Prime decomposition
The prime decomposition of a number is defined as a list of prime numbers which when all multiplied together, are equal to that number.
You are encouraged to solve this task according to the task description, using any language you may know.
- Example
12 = 2 × 2 × 3, so its prime decomposition is {2, 2, 3}
- Task
Write a function which returns an array or collection which contains the prime decomposition of a given number greater than 1.
If your language does not have an isPrime-like function available, you may assume that you have a function which determines whether a number is prime (note its name before your code).
If you would like to test code from this task, you may use code from trial division or the Sieve of Eratosthenes.
Note: The program must not be limited by the word size of your computer or some other artificial limit; it should work for any number regardless of size (ignoring the physical limits of RAM etc).
- Related tasks
- count in factors
- factors of an integer
- Sieve of Eratosthenes
- primality by trial division
- factors of a Mersenne number
- trial factoring of a Mersenne number
- partition an integer X into N primes
- sequence of primes by Trial Division
11l
F decompose(BigInt number)
[BigInt] result
V n = number
BigInt i = 2
L n % i == 0
result.append(i)
n I/= i
i = 3
L n >= i * i
L n % i == 0
result.append(i)
n I/= i
i += 2
I n != 1
result.append(n)
R result
L(i) 2..9
print(decompose(i))
print(decompose(1023 * 1024))
print(decompose(2 * 3 * 5 * 7 * 11 * 11 * 13 * 17))
print(decompose(BigInt(16860167264933) * 179951))- Output:
[2] [3] [2, 2] [5] [2, 3] [7] [2, 2, 2] [3, 3] [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 11, 31] [2, 3, 5, 7, 11, 11, 13, 17] [179951, 16860167264933]
360 Assembly
For maximum compatibility, this program uses only the basic instruction set.
PRIMEDE CSECT
USING PRIMEDE,R13
B 80(R15) skip savearea
DC 17F'0' savearea
DC CL8'PRIMEDE'
STM R14,R12,12(R13)
ST R13,4(R15)
ST R15,8(R13)
LR R13,R15 end prolog
LA R2,0
LA R3,1023
LA R4,1024
MR R2,R4
ST R3,N n=1023*1024
LA R5,WBUFFER
LA R6,0
L R1,N n
XDECO R1,0(R5)
LA R5,12(R5)
MVC 0(3,R5),=C' : '
LA R5,3(R5)
LA R0,2
ST R0,I i=2
WHILE1 EQU * do while(i<=n/2)
L R2,N
SRA R2,1
L R4,I
CR R4,R2 i<=n/2
BH EWHILE1
WHILE2 EQU * do while(n//i=0)
L R3,N
LA R2,0
D R2,I
LTR R2,R2 n//i=0
BNZ EWHILE2
ST R3,N n=n/i
ST R3,M m=n
L R1,I i
XDECO R1,WDECO
MVC 0(5,R5),WDECO+7
LA R5,5(R5)
MVI OK,X'01' ok
B WHILE2
EWHILE2 EQU *
L R4,I
CH R4,=H'2' if i=2 then
BNE NE2
LA R0,3
ST R0,I i=3
B EIFNE2
NE2 L R2,I else
LA R2,2(R2)
ST R2,I i=i+2
EIFNE2 B WHILE1
EWHILE1 EQU *
CLI OK,X'01' if ^ok then
BE NOTPRIME
MVC 0(7,R5),=C'[prime]'
LA R5,7(R5)
B EPRIME
NOTPRIME L R1,M m
XDECO R1,WDECO
MVC 0(5,R5),WDECO+7
EPRIME XPRNT WBUFFER,80 put
L R13,4(0,R13) epilog
LM R14,R12,12(R13)
XR R15,R15
BR R14
N DS F
I DS F
M DS F
OK DC X'00'
WBUFFER DC CL80' '
WDECO DS CL16
YREGS
END PRIMEDE- Output:
1047552 : 2 2 2 2 2 2 2 2 2 2 3 11 31
AArch64 Assembly
/* ARM assembly AARCH64 Raspberry PI 3B */
/* program primeDecomp64.s */
/*******************************************/
/* Constantes file */
/*******************************************/
/* for this file see task include a file in language AArch64 assembly*/
.include "../includeConstantesARM64.inc"
