='right(c, w)     _    "d="right(d, w)     _    'x='right($.s, w+1)
      if #<N  then return                        /*return, keep searching for more sols.*/
      exit #                                     /*stick a fork in it,  we're all done. */
output   when using the default input:
──────────────────────────── aⁿ+bⁿ+cⁿ+dⁿ==xⁿ  where ⁿ=5 ─────────────────────────────
      solution 1:       a= 27       b= 84       c=110       d=133       x= 144
output   when using the input of:     1   4000   999
─────────────────────────────── aⁿ+bⁿ+cⁿ+dⁿ==xⁿ  where ⁿ=5 ───────────────────────────────
      solution   1:       a=  27       b=  84       c= 110       d= 133       x=  144
      solution   2:       a=  54       b= 168       c= 220       d= 266       x=  288
      solution   3:       a=  81       b= 252       c= 330       d= 399       x=  432
      solution   4:       a= 108       b= 336       c= 440       d= 532       x=  576
      solution   5:       a= 135       b= 420       c= 550       d= 665       x=  720
      solution   6:       a= 162       b= 504       c= 660       d= 798       x=  864
      solution   7:       a= 189       b= 588       c= 770       d= 931       x= 1008
      solution   8:       a= 216       b= 672       c= 880       d=1064       x= 1152
      solution   9:       a= 243       b= 756       c= 990       d=1197       x= 1296
      solution  10:       a= 270       b= 840       c=1100       d=1330       x= 1440
      solution  11:       a= 297       b= 924       c=1210       d=1463       x= 1584
      solution  12:       a= 324       b=1008       c=1320       d=1596       x= 1728
      solution  13:       a= 351       b=1092       c=1430       d=1729       x= 1872
      solution  14:       a= 378       b=1176       c=1540       d=1862       x= 2016
      solution  15:       a= 405       b=1260       c=1650       d=1995       x= 2160
      solution  16:       a= 432       b=1344       c=1760       d=2128       x= 2304
      solution  17:       a= 459       b=1428       c=1870       d=2261       x= 2448
      solution  18:       a= 486       b=1512       c=1980       d=2394       x= 2592
      solution  19:       a= 513       b=1596       c=2090       d=2527       x= 2736
      solution  20:       a= 540       b=1680       c=2200       d=2660       x= 2880
      solution  21:       a= 567       b=1764       c=2310       d=2793       x= 3024
      solution  22:       a= 594       b=1848       c=2420       d=2926       x= 3168
      solution  23:       a= 621       b=1932       c=2530       d=3059       x= 3312
      solution  24:       a= 648       b=2016       c=2640       d=3192       x= 3456
      solution  25:       a= 675       b=2100       c=2750       d=3325       x= 3600
      solution  26:       a= 702       b=2184       c=2860       d=3458       x= 3744
      solution  27:       a= 729       b=2268       c=2970       d=3591       x= 3888

lightning-fast computation

Programming note:   it can be observed from the 1st REXX program's output   (2nd example)   that all of the index solutions are just multiples of the 1st known set:

     for A,   a multiple of   27
     for B,   a multiple of   84      
     for C,   a multiple of  110
     for D,   a multiple of  133
     for X,   a multiple of  144

where "a multiple" is some positive integer.


Intrepid and resourceful Rosetta Code userid   Pat Garrett   has found that in a research paper that   Jim Frye   found another solution:

555   +   31835   +   289695   +   852825     =     853595

The paper can be seen at:   A variety of Euler's conjecture.


So this 2nd known set was added to the program below.

So, index solutions are also multiples of the 2nd known set:

     for A,   a multiple of      55
     for B,   a multiple of   3,183      
     for C,   a multiple of  28,969
     for D,   a multiple of  85,282
     for X,   a multiple of  85,359


Execution time for computing/displaying/writing   200   numbers on a slow PC is about   1/16   second.

Execution time on a slow PC for computing (alone)   for   6,000   numbers is less than   one   second.

Execution time on a fast PC for computing (alone)   for   23,686   numbers is exactly   1.00   seconds.

/*REXX program shows unique positive integers for ────────── aⁿ+bⁿ+cⁿ+dⁿ==xⁿ  where n=5 */
numeric digits 1000                              /*ensure enough decimal digs for powers*/
parse arg N oFID .                               /*obtain optional arguments from the CL*/
if N=='' | N==","  then N= 200                   /*Not specified?  Then use the default.*/
if oFID==''|oFID==","  then oFID= 'EULERSUM.OUT' /* "      "         "   "   "     "    */
tell= N>=0                                       /*if N is ≥ 0, show output to terminal.*/
N= abs(N)                                        /*use the absolute value of  N.        */
                      a.1=  27  ;   a.2=    55   /*the   A   values for the two sets.   */
                      b.1=  84  ;   b.2=  3183   /* "    B      "    "   "   "    "     */
                      c.1= 110  ;   c.2= 28969   /* "    C      "    "   "   "    "     */
                      d.1= 133  ;   d.2= 85282   /* "    D      "    "   "   "    "     */
                      x.1= 144  ;   x.2= 85359   /* "    X      "    "   "   "    "     */
w= length( commas(N * x.2) )                     /*W:  used to align displayed numbers. */
$= center(' 'subword( sourceLine(1), 9, 3)" ", 70 +5*w, '─')           /*create a title.*/
call show                                        /*show a title  (from 1st line of pgm).*/
pad= left('',5)                                  /*used for padding (spacing) the output*/
oo= 1;   tt= 1                                   /*a counter for the  A.1  &  A.2  sets.*/
#= 0                                             /*count of number of solutions so far. */
       do j=1  until #>N                         /*step through the possible solutions. */
       one= a.1 * oo                             /*calculate the 1st set's  A.1  value. */
       two= a.2 * tt                             /*    "      "  2nd   "    A.2    "    */
       use= min(one, two)                        /*pick which "set" that is to be used. */
       #= # + 1                                  /*bump counter for number of solutions.*/
       if one==use  then do;      mult=oo;      oo= oo + 1;      which= 1;      end
       if two==use  then do;      mult=tt;      tt= tt + 1;      which= 2;      end
       $= pad  'solution'  right(#,length(N))":  "  'a='right( commas(a.which * mult), w),
                                            pad     'b='right( commas(b.which * mult), w),
                                            pad     'c='right( commas(c.which * mult), w),
                                            pad     'd='right( commas(d.which * mult), w),
                                            pad     'x='right( commas(x.which * mult), w)
       call show                                 /*write; maybe show output to terminal.*/
       res= (x.which * mult) **5                 /*compute the sum of the  right  side. */
       sum= (a.which * mult) **5   +   ,         /*   "     "   "   "  "    left    "   */
            (b.which * mult) **5   +   ,
            (c.which * mult) **5   +   ,
            (d.which * mult) **5
       if sum==res  then iterate                 /*All is kosher?   Then keep truckin'. */
       $= "***error*** the left side sum   doesn't   equal the right side result (X**5)."
       tell=1;  call show;  exit 13              /*force telling of error to terminal.  */
       end   /*j*/
tell=1;                                                                          call show
$= pad ' Showed '   commas(N)   " solutions,  output written to file: " oFID;    call show
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
commas: parse arg _;  do jc=length(_)-3  to 1  by -3; _=insert(',', _, jc); end;  return _
show:   if tell  then say $;  call lineout oFID, $;  $=;  return  /*show and/or write it*/
output   when using the default input of:     200

(Shown at three-quarter size.)

────────────────────────────────────────────── aⁿ+bⁿ+cⁿ+dⁿ==xⁿ  where n=5 ──────────────────────────────────────────────
      solution   1:   a=        27       b=        84       c=       110       d=       133       x=       144
      solution   2:   a=        54       b=       168       c=       220       d=       266       x=       288
      solution   3:   a=        55       b=     3,183       c=    28,969       d=    85,282       x=    85,359
      solution   4:   a=        81       b=       252       c=       330       d=       399       x=       432
      solution   5:   a=       108       b=       336       c=       440       d=       532       x=       576
      solution   6:   a=       110       b=     6,366       c=    57,938       d=   170,564       x=   170,718
      solution   7:   a=       135       b=       420       c=       550       d=       665       x=       720
      solution   8:   a=       162       b=       504       c=       660       d=       798       x=       864
      solution   9:   a=       165       b=     9,549       c=    86,907       d=   255,846       x=   256,077
      solution  10:   a=       189       b=       588       c=       770       d=       931       x=     1,008
      solution  11:   a=       216       b=       672       c=       880       d=     1,064       x=     1,152
      solution  12:   a=       220       b=    12,732       c=   115,876       d=   341,128       x=   341,436
      solution  13:   a=       243       b=       756       c=       990       d=     1,197       x=     1,296
      solution  14:   a=       270       b=       840       c=     1,100       d=     1,330       x=     1,440
      solution  15:   a=       275       b=    15,915       c=   144,845       d=   426,410       x=   426,795
      solution  16:   a=       297       b=       924       c=     1,210       d=     1,463       x=     1,584
      solution  17:   a=       324       b=     1,008       c=     1,320       d=     1,596       x=     1,728
      solution  18:   a=       330       b=    19,098       c=   173,814       d=   511,692       x=   512,154
      solution  19:   a=       351       b=     1,092       c=     1,430       d=     1,729       x=     1,872
      solution  20:   a=       378       b=     1,176       c=     1,540       d=     1,862       x=     2,016
      solution  21:   a=       385       b=    22,281       c=   202,783       d=   596,974       x=   597,513
      solution  22:   a=       405       b=     1,260       c=     1,650       d=     1,995       x=     2,160
      solution  23:   a=       432       b=     1,344       c=     1,760       d=     2,128       x=     2,304
      solution  24:   a=       440       b=    25,464       c=   231,752       d=   682,256       x=   682,872
      solution  25:   a=       459       b=     1,428       c=     1,870       d=     2,261       x=     2,448
      solution  26:   a=       486       b=     1,512       c=     1,980       d=     2,394       x=     2,592
      solution  27:   a=       495       b=    28,647       c=   260,721       d=   767,538       x=   768,231
      solution  28:   a=       513       b=     1,596       c=     2,090       d=     2,527       x=     2,736
      solution  29:   a=       540       b=     1,680       c=     2,200       d=     2,660       x=     2,880
      solution  30:   a=       550       b=    31,830       c=   289,690       d=   852,820       x=   853,590
      solution  31:   a=       567       b=     1,764       c=     2,310       d=     2,793       x=     3,024
      solution  32:   a=       594       b=     1,848       c=     2,420       d=     2,926       x=     3,168
      solution  33:   a=       605       b=    35,013       c=   318,659       d=   938,102       x=   938,949
      solution  34:   a=       621       b=     1,932       c=     2,530       d=     3,059       x=     3,312
      solution  35:   a=       648       b=     2,016       c=     2,640       d=     3,192       x=     3,456
      solution  36:   a=       660       b=    38,196       c=   347,628       d= 1,023,384       x= 1,024,308
      solution  37:   a=       675       b=     2,100       c=     2,750       d=     3,325       x=     3,600
      solution  38:   a=       702       b=     2,184       c=     2,860       d=     3,458       x=     3,744
      solution  39:   a=       715       b=    41,379       c=   376,597       d= 1,108,666       x= 1,109,667
      solution  40:   a=       729       b=     2,268       c=     2,970       d=     3,591       x=     3,888
      solution  41:   a=       756       b=     2,352       c=     3,080       d=     3,724       x=     4,032
      solution  42:   a=       770       b=    44,562       c=   405,566       d= 1,193,948       x= 1,195,026
      solution  43:   a=       783       b=     2,436       c=     3,190       d=     3,857       x=     4,176
      solution  44:   a=       810       b=     2,520       c=     3,300       d=     3,990       x=     4,320
      solution  45:   a=       825       b=    47,745       c=   434,535       d= 1,279,230       x= 1,280,385
      solution  46:   a=       837       b=     2,604       c=     3,410       d=     4,123       x=     4,464
      solution  47:   a=       864       b=     2,688       c=     3,520       d=     4,256       x=     4,608
      solution  48:   a=       880       b=    50,928       c=   463,504       d= 1,364,512       x= 1,365,744
      solution  49:   a=       891       b=     2,772       c=     3,630       d=     4,389       x=     4,752
      solution  50:   a=       918       b=     2,856       c=     3,740       d=     4,522       x=     4,896
      solution  51:   a=       935       b=    54,111       c=   492,473       d= 1,449,794       x= 1,451,103
      solution  52:   a=       945       b=     2,940       c=     3,850       d=     4,655       x=     5,040
      solution  53:   a=       972       b=     3,024       c=     3,960       d=     4,788       x=     5,184
      solution  54:   a=       990       b=    57,294       c=   521,442       d= 1,535,076       x= 1,536,462
      solution  55:   a=       999       b=     3,108       c=     4,070       d=     4,921       x=     5,328
      solution  56:   a=     1,026       b=     3,192       c=     4,180       d=     5,054       x=     5,472
      solution  57:   a=     1,045       b=    60,477       c=   550,411       d= 1,620,358       x= 1,621,821
      solution  58:   a=     1,053       b=     3,276       c=     4,290       d=     5,187       x=     5,616
      solution  59:   a=     1,080       b=     3,360       c=     4,400       d=     5,320       x=     5,760
      solution  60:   a=     1,100       b=    63,660       c=   579,380       d= 1,705,640       x= 1,707,180
      solution  61:   a=     1,107       b=     3,444       c=     4,510       d=     5,453       x=     5,904
      solution  62:   a=     1,134       b=     3,528       c=     4,620       d=     5,586       x=     6,048
      solution  63:   a=     1,155       b=    66,843       c=   608,349       d= 1,790,922       x= 1,792,539
      solution  64:   a=     1,161       b=     3,612       c=     4,730       d=     5,719       x=     6,192
      solution  65:   a=     1,188       b=     3,696       c=     4,840       d=     5,852       x=     6,336
      solution  66:   a=     1,210       b=    70,026       c=   637,318       d= 1,876,204       x= 1,877,898