.equ NBFACT, 100
/*******************************************/
/* Structures */
/********************************************/
/* structurea area factors */
.struct 0
fac_value: // factor
.struct fac_value + 8
fac_number: // number of identical factors
.struct fac_number + 8
fac_end:
/*******************************************/
/* Initialized data */
/*******************************************/
.data
szMessStartPgm: .asciz "Program start \n"
szMessEndPgm: .asciz "Program normal end.\n"
szMessNotPrime: .asciz "Not prime.\n"
szMessPrime: .asciz "Prime\n"
szCarriageReturn: .asciz "\n"
szSpaces: .asciz " "
szMessNumber: .asciz " The factors of @ are :\n"
/*******************************************/
/* UnInitialized data */
/*******************************************/
.bss
sZoneConv: .skip 32
.align 4
tbZoneDecom: .skip fac_end * NBFACT
/*******************************************/
/* code section */
/*******************************************/
.text
.global main
main: // program start
ldr x0,qAdrszMessStartPgm // display start message
bl affichageMess
ldr x20,qVal
//mov x20,17
mov x0,x20
ldr x1,qAdrtbZoneDecom
bl decompFact // decomposition
cmp x0,#0
beq 1f
mov x2,x0
mov x0,x20
ldr x1,qAdrtbZoneDecom
bl displayFactors // display factors
b 2f
1:
ldr x0,qAdrszMessPrime // prime
bl affichageMess
2:
ldr x0,qAdrszMessEndPgm // display end message
bl affichageMess
100: // standard end of the program
mov x0,0 // return code
mov x8,EXIT // request to exit program
svc 0 // perform system call
qAdrszMessStartPgm: .quad szMessStartPgm
qAdrszMessEndPgm: .quad szMessEndPgm
qAdrszCarriageReturn: .quad szCarriageReturn
qAdrszMessNotPrime: .quad szMessNotPrime
qAdrszMessPrime: .quad szMessPrime
qAdrtbZoneDecom: .quad tbZoneDecom
//qVal: .quad 2 <<31
qVal: .quad 1047552 // test not prime
//qVal: .quad 1429671721 // test not prime (37811 * 37811)
/******************************************************************/
/* prime decomposition */
/******************************************************************/
/* x0 contains the number */
/* x1 contains address factors array */
/* REMARK no save register x9-x19 */
decompFact:
stp x1,lr,[sp,-16]! // save registers
mov x12,x0 // save number
bl isPrime // prime ?
cbnz x0,12f // yes -> no decomposition
mov x19,fac_end // element area size
mov x18,0 // raz indice
mov x16,0 // prev divisor
mov x17,0 // number of identical divisors
mov x13,2 // first divisor
2:
cmp x12,1
beq 10f
udiv x14,x12,x13 // division
msub x15,x14,x13,x12 // remainder = x12 -(x13*x14)
cbnz x15,5f // if remainder <> zero x13 not divisor
mov x12,x14 // quotient -> new dividende
cmp x13,x16 // same divisor ?
beq 4f // yes
cbz x16,3f // yes it is first divisor ?
madd x11,x18,x19,x1 // no -> store prev divisor in the area
str x16,[x11,fac_value]
str x17,[x11,fac_number] // and store number of same factor
add x18,x18,1 // increment indice
mov x17,0 // raz number of same factor
3:
mov x16,x13 // save new divisor
4:
add x17,x17,1 // increment number of same factor
mov x0,x12 // the new dividende is prime ?
bl isPrime
cbnz x0,10f // yes
b 2b // else loop
5: // divisor is not a factor
cmp x13,2 // begin ?
cinc x13,x13,ne // if divisor <> 2 add 1
add x13,x13,1
b 2b // and loop
10: // new dividende is prime
cmp x16,x12 // divisor = dividende ?