      solution  67:   a=     1,215       b=     3,780       c=     4,950       d=     5,985       x=     6,480
      solution  68:   a=     1,242       b=     3,864       c=     5,060       d=     6,118       x=     6,624
      solution  69:   a=     1,265       b=    73,209       c=   666,287       d= 1,961,486       x= 1,963,257
      solution  70:   a=     1,269       b=     3,948       c=     5,170       d=     6,251       x=     6,768
      solution  71:   a=     1,296       b=     4,032       c=     5,280       d=     6,384       x=     6,912
      solution  72:   a=     1,320       b=    76,392       c=   695,256       d= 2,046,768       x= 2,048,616
      solution  73:   a=     1,323       b=     4,116       c=     5,390       d=     6,517       x=     7,056
      solution  74:   a=     1,350       b=     4,200       c=     5,500       d=     6,650       x=     7,200
      solution  75:   a=     1,375       b=    79,575       c=   724,225       d= 2,132,050       x= 2,133,975
      solution  76:   a=     1,377       b=     4,284       c=     5,610       d=     6,783       x=     7,344
      solution  77:   a=     1,404       b=     4,368       c=     5,720       d=     6,916       x=     7,488
      solution  78:   a=     1,430       b=    82,758       c=   753,194       d= 2,217,332       x= 2,219,334
      solution  79:   a=     1,431       b=     4,452       c=     5,830       d=     7,049       x=     7,632
      solution  80:   a=     1,458       b=     4,536       c=     5,940       d=     7,182       x=     7,776
      solution  81:   a=     1,485       b=    85,941       c=   782,163       d= 2,302,614       x= 2,304,693
      solution  82:   a=     1,512       b=     4,704       c=     6,160       d=     7,448       x=     8,064
      solution  83:   a=     1,539       b=     4,788       c=     6,270       d=     7,581       x=     8,208
      solution  84:   a=     1,540       b=    89,124       c=   811,132       d= 2,387,896       x= 2,390,052
      solution  85:   a=     1,566       b=     4,872       c=     6,380       d=     7,714       x=     8,352
      solution  86:   a=     1,593       b=     4,956       c=     6,490       d=     7,847       x=     8,496
      solution  87:   a=     1,595       b=    92,307       c=   840,101       d= 2,473,178       x= 2,475,411
      solution  88:   a=     1,620       b=     5,040       c=     6,600       d=     7,980       x=     8,640
      solution  89:   a=     1,647       b=     5,124       c=     6,710       d=     8,113       x=     8,784
      solution  90:   a=     1,650       b=    95,490       c=   869,070       d= 2,558,460       x= 2,560,770
      solution  91:   a=     1,674       b=     5,208       c=     6,820       d=     8,246       x=     8,928
      solution  92:   a=     1,701       b=     5,292       c=     6,930       d=     8,379       x=     9,072
      solution  93:   a=     1,705       b=    98,673       c=   898,039       d= 2,643,742       x= 2,646,129
      solution  94:   a=     1,728       b=     5,376       c=     7,040       d=     8,512       x=     9,216
      solution  95:   a=     1,755       b=     5,460       c=     7,150       d=     8,645       x=     9,360
      solution  96:   a=     1,760       b=   101,856       c=   927,008       d= 2,729,024       x= 2,731,488
      solution  97:   a=     1,782       b=     5,544       c=     7,260       d=     8,778       x=     9,504
      solution  98:   a=     1,809       b=     5,628       c=     7,370       d=     8,911       x=     9,648
      solution  99:   a=     1,815       b=   105,039       c=   955,977       d= 2,814,306       x= 2,816,847
      solution 100:   a=     1,836       b=     5,712       c=     7,480       d=     9,044       x=     9,792
      solution 101:   a=     1,863       b=     5,796       c=     7,590       d=     9,177       x=     9,936
      solution 102:   a=     1,870       b=   108,222       c=   984,946       d= 2,899,588       x= 2,902,206
      solution 103:   a=     1,890       b=     5,880       c=     7,700       d=     9,310       x=    10,080
      solution 104:   a=     1,917       b=     5,964       c=     7,810       d=     9,443       x=    10,224
      solution 105:   a=     1,925       b=   111,405       c= 1,013,915       d= 2,984,870       x= 2,987,565
      solution 106:   a=     1,944       b=     6,048       c=     7,920       d=     9,576       x=    10,368
      solution 107:   a=     1,971       b=     6,132       c=     8,030       d=     9,709       x=    10,512
      solution 108:   a=     1,980       b=   114,588       c= 1,042,884       d= 3,070,152       x= 3,072,924
      solution 109:   a=     1,998       b=     6,216       c=     8,140       d=     9,842       x=    10,656
      solution 110:   a=     2,025       b=     6,300       c=     8,250       d=     9,975       x=    10,800
      solution 111:   a=     2,035       b=   117,771       c= 1,071,853       d= 3,155,434       x= 3,158,283
      solution 112:   a=     2,052       b=     6,384       c=     8,360       d=    10,108       x=    10,944
      solution 113:   a=     2,079       b=     6,468       c=     8,470       d=    10,241       x=    11,088
      solution 114:   a=     2,090       b=   120,954       c= 1,100,822       d= 3,240,716       x= 3,243,642
      solution 115:   a=     2,106       b=     6,552       c=     8,580       d=    10,374       x=    11,232
      solution 116:   a=     2,133       b=     6,636       c=     8,690       d=    10,507       x=    11,376
      solution 117:   a=     2,145       b=   124,137       c= 1,129,791       d= 3,325,998       x= 3,329,001
      solution 118:   a=     2,160       b=     6,720       c=     8,800       d=    10,640       x=    11,520
      solution 119:   a=     2,187       b=     6,804       c=     8,910       d=    10,773       x=    11,664
      solution 120:   a=     2,200       b=   127,320       c= 1,158,760       d= 3,411,280       x= 3,414,360
      solution 121:   a=     2,214       b=     6,888       c=     9,020       d=    10,906       x=    11,808
      solution 122:   a=     2,241       b=     6,972       c=     9,130       d=    11,039       x=    11,952
      solution 123:   a=     2,255       b=   130,503       c= 1,187,729       d= 3,496,562       x= 3,499,719
      solution 124:   a=     2,268       b=     7,056       c=     9,240       d=    11,172       x=    12,096
      solution 125:   a=     2,295       b=     7,140       c=     9,350       d=    11,305       x=    12,240
      solution 126:   a=     2,310       b=   133,686       c= 1,216,698       d= 3,581,844       x= 3,585,078
      solution 127:   a=     2,322       b=     7,224       c=     9,460       d=    11,438       x=    12,384
      solution 128:   a=     2,349       b=     7,308       c=     9,570       d=    11,571       x=    12,528
      solution 129:   a=     2,365       b=   136,869       c= 1,245,667       d= 3,667,126       x= 3,670,437
      solution 130:   a=     2,376       b=     7,392       c=     9,680       d=    11,704       x=    12,672
      solution 131:   a=     2,403       b=     7,476       c=     9,790       d=    11,837       x=    12,816
      solution 132:   a=     2,420       b=   140,052       c= 1,274,636       d= 3,752,408       x= 3,755,796
      solution 133:   a=     2,430       b=     7,560       c=     9,900       d=    11,970       x=    12,960
      solution 134:   a=     2,457       b=     7,644       c=    10,010       d=    12,103       x=    13,104
      solution 135:   a=     2,475       b=   143,235       c= 1,303,605       d= 3,837,690       x= 3,841,155
      solution 136:   a=     2,484       b=     7,728       c=    10,120       d=    12,236       x=    13,248
      solution 137:   a=     2,511       b=     7,812       c=    10,230       d=    12,369       x=    13,392
      solution 138:   a=     2,530       b=   146,418       c= 1,332,574       d= 3,922,972       x= 3,926,514
      solution 139:   a=     2,538       b=     7,896       c=    10,340       d=    12,502       x=    13,536
      solution 140:   a=     2,565       b=     7,980       c=    10,450       d=    12,635       x=    13,680
      solution 141:   a=     2,585       b=   149,601       c= 1,361,543       d= 4,008,254       x= 4,011,873
      solution 142:   a=     2,592       b=     8,064       c=    10,560       d=    12,768       x=    13,824
      solution 143:   a=     2,619       b=     8,148       c=    10,670       d=    12,901       x=    13,968
      solution 144:   a=     2,640       b=   152,784       c= 1,390,512       d= 4,093,536       x= 4,097,232
      solution 145:   a=     2,646       b=     8,232       c=    10,780       d=    13,034       x=    14,112
      solution 146:   a=     2,673       b=     8,316       c=    10,890       d=    13,167       x=    14,256
      solution 147:   a=     2,695       b=   155,967       c= 1,419,481       d= 4,178,818       x= 4,182,591
      solution 148:   a=     2,700       b=     8,400       c=    11,000       d=    13,300       x=    14,400
      solution 149:   a=     2,727       b=     8,484       c=    11,110       d=    13,433       x=    14,544
      solution 150:   a=     2,750       b=   159,150       c= 1,448,450       d= 4,264,100       x= 4,267,950
      solution 151:   a=     2,754       b=     8,568       c=    11,220       d=    13,566       x=    14,688
      solution 152:   a=     2,781       b=     8,652       c=    11,330       d=    13,699       x=    14,832
      solution 153:   a=     2,805       b=   162,333       c= 1,477,419       d= 4,349,382       x= 4,353,309
      solution 154:   a=     2,808       b=     8,736       c=    11,440       d=    13,832       x=    14,976
      solution 155:   a=     2,835       b=     8,820       c=    11,550       d=    13,965       x=    15,120
      solution 156:   a=     2,860       b=   165,516       c= 1,506,388       d= 4,434,664       x= 4,438,668
      solution 157:   a=     2,862       b=     8,904       c=    11,660       d=    14,098       x=    15,264
      solution 158:   a=     2,889       b=     8,988       c=    11,770       d=    14,231       x=    15,408
      solution 159:   a=     2,915       b=   168,699       c= 1,535,357       d= 4,519,946       x= 4,524,027
      solution 160:   a=     2,916       b=     9,072       c=    11,880       d=    14,364       x=    15,552
      solution 161:   a=     2,943       b=     9,156       c=    11,990       d=    14,497       x=    15,696
      solution 162:   a=     2,970       b=   171,882       c= 1,564,326       d= 4,605,228       x= 4,609,386
      solution 163:   a=     2,997       b=     9,324       c=    12,210       d=    14,763       x=    15,984
      solution 164:   a=     3,024       b=     9,408       c=    12,320       d=    14,896       x=    16,128
      solution 165:   a=     3,025       b=   175,065       c= 1,593,295       d= 4,690,510       x= 4,694,745
      solution 166:   a=     3,051       b=     9,492       c=    12,430       d=    15,029       x=    16,272
      solution 167:   a=     3,078       b=     9,576       c=    12,540       d=    15,162       x=    16,416
      solution 168:   a=     3,080       b=   178,248       c= 1,622,264       d= 4,775,792       x= 4,780,104
      solution 169:   a=     3,105       b=     9,660       c=    12,650       d=    15,295       x=    16,560
      solution 170:   a=     3,132       b=     9,744       c=    12,760       d=    15,428       x=    16,704
      solution 171:   a=     3,135       b=   181,431       c= 1,651,233       d= 4,861,074       x= 4,865,463
      solution 172:   a=     3,159       b=     9,828       c=    12,870       d=    15,561       x=    16,848
      solution 173:   a=     3,186       b=     9,912       c=    12,980       d=    15,694       x=    16,992
      solution 174:   a=     3,190       b=   184,614       c= 1,680,202       d= 4,946,356       x= 4,950,822
      solution 175:   a=     3,213       b=     9,996       c=    13,090       d=    15,827       x=    17,136
      solution 176:   a=     3,240       b=    10,080       c=    13,200       d=    15,960       x=    17,280
      solution 177:   a=     3,245       b=   187,797       c= 1,709,171       d= 5,031,638       x= 5,036,181
      solution 178:   a=     3,267       b=    10,164       c=    13,310       d=    16,093       x=    17,424
      solution 179:   a=     3,294       b=    10,248       c=    13,420       d=    16,226       x=    17,568
      solution 180:   a=     3,300       b=   190,980       c= 1,738,140       d= 5,116,920       x= 5,121,540
      solution 181:   a=     3,321       b=    10,332       c=    13,530       d=    16,359       x=    17,712
      solution 182:   a=     3,348       b=    10,416       c=    13,640       d=    16,492       x=    17,856
      solution 183:   a=     3,355       b=   194,163       c= 1,767,109       d= 5,202,202       x= 5,206,899
      solution 184:   a=     3,375       b=    10,500       c=    13,750       d=    16,625       x=    18,000
      solution 185:   a=     3,402       b=    10,584       c=    13,860       d=    16,758       x=    18,144
      solution 186:   a=     3,410       b=   197,346       c= 1,796,078       d= 5,287,484       x= 5,292,258
      solution 187:   a=     3,429       b=    10,668       c=    13,970       d=    16,891       x=    18,288
      solution 188:   a=     3,456       b=    10,752       c=    14,080       d=    17,024       x=    18,432
      solution 189:   a=     3,465       b=   200,529       c= 1,825,047       d= 5,372,766       x= 5,377,617
      solution 190:   a=     3,483       b=    10,836       c=    14,190       d=    17,157       x=    18,576
      solution 191:   a=     3,510       b=    10,920       c=    14,300       d=    17,290       x=    18,720
      solution 192:   a=     3,520       b=   203,712       c= 1,854,016       d= 5,458,048       x= 5,462,976
      solution 193:   a=     3,537       b=    11,004       c=    14,410       d=    17,423       x=    18,864
      solution 194:   a=     3,564       b=    11,088       c=    14,520       d=    17,556       x=    19,008
      solution 195:   a=     3,575       b=   206,895       c= 1,882,985       d= 5,543,330       x= 5,548,335
      solution 196:   a=     3,591       b=    11,172       c=    14,630       d=    17,689       x=    19,152
      solution 197:   a=     3,618       b=    11,256       c=    14,740       d=    17,822       x=    19,296
      solution 198:   a=     3,630       b=   210,078       c= 1,911,954       d= 5,628,612       x= 5,633,694
      solution 199:   a=     3,645       b=    11,340       c=    14,850       d=    17,955       x=    19,440
      solution 200:   a=     3,672       b=    11,424       c=    14,960       d=    18,088       x=    19,584