cinc x17,x17,eq //add 1 if last dividende = diviseur
madd x11,x18,x19,x1
str x16,[x11,fac_value] // store divisor in area
str x17,[x11,fac_number] // and store number
add x18,x18,1 // increment indice
cmp x16,x12 //store last dividende if <> diviseur
beq 11f
madd x11,x18,x19,x1
str x12,[x11,fac_value] // sinon stockage dans la table
mov x17,1
str x17,[x11,fac_number] // store 1 in number
add x18,x18,1
11:
mov x0,x18 // return nb factors
b 100f
12:
mov x0,#0 // number is prime
b 100f
100:
ldp x1,lr,[sp],16 // restaur des 2 registres
ret // retour adresse lr x30
/******************************************************************/
/* prime decomposition */
/******************************************************************/
/* x0 contains the number */
/* x1 contains address factors array */
/* x2 number of factors */
displayFactors:
stp x1,lr,[sp,-16]! // save registres
mov x19,fac_end // element area size
mov x13,x1 // save area address
ldr x1,qAdrsZoneConv // load zone conversion address
bl conversion10
ldr x0,qAdrszMessNumber
bl strInsertAtCharInc // insert result at Second @ character
bl affichageMess
mov x9,0 // indice
1:
madd x10,x9,x19,x13 // compute address area element
ldr x0,[x10,fac_value]
ldr x12,[x10,fac_number]
bl conversion10 // decimal conversion
2:
mov x0,x1
bl affichageMess
ldr x0,qAdrszSpaces
bl affichageMess
subs x12,x12,#1
bgt 2b
add x9,x9,1
cmp x9,x2
blt 1b
ldr x0,qAdrszCarriageReturn
bl affichageMess
100:
ldp x1,lr,[sp],16 // restaur des 2 registres
ret // retour adresse lr x30
qAdrsZoneConv: .quad sZoneConv
qAdrszSpaces: .quad szSpaces
qAdrszMessNumber: .quad szMessNumber
/******************************************************************/
/* test if number is prime */
/******************************************************************/
/* x0 contains the number */
/* x0 return 1 if prime else return 0 */
isPrime:
stp x1,lr,[sp,-16]! // save registers
cmp x0,1 // <= 1 ?
ble 98f
cmp x0,3 // 2 and 3 prime
ble 97f
tst x0,1 // even ?
beq 98f
mov x9,3 // first divisor
1:
udiv x11,x0,x9
msub x10,x11,x9,x0 // compute remainder
cbz x10,98f // end if zero
add x9,x9,#2 // increment divisor
cmp x9,x11 // divisors<=quotient ?
ble 1b // loop
97:
mov x0,1 // return prime
b 100f
98:
mov x0,0 // not prime
b 100f
100:
ldp x1,lr,[sp],16 // restaur 2 registers
ret // return to address lr x30
/********************************************************/
/* File Include fonctions */
/********************************************************/
/* for this file see task include a file in language AArch64 assembly */
.include "../includeARM64.inc"- Output:
Program start The factors of 1047552 are : 2 2 2 2 2 2 2 2 2 2 3 11 31 Program normal end.
ABAP
class ZMLA_ROSETTA definition
public
create public .
public section.
types:
enumber TYPE N LENGTH 60,
listof_enumber TYPE TABLE OF enumber .
class-methods FACTORS
importing
value(N) type ENUMBER
exporting
value(ORET) type LISTOF_ENUMBER .
protected section.
private section.
ENDCLASS.
CLASS ZMLA_ROSETTA IMPLEMENTATION.
* <SIGNATURE>---------------------------------------------------------------------------------------+
* | Static Public Method ZMLA_ROSETTA=>FACTORS
* +-------------------------------------------------------------------------------------------------+
* | [--->] N TYPE ENUMBER
* | [<---] ORET TYPE LISTOF_ENUMBER
* +--------------------------------------------------------------------------------------</SIGNATURE>
method FACTORS.
CLEAR oret.
WHILE n mod 2 = 0.
n = n / 2.
APPEND 2 to oret.
ENDWHILE.
DATA: lim type enumber,
i type enumber.
lim = sqrt( n ).
i = 3.
WHILE i <= lim.
WHILE n mod i = 0.