       Showed  200  solutions,  output written to file:  EULERSUM.OUT

Ring

# Project : Euler's sum of powers conjecture

max=250
for w = 1 to max
     for x = 1 to w
          for y = 1 to x
               for z = 1 to y
                    sum = pow(w,5) + pow(x,5) + pow(y,5) + pow(z,5)
                    s1  = floor(pow(sum,0.2))
                    if sum = pow(s1,5) 
                       see "" + w + "^5 + " + x + "^5 + " + y + "^5 + " + z + "^5 = " + s1 + "^5" 
                    ok
               next 
          next
     next 
next

Output:

133^5 + 110^5 + 84^5 + 27^5 = 144^5

Ruby

Brute force:

power5 = (1..250).each_with_object({}){|i,h| h[i**5]=i}
result = power5.keys.repeated_combination(4).select{|a| power5[a.inject(:+)]}
puts result.map{|a| a.map{|i| "#{power5[i]}**5"}.join(' + ') + " = #{power5[a.inject(:+)]}**5"}
Output:
27**5 + 84**5 + 110**5 + 133**5 = 144**5

Faster version:

Translation of: Python
p5, sum2, max = {}, {}, 250
(1..max).each do |i|
  p5[i**5] = i
  (i..max).each{|j| sum2[i**5 + j**5] = [i,j]}
end

result = {}
sk = sum2.keys.sort
p5.keys.sort.each do |p|
  sk.each do |s|
    break if p <= s
    result[(sum2[s] + sum2[p-s]).sort] = p5[p]  if sum2[p - s]
  end
end
result.each{|k,v| puts k.map{|i| "#{i}**5"}.join(' + ') + " = #{v}**5"}

The output is the same above.

Rust

const MAX_N : u64 = 250;

fn eulers_sum_of_powers() -> (usize, usize, usize, usize, usize) {
    let pow5: Vec<u64> = (0..MAX_N).map(|i| i.pow(5)).collect();
    let pow5_to_n = |pow| pow5.binary_search(&pow);

    for x0 in 1..MAX_N as usize {
        for x1 in 1..x0 {
            for x2 in 1..x1 {
                for x3 in 1..x2 {
                    let pow_sum = pow5[x0] + pow5[x1] + pow5[x2] + pow5[x3];
                    if let Ok(n) = pow5_to_n(pow_sum) {
                        return (x0, x1, x2, x3, n)
                    }
                }
            }
        }
    }

    panic!();
}

fn main() {
	let (x0, x1, x2, x3, y) = eulers_sum_of_powers();
	println!("{}^5 + {}^5 + {}^5 + {}^5 == {}^5", x0, x1, x2, x3, y)
}
Output:
133^5 + 110^5 + 84^5 + 27^5 == 144^5

Scala

Functional programming

import scala.collection.Searching.{Found, search}

object EulerSopConjecture extends App {

  val (maxNumber, fifth) = (250, (1 to 250).map { i => math.pow(i, 5).toLong })

  def binSearch(fact: Int*) = fifth.search(fact.map(f => fifth(f)).sum)

  def sop = (0 until maxNumber)
    .flatMap(a => (a until maxNumber)
      .flatMap(b => (b until maxNumber)
        .flatMap(c => (c until maxNumber)
          .map { case x$1@d => (binSearch(a, b, c, d), x$1) }
          .withFilter { case (f, _) => f.isInstanceOf[Found] }
          .map { case (f, d) => (a + 1, b + 1, c + 1, d + 1, f.insertionPoint + 1) }))).take(1)
    .map { case (a, b, c, d, f) => s"$a⁵ + $b⁵ + $c⁵ + $d⁵ = $f⁵" }

  println(sop)

}
Output:

Vector(27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵)

Seed7

$ include "seed7_05.s7i";

const func integer: binarySearch (in array integer: arr, in integer: aKey) is func
  result
    var integer: index is 0;
  local
    var integer: low is 1;
    var integer: high is 0;
    var integer: middle is 0;
  begin
    high := length(arr);
    while index = 0 and low <= high do
      middle := (low + high) div 2;
      if aKey < arr[middle] then
        high := pred(middle);
      elsif aKey > arr[middle] then
        low := succ(middle);
      else
        index := middle;
      end if;
    end while;
  end func;

const proc: main is func
  local
    var array integer: p5 is 249 times 0;
    var integer: i is 0;
    var integer: x0 is 0;
    var integer: x1 is 0;
    var integer: x2 is 0;
    var integer: x3 is 0;
    var integer: sum is 0;
    var integer: y is 0;
    var boolean: found is FALSE;
  begin
    for i range 1 to 249 do
      p5[i] := i ** 5;
    end for;
    for x0 range 1 to 249 until found do
      for x1 range 1 to pred(x0) until found do
        for x2 range 1 to pred(x1) until found do
          for x3 range 1 to pred(x2) until found do
            sum := p5[x0] + p5[x1] + p5[x2] + p5[x3];
            y := binarySearch(p5, sum);
            if y > 0 then
              writeln(x0 <& "**5 + " <& x1 <& "**5 + " <& x2 <& "**5 + " <& x3 <& "**5 = " <& y <& "**5");
              found := TRUE;
            end if;
          end for;
        end for;
      end for;
    end for;
    if not found then
      writeln("No solution was found");
    end if;
  end func;
Output:
133**5 + 110**5 + 84**5 + 27**5 = 144**5

SenseTalk

findEulerSumOfPowers
to findEulerSumOfPowers
    set MAX_NUMBER to 250
    set possibleValues to 1..MAX_NUMBER
    set possible5thPowers to each item of possibleValues to the power of 5
    repeat for x0 in 1..250
        repeat for x1 in 1..x0
            repeat for x2 in 1..x1
                repeat for x3 in 1..x2
                    set possibleSum to item x0 of possible5thPowers \
                            plus item x1 of possible5thPowers \
                            plus item x2 of possible5thPowers \
                            plus item x3 of possible5thPowers
                    if possibleSum is in possible5thPowers
                        put x0 & "^5 + " & x1 & "^5 + " & x2 & "^5 + " & x3 & "^5 = " & the item number of possibleSum within possible5thPowers & "^5"
                        return
                    end if
                end repeat
            end repeat
        end repeat
    end repeat
end findEulerSumOfPowers

Sidef

Translation of: Raku
define range = (1 ..^ 250)

var p5 = Hash()
var sum2 = Hash()

for i in (range) {
    p5{i**5} = i
    for j in (range) {
        sum2{i**5 + j**5} = [i, j]
    }
}

var sk = sum2.keys.map{ Num(_) }.sort

for p in (p5.keys.map{ Num(_) }.sort) {

    var s = sk.first {|s|
        p > s && sum2.exists(p-s)
    } \\ next

    var t = (sum2{s} + sum2{p-s} -> map{|n| "#{n}⁵" }.join(' + '))
    say "#{t} = #{p5{p}}⁵"
    break
}
Output:
84⁵ + 27⁵ + 133⁵ + 110⁵ =  144⁵

Swift

Translation of: Rust
extension BinaryInteger {
  @inlinable
  public func power(_ n: Self) -> Self {
    return stride(from: 0, to: n, by: 1).lazy.map({_ in self }).reduce(1, *)
  }
}

func sumOfPowers(maxN: Int = 250) -> (Int, Int, Int, Int, Int) {
  let pow5 = (0..<maxN).map({ $0.power(5) })
  let pow5ToN = {n in pow5.firstIndex(of: n)}

  for x0 in 1..<maxN {
    for x1 in 1..<x0 {
      for x2 in 1..<x1 {
        for x3 in 1..<x2 {
          let powSum = pow5[x0] + pow5[x1] + pow5[x2] + pow5[x3]

          if let idx = pow5ToN(powSum) {
            return (x0, x1, x2, x3, idx)
          }
        }
      }
    }
  }

  fatalError("Did not find solution")
}

let (x0, x1, x2, x3, y) = sumOfPowers()

print("\(x0)^5 + \(x1)^5 + \(x2)^5 \(x3)^5 = \(y)^5")
Output:
133^5 + 110^5 + 84^5 + 27^5 = 144^5