APPEND i to oret.
n = n / i.
lim = sqrt( n ).
ENDWHILE.
i = i + 2.
ENDWHILE.
IF n > 1.
APPEND n to oret.
ENDIF.
endmethod.
ENDCLASS.
ACL2
(include-book "arithmetic-3/top" :dir :system)
(defun prime-factors-r (n i)
(declare (xargs :mode :program))
(cond ((or (zp n) (zp (- n i)) (zp i) (< i 2) (< n 2))
(list n))
((= (mod n i) 0)
(cons i (prime-factors-r (floor n i) 2)))
(t (prime-factors-r n (1+ i)))))
(defun prime-factors (n)
(declare (xargs :mode :program))
(prime-factors-r n 2))
Ada
The solution is generic.
The package Prime_Numbers is instantiated by a type that supports necessary operations +, *, /, mod, >. The constants 0, 1, 2 are parameters too, because the type might have no literals. The same package is used for Almost prime#Ada, Semiprime#Ada, Count in factors#Ada, Primality by Trial Division#Ada, Sequence of primes by Trial Division#Ada, and Ulam_spiral_(for_primes)#Ada.
This is the specification of the generic package Prime_Numbers:
generic
type Number is private;
Zero : Number;
One : Number;
Two : Number;
with function "+" (X, Y : Number) return Number is <>;
with function "*" (X, Y : Number) return Number is <>;
with function "/" (X, Y : Number) return Number is <>;
with function "mod" (X, Y : Number) return Number is <>;
with function ">" (X, Y : Number) return Boolean is <>;
package Prime_Numbers is
type Number_List is array (Positive range <>) of Number;
function Decompose (N : Number) return Number_List;
function Is_Prime (N : Number) return Boolean;
end Prime_Numbers;
The function Decompose first estimates the maximal result length as log2 of the argument. Then it allocates the result and starts to enumerate divisors. It does not care to check if the divisors are prime, because non-prime divisors will be automatically excluded.
This is the implementation of the generic package Prime_Numbers:
package body Prime_Numbers is
-- auxiliary (internal) functions
function First_Factor (N : Number; Start : Number) return Number is
K : Number := Start;
begin
while ((N mod K) /= Zero) and then (N > (K*K)) loop
K := K + One;
end loop;
if (N mod K) = Zero then
return K;
else
return N;
end if;
end First_Factor;
function Decompose (N : Number; Start : Number) return Number_List is
F: Number := First_Factor(N, Start);
M: Number := N / F;
begin
if M = One then -- F is the last factor
return (1 => F);
else
return F & Decompose(M, Start);
end if;
end Decompose;
-- functions visible from the outside
function Decompose (N : Number) return Number_List is (Decompose(N, Two));
function Is_Prime (N : Number) return Boolean is
(N > One and then First_Factor(N, Two)=N);
end Prime_Numbers;
In the example provided, the package Prime_Numbers is instantiated with plain integer type:
with Prime_Numbers, Ada.Text_IO;
procedure Test_Prime is
package Integer_Numbers is new
Prime_Numbers (Natural, 0, 1, 2);
use Integer_Numbers;
procedure Put (List : Number_List) is
begin
for Index in List'Range loop
Ada.Text_IO.Put (Positive'Image (List (Index)));
end loop;
end Put;
begin
Put (Decompose (12));
end Test_Prime;
- Output:
(decomposition of 12)
2 2 3
ALGOL 68
- note: This specimen retains the original Python coding style.