Tcl

proc doit {{badx 250} {complete 0}} {
    ## NB: $badx is exclusive upper limit, and also limits y!
    for {set y 1} {$y < $badx} {incr y} {
        set s [expr {$y ** 5}]
        set r5($s) $y           ;# fifth roots of valid sums
    }
    for {set a 1} {$a < $badx} {incr a} {
        set suma [expr {$a ** 5}]
        for {set b 1} {$b <= $a} {incr b} {
            set sumb [expr {$suma + ($b ** 5)}]
            for {set c 1} {$c <= $b} {incr c} {
                set sumc [expr {$sumb + ($c ** 5)}]
                for {set d 1} {$d <= $c} {incr d} {
                    set sumd [expr {$sumc + ($d ** 5)}]
                    if {[info exists r5($sumd)]} {
                        set e $r5($sumd)
                        puts "$e^5 = $a^5 + $b^5 + $c^5 + $d^5"
                        if {!$complete} {
                            return
                        }
                    }
                }
            }
        }
    }
    puts "search complete (x < $badx)"
}
doit
Output:
144^5 = 133^5 + 110^5 + 84^5 + 27^5

real 0m2.387s

UNIX Shell

Works with: Bourne Again SHell
Works with: Korn Shell
Works with: Zsh

Shell is not the go-to language for number-crunching, but if you're going to use a shell, it looks like ksh is the fastest option, at about 8x faster than bash and 2x faster than zsh.

MAX=250
pow5=()
for (( i=1; i<MAX; ++i )); do
  pow5[i]=$(( i*i*i*i*i ))
done
for (( a=1; a<MAX; ++a )); do
  for (( b=a+1; b<MAX; ++b )); do
    for (( c=b+1; c<MAX; ++c )); do
      for (( d=c+1; d<MAX; ++d )); do
        (( sum=pow5[a]+pow5[b]+pow5[c]+pow5[d] ))
        (( low=d+3 ))
        (( high=MAX ))
        while (( low <= high )); do
          (( guess=(low+high)/2 ))
          if (( pow5[guess]  == sum )); then
            printf 'Found example: %d⁵+%d⁵+%d⁵+%d⁵=%d⁵\n' "$a" "$b" "$c" "$d" "$guess"
            exit 0
          elif (( pow5[guess] < sum )); then
            (( low=guess+1 ))
          else
            (( high=guess-1 ))
          fi
        done
      done
    done
  done
done
printf 'No examples found.\n'
exit 1
Output:
$ time bash esop.sh
Found example: 27⁵+84⁵+110⁵+133⁵=144⁵
bash esop.sh  6953.75s user 37.53s system 99% cpu 1:57:02.41 total
$ time ksh esop.sh
Found example: 27⁵+84⁵+110⁵+133⁵=144⁵
ksh esop.sh  855.66s user 5.30s system 99% cpu 14:26.78 total
$ time zsh esop.sh
Found example: 27⁵+84⁵+110⁵+133⁵=144⁵
zsh esop.sh  1969.48s user 250.82s system 99% cpu 37:11.62 total

VBScript

Translation of: ERRE
Max=250

For X0=1 To Max
	For X1=1 To X0
		For X2=1 To X1
			For X3=1 To X2
				Sum=fnP5(X0)+fnP5(X1)+fnP5(X2)+fnP5(X3)
				S1=Int(Sum^0.2)
				If Sum=fnP5(S1) Then
					WScript.StdOut.Write X0 & " " & X1 & " " & X2 & " " & X3 & " " & S1
					WScript.Quit
				End If
			Next
		Next
	Next
Next

Function fnP5(n)
	fnP5 = n ^ 5
End Function
Output:
133 110 84 27 144

Visual Basic .NET

Works with: Visual Basic .NET version 2011
' Euler's sum of powers of 4 conjecture - Patrice Grandin - 17/05/2020

' x1^4  +  x2^4  +  x3^4  +  x4^4  =  x5^4

' Project\Add reference\Assembly\Framework   System.Numerics
Imports System.Numerics 'BigInteger

Public Class EulerPower4Sum

    Private Sub MyForm_Load(sender As Object, e As EventArgs) Handles MyBase.Load
        Dim t1, t2 As DateTime
        t1 = Now
        EulerPower45Sum()   '16.7 sec
        'EulerPower44Sum()   '633 years !!
        t2 = Now
        Console.WriteLine((t2 - t1).TotalSeconds & " sec")
    End Sub 'Load

    Private Sub EulerPower45Sum()
        '30^4 + 120^4 + 272^4 + 315^4 = 353^4
        Const MaxN = 360
        Dim i, j, i1, i2, i3, i4, i5 As Int32
        Dim p4(MaxN), n, sumx As Int64
        Debug.Print(">EulerPower45Sum")
        For i = 1 To MaxN
            n = 1
            For j = 1 To 4
                n *= i
            Next j
            p4(i) = n
        Next i
        For i1 = 1 To MaxN
            If i1 Mod 5 = 0 Then Debug.Print(">i1=" & i1)
            For i2 = i1 To MaxN
                For i3 = i2 To MaxN
                    For i4 = i3 To MaxN
                        sumx = p4(i1) + p4(i2) + p4(i3) + p4(i4)
                        i5 = i4 + 1
                        While i5 <= MaxN AndAlso p4(i5) <= sumx
                            If p4(i5) = sumx Then
                                Debug.Print(i1 & " " & i2 & " " & i3 & " " & i4 & " " & i5)
                                Exit Sub
                            End If
                            i5 += 1
                        End While
                    Next i4
                Next i3
            Next i2
        Next i1
        Debug.Print("Not found!")
    End Sub 'EulerPower45Sum