#IF long int possible THEN #
MODE LINT = LONG INT;
LINT lmax int = long max int;
OP LLENG = (INT i)LINT: LENG i,
LSHORTEN = (LINT i)INT: SHORTEN i;
#ELSE
MODE LINT = INT;
LINT lmax int = max int;
OP LLENG = (INT i)LINT: i,
LSHORTEN = (LINT i)INT: i;
FI#
OP LLONG = (INT i)LINT: LLENG i;
MODE YIELDLINT = PROC(LINT)VOID;
PROC (LINT, YIELDLINT)VOID gen decompose;
INT upb cache = bits width;
BITS cache := 2r0;
BITS cached := 2r0;
PROC is prime = (LINT n)BOOL: (
BOOL
has factor := FALSE,
out := TRUE;
# FOR LINT factor IN # gen decompose(n, # ) DO ( #
## (LINT factor)VOID:(
IF has factor THEN out := FALSE; GO TO done FI;
has factor := TRUE
# OD # ));
done: out
);
PROC is prime cached := (LINT n)BOOL: (
LINT l half n = n OVER LLONG 2 - LLONG 1;
IF l half n <= LLENG upb cache THEN
INT half n = LSHORTEN l half n;
IF half n ELEM cached THEN
BOOL(half n ELEM cache)
ELSE
BOOL out = is prime(n);
BITS mask = 2r1 SHL (upb cache - half n);
cached := cached OR mask;
IF out THEN cache := cache OR mask FI;
out
FI
ELSE
is prime(n) # above useful cache limit #
FI
);
PROC gen primes := (YIELDLINT yield)VOID:(
yield(LLONG 2);
LINT n := LLONG 3;
WHILE n < l maxint - LLONG 2 DO
yield(n);
n +:= LLONG 2;
WHILE n < l maxint - LLONG 2 AND NOT is prime cached(n) DO
n +:= LLONG 2
OD
OD
);
# PROC # gen decompose := (LINT in n, YIELDLINT yield)VOID: (
LINT n := in n;
# FOR LINT p IN # gen primes( # ) DO ( #
## (LINT p)VOID:
IF p*p > n THEN
GO TO done
ELSE
WHILE n MOD p = LLONG 0 DO
yield(p);
n := n OVER p
OD
FI
# OD # );
done:
IF n > LLONG 1 THEN
yield(n)
FI
);
main:(
# FOR LINT m IN # gen primes( # ) DO ( #
## (LINT m)VOID:(
LINT p = LLONG 2 ** LSHORTEN m - LLONG 1;
print(("2**",whole(m,0),"-1 = ",whole(p,0),", with factors:"));
# FOR LINT factor IN # gen decompose(p, # ) DO ( #
## (LINT factor)VOID:
print((" ",whole(factor,0)))
# OD # );
print(new line);
IF m >= LLONG 59 THEN GO TO done FI
# OD # ));
done: EMPTY
)- Output:
2**2-1 = 3, with factors: 3 2**3-1 = 7, with factors: 7 2**5-1 = 31, with factors: 31 2**7-1 = 127, with factors: 127 2**11-1 = 2047, with factors: 23 89 2**13-1 = 8191, with factors: 8191 2**17-1 = 131071, with factors: 131071 2**19-1 = 524287, with factors: 524287 2**23-1 = 8388607, with factors: 47 178481 2**29-1 = 536870911, with factors: 233 1103 2089 2**31-1 = 2147483647, with factors: 2147483647 2**37-1 = 137438953471, with factors: 223 616318177 2**41-1 = 2199023255551, with factors: 13367 164511353 2**43-1 = 8796093022207, with factors: 431 9719 2099863 2**47-1 = 140737488355327, with factors: 2351 4513 13264529 2**53-1 = 9007199254740991, with factors: 6361 69431 20394401 2**59-1 = 576460752303423487, with factors: 179951 3203431780337
Note: ALGOL 68G took 49,109,599 BogoMI and ELLA ALGOL 68RS took 1,127,634 BogoMI to complete the example.
ALGOL-M
Sadly, ALGOL-M does not allow arrays to be passed as parameters to procedures or functions, so the routine must store its results in (and know the name of) the external array used for that purpose.
BEGIN
INTEGER I, K, NFOUND;
INTEGER ARRAY FACTORS[1:16];
COMMENT - RETURN P MOD Q;
INTEGER FUNCTION MOD (P, Q);
INTEGER P, Q;
BEGIN
MOD := P - Q * (P / Q);
END;
COMMENT
FIND THE PRIME FACTORS OF N AND STORE IN THE EXTERNAL
ARRAY "FACTORS", RETURNING THE NUMBER FOUND. IF N IS
PRIME, IT WILL BE STORED AS THE FIRST AND ONLY FACTOR;
INTEGER FUNCTION PRIMEFACTORS(N);
INTEGER N;
BEGIN
INTEGER P, COUNT;
P := 2;
COUNT := 1;
WHILE N >= P * P DO
BEGIN
IF MOD(N, P) = 0 THEN
BEGIN
FACTORS[COUNT] := P;
COUNT := COUNT + 1;
N := N / P;
END
ELSE
P := P + 1;
END;
FACTORS[COUNT] := N;
PRIMEFACTORS := COUNT;
END;
COMMENT -- EXERCISE THE ROUTINE;
FOR I := 77 STEP 2 UNTIL 99 DO
BEGIN
WRITE(I,":");
NFOUND := PRIMEFACTORS(I);
COMMENT - PRINT OUT THE FACTORS THAT WERE FOUND;
FOR K := 1 STEP 1 UNTIL NFOUND DO
BEGIN
WRITEON(FACTORS[K]);
END;
END;
END- Output:
77: 7 11 79: 79 81: 3 3 3 3 83: 83 85: 5 17 87: 3 29 89: 89 91: 7 13 93: 3 31 95: 5 19 97: 97 99: 3 3 11
ALGOL W
Algol W procedures can't return arrays, so an array to store the factors in must be passed as a parameter.
begin % find the prime decompositionmtion of some integers %
% increments n and returns the new value %
integer procedure inc ( integer value result n ) ; begin n := n + 1; n end;
% divides n by d and returns the result %
integer procedure over ( integer value result n
; integer value d
) ; begin n := n div d; n end;
% sets the elements of f to the prime factors of n %
% the bounds of f should be 0 :: x where x is large enough to hold %
% all the factors, f( 0 ) will contain 6he number of factors %
procedure decompose ( integer value n; integer array f ( * ) ) ;
begin
integer d, v;
f( 0 ) := 0;
v := abs n;
if v > 0 and v rem 2 = 0 then begin
f( inc( f( 0 ) ) ) := 2;
while over( v, 2 ) > 0 and v rem 2 = 0 do f( inc( f( 0 ) ) ) := 2;
end if_2_divides_v ;
d := 3;
while d * d <= v do begin
if v rem d = 0 then begin
f( inc( f( 0 ) ) ) := d;
while over( v, d ) > 0 and v rem d = 0 do f( inc( f( 0 ) ) ) := d;
end if_d_divides_v ;
d := d + 2
end while_d_squared_le_v ;
if v > 1 then f( inc( f( 0 ) ) ) := v
end factorise ;
% some test cases %
for n := 0, 1, 7, 31, 127, 2047, 8191, 131071, 524287, 2520, 32767, 8855, 441421750 do begin
integer array f( 0 :: 20 );
decompose( n, f );
write( s_w := 0, n, ": " );
for fPos := 1 until f( 0 ) do writeon( i_w := 1, s_w := 0, " ", f( fPos ) );
end for_n ;
end.- Output:
0:
1:
7: 7
31: 31
127: 127
2047: 23 89
8191: 8191
131071: 131071
524287: 524287
2520: 2 2 2 3 3 5 7
32767: 7 31 151
8855: 5 7 11 23
441421750: 2 5 5 5 7 11 23 997
Arturo
decompose: function [num][
facts: to [:string] factors.prime num
print [
pad.right (to :string num) ++ " = " ++ join.with:" x " facts 30
"{"++ (join.with:", " unique facts) ++ "}"
]
]
loop 2..40 => decompose