    Private Sub EulerPower44Sum()
        '95800^4 + 217519^4 + 414560^4 = 422481^4
        Const MaxN = 500000   '500000^4 => decimal(23) => binary(76) !!
        Dim i, j, i1, i2, i3, i4 As Int32
        Dim p4(MaxN), n, sumx As BigInteger
        Dim t0 As DateTime
        Debug.Print(">EulerPower44Sum")
        For i = 1 To MaxN
            n = 1
            For j = 1 To 4
                n *= i
            Next j
            p4(i) = n
        Next i
        t0 = Now
        For i1 = 1 To MaxN
            Debug.Print(">i1=" & i1)
            For i2 = i1 To MaxN
                If i2 Mod 100 = 0 Then Debug.Print(">i1=" & i1 & " i2=" & i2 & " " & Int((Now - t0).TotalSeconds) & " sec")
                For i3 = i2 To MaxN
                    sumx = p4(i1) + p4(i2) + p4(i3)
                    i4 = i3 + 1
                    While i4 <= MaxN AndAlso p4(i4) <= sumx
                        If p4(i4) = sumx Then
                            Debug.Print(i1 & " " & i2 & " " & i3 & " " & i4)
                            Exit Sub
                        End If
                        i4 += 1
                    End While
                Next i3
            Next i2
        Next i1
        Debug.Print("Not found!")
    End Sub 'EulerPower44Sum

End Class
Output:
133 110 84 27 144

Wren

Translation of: C
var start = System.clock
var n = 250
var m = 30

var p5 = List.filled(n+m+1, 0)
var s = 0
while (s < n) {
    var sq = s * s
    p5[s] = sq * sq * s
    s = s + 1
}
var max = p5[n-1]
while (s < p5.count) {
    p5[s] = max + 1
    s = s + 1
}
for (a in 1...n-3) {
    for (b in a + 1...n-2) {
        for (c in b + 1...n-1) {
            var d = c + 1
            var t = p5[a] + p5[b] + p5[c]
            var e = d + (t % m)
            s = t + p5[d]
            while (s <= max) {
                e = e - m
                while (p5[e+m] <= s) e = e + m 
                if (p5[e] == s) {
                    System.print("%(a)⁵ + %(b)⁵ + %(c)⁵ + %(d)⁵ = %(e)⁵")
                    System.print("Took %(System.clock - start) seconds")
                    return
                }
                d = d + 1
                e = e + 1
                s = t + p5[d]
            }
        }
    }
}
Output:

Timing is for an Intel Core i7-8565U machine running Wren 0.4.0 on Ubuntu 22.04.

27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵
Took 7.168733 seconds

XPL0

Runs in 50.3 seconds on Pi4.

func real Pow5(N);
int  N;
real X, P;
[X:= float(N);
P:= X*X;
P:= P*P;
return P*X;
];

int  X0, X1, X2, X3, Y;
real SP;
[for X0:= 1 to 250 do
  for X1:= 1 to X0-1 do
    for X2:= 1 to X1-1 do
      for X3:= 1 to X2-1 do
        [SP:= Pow5(X0) + Pow5(X1) + Pow5(X2) + Pow5(X3);
        for Y:= X0+1 to 250 do
          if Pow5(Y) = SP then
                [IntOut(0, X0);  Text(0, "^^5 + ");
                 IntOut(0, X1);  Text(0, "^^5 + ");
                 IntOut(0, X2);  Text(0, "^^5 + ");
                 IntOut(0, X3);  Text(0, "^^5 = ");
                 IntOut(0, Y);   Text(0, "^^5^m^j");
                 exit;
                ];
        ];
]
Output:
133^5 + 110^5 + 84^5 + 27^5 = 144^5

Zig

Translation of: Go
const std = @import("std");
const stdout = std.io.getStdOut().outStream();

pub fn main() !void {
    var pow5: [250]i64 = undefined;
    for (pow5) |*e, i| {
        const n = @intCast(i64, i);
        e.* = n * n * n * n * n;
    }
    var x0: u16 = 4;
    while (x0 < pow5.len) : (x0 += 1) {
        var x1: u16 = 3;
        while (x1 < x0) : (x1 += 1) {
            var x2: u16 = 2;
            while (x2 < x1) : (x2 += 1) {
                var x3: u16 = 1;
                while (x3 < x2) : (x3 += 1) {
                    const sum = pow5[x0] + pow5[x1] + pow5[x2] + pow5[x3];
                    var y: u16 = x0 + 1;
                    while (y < pow5.len) : (y += 1) if (sum == pow5[y]) {
                        try stdout.print("{}⁵ + {}⁵ + {}⁵ + {}⁵ = {}⁵\n", .{ x0, x1, x2, x3, y });
                        return;
                    };
                }
            }
        }
    }

    try stdout.print("Sorry, no solution found.\n", .{});
}
Output:
133⁵ + 110⁵ + 84⁵ + 27⁵ = 144⁵

zkl

Uses two look up tables for efficiency. Counts from 0 for ease of coding.

pow5s:=[1..249].apply("pow",5); // (1^5, 2^5, 3^5 .. 249^5)
pow5r:=pow5s.enumerate().apply("reverse").toDictionary(); // [144^5:144, ...]
foreach x0,x1,x2,x3 in (249,x0,x1,x2){
   sum:=pow5s[x0] + pow5s[x1] + pow5s[x2] + pow5s[x3];
   if(pow5r.holds(sum))
      println("%d^5 + %d^5 + %d^5 + %d^5 = %d^5"
          .fmt(x3+1,x2+1,x1+1,x0+1,pow5r[sum]+1));
      break(4);  // the foreach is actually four loops
}
Output:
27^5 + 84^5 + 110^5 + 133^5 = 144^5

Using the Python technique of caching double sums results in a 5x speed up [to the first/only solution]; actually the speed up is close to 25x but creating the caches dominates the runtime to the first solution.

Translation of: Python
p5,sum2:=Dictionary(),Dictionary();
foreach i in ([1..249]){
   p5[i.pow(5)]=i;
   foreach j in ([i..249]){ sum2[i.pow(5) + j.pow(5)]=T(i,j) } // 31,125 keys
}

sk:=sum2.keys.apply("toInt").copy().sort(); // RW list sorts faster than a RO one
foreach p,s in (p5.keys.apply("toInt"),sk){
   if(p<=s) break;
   if(sum2.holds(p - s)){
      println("%d^5 + %d^5 + %d^5 + %d^5 = %d^5"
          .fmt(sum2[s].xplode(),sum2[p - s].xplode(),p5[p]));
      break(2);  // or get permutations
   }
}

Note: dictionary keys are always strings and copying a read only list creates a read write list.

Output:
27^5 + 84^5 + 110^5 + 133^5 = 144^5