- Output:
2 = 2 {2}
3 = 3 {3}
4 = 2 x 2 {2}
5 = 5 {5}
6 = 2 x 3 {2, 3}
7 = 7 {7}
8 = 2 x 2 x 2 {2}
9 = 3 x 3 {3}
10 = 2 x 5 {2, 5}
11 = 11 {11}
12 = 2 x 2 x 3 {2, 3}
13 = 13 {13}
14 = 2 x 7 {2, 7}
15 = 3 x 5 {3, 5}
16 = 2 x 2 x 2 x 2 {2}
17 = 17 {17}
18 = 2 x 3 x 3 {2, 3}
19 = 19 {19}
20 = 2 x 2 x 5 {2, 5}
21 = 3 x 7 {3, 7}
22 = 2 x 11 {2, 11}
23 = 23 {23}
24 = 2 x 2 x 2 x 3 {2, 3}
25 = 5 x 5 {5}
26 = 2 x 13 {2, 13}
27 = 3 x 3 x 3 {3}
28 = 2 x 2 x 7 {2, 7}
29 = 29 {29}
30 = 2 x 3 x 5 {2, 3, 5}
31 = 31 {31}
32 = 2 x 2 x 2 x 2 x 2 {2}
33 = 3 x 11 {3, 11}
34 = 2 x 17 {2, 17}
35 = 5 x 7 {5, 7}
36 = 2 x 2 x 3 x 3 {2, 3}
37 = 37 {37}
38 = 2 x 19 {2, 19}
39 = 3 x 13 {3, 13}
40 = 2 x 2 x 2 x 5 {2, 5}
AutoHotkey
MsgBox % factor(8388607) ; 47 * 178481
factor(n)
{
if (n = 1)
return
f = 2
while (f <= n)
{
if (Mod(n, f) = 0)
{
next := factor(n / f)
return, % f "`n" next
}
f++
}
}
Optimized Version
prime_numbers(n) {
if (n <= 3)
return [n]
ans := []
done := false
while !done
{
if !Mod(n,2){
ans.push(2)
n /= 2
continue
}
if !Mod(n,3) {
ans.push(3)
n /= 3
continue
}
if (n = 1)
return ans
sr := sqrt(n)
done := true
; try to divide the checked number by all numbers till its square root.
i := 6
while (i <= sr+6){
if !Mod(n, i-1) { ; is n divisible by i-1?
ans.push(i-1)
n /= i-1
done := false
break
}
if !Mod(n, i+1) { ; is n divisible by i+1?
ans.push(i+1)
n /= i+1
done := false
break
}
i += 6
}
}
ans.push(n)
return ans
}
Examples:
num := 8388607, output := ""
for i, p in prime_numbers(num)
output .= p " * "
MsgBox % num " = " Trim(output, " * ")
return
- Output:
8388607 = 47 * 178481
AWK
As the examples show, pretty large numbers can be factored in tolerable time:
# Usage: awk -f primefac.awk
function pfac(n, r, f){
r = ""; f = 2
while (f <= n) {
while(!(n % f)) {
n = n / f
r = r " " f
}
f = f + 2 - (f == 2)
}
return r
}
# For each line of input, print the prime factors.
{ print pfac($1) }
- Output:
entering input on stdin
$ 36 2 2 3 3 77 7 11 536870911 233 1103 2089 8796093022207 431 9719 2099863
BASIC
ANSI BASIC
100 PROGRAM PrimeDecomposition
110 REM -(2^31) has most prime factors (31 twos) than other 32-bit signed integer.
120 DIM Facs(0 TO 30)
130 INPUT PROMPT "Enter a number: ": N
140 CALL CalcFacs(N, Facs, FacsCnt)
150 REM There is at least one factor
160 FOR I = 0 TO FacsCnt - 1
170 PRINT Facs(I);
180 NEXT I
190 PRINT
200 END
210 REM **
220 EXTERNAL SUB CalcFacs(N, Facs(), FacsCnt)
230 LET N = ABS(N)
240 LET FacsCnt = 0
250 IF N >= 2 THEN
260 LET I = 2
270 DO WHILE I * I <= N
280 IF MOD(N, I) = 0 THEN
290 LET N = INT(N / I)
300 LET Facs(FacsCnt) = I
310 LET FacsCnt = FacsCnt + 1
320 LET I = 2
330 ELSE
340 LET I = I + 1
350 END IF
360 LOOP
370 LET Facs(FacsCnt) = N
380 LET FacsCnt = FacsCnt + 1
390 END IF
400 END SUB
- Output:
3 runs.
Enter a number: 32 2 2 2 2 2
Enter a number: 2520 2 2 2 3 3 5 7
Enter a number: 13 13
Applesoft BASIC
9040 PF(0) = 0 : SC = 0
9050 FOR CA = 2 TO INT( SQR(I))
9060 IF I = 1 THEN RETURN
9070 IF INT(I / CA) * CA = I THEN GOSUB 9200 : GOTO 9060
9080 CA = CA + SC : SC = 1
9090 NEXT CA
9100 IF I = 1 THEN RETURN
9110 CA = I
9200 PF(0) = PF(0) + 1
9210 PF(PF(0)) = CA
9220 I = I / CA
9230 RETURNASIC
REM Prime decomposition
DIM Facs(14)
REM -(2^15) has most prime factors (15 twos) than other 16-bit signed integer.
PRINT "Enter a number";
INPUT N
GOSUB CalcFacs:
FacsCntM1 = FacsCnt - 1
FOR I = 0 TO FacsCntM1
PRINT Facs(I);
NEXT I
PRINT
END
CalcFacs:
N = ABS(N)
FacsCnt = 0
IF N >= 2 THEN
I = 2
SqrI = I * I
WHILE SqrI <= N
NModI = N MOD I
IF NModI = 0 THEN
N = N / I
Facs(FacsCnt) = I
FacsCnt = FacsCnt + 1
I = 2
ELSE
I = I + 1
ENDIF
SqrI = I * I
WEND
Facs(FacsCnt) = N
FacsCnt = FacsCnt + 1
ENDIF
RETURN
- Output:
3 runs.
Enter a number?32
2 2 2 2 2
Enter a number?2520
2 2 2 3 3 5 7
Enter a number?13
13
Commodore BASIC
It's not easily possible to have arbitrary precision integers in PET basic, so here is at least a version using built-in data types (reals). On return from the subroutine starting at 9000 the global array pf contains the number of factors followed by the factors themselves:
9000 REM ----- function generate
9010 REM in ... i ... number
9020 REM out ... pf() ... factors
9030 REM mod ... ca ... pf candidate
9040 pf(0)=0 : ca=2 : REM special case
9050 IF i=1 THEN RETURN
9060 IF INT(i/ca)*ca=i THEN GOSUB 9200 : GOTO 9050
9070 FOR ca=3 TO INT( SQR(i)) STEP 2
9080 IF i=1 THEN RETURN
9090 IF INT(i/ca)*ca=i THEN GOSUB 9200 : GOTO 9080
9100 NEXT
9110 IF i>1 THEN ca=i : GOSUB 9200
9120 RETURN
9200 pf(0)=pf(0)+1
9210 pf(pf(0))=ca
9220 i=i/ca
9230 RETURN
Craft Basic
define limit = 20, loops = 0
dim list[limit]
input "loops?", loops
for x = 1 to loops
let n = x
print n, " : ",
gosub collectprimefactors
for y = 0 to c
if list[y] then
print list[y], " ",
let list[y] = 0
endif
next y
print ""
next x
end
sub collectprimefactors
let c = 0
do
if int(n mod 2) = 0 then
let n = int(n / 2)
let list[c] = 2
let c = c + 1
endif
wait
loop int(n mod 2) = 0
for i = 3 to sqrt(n) step 2
do
if int(n mod i) = 0 then
let n = int(n / i)
let list[c] = i
let c = c + 1
endif
wait
loop int(n mod i) = 0
next i
if n > 2 then
let list[c] = n
let c = c + 1
endif
return
- Output:
loops? 20 1 : 2 : 2 3 : 3 4 : 2 2 5 : 5 6 : 2 3 7 : 7 8 : 2 2 2 9 : 3 3 10 : 2 5 11 : 11 12 : 2 2 3 13 : 13 14 : 2 7 15 : 3 5 16 : 2 2 2 2 17 : 17 18 : 2 3 3 19 : 19 20 : 2 2 5
FreeBASIC
' FB 1.05.0 Win64
Function isPrime(n As Integer) As Boolean
If n Mod 2 = 0 Then Return n = 2
If n Mod 3 = 0 Then Return n = 3
Dim d As Integer = 5
While d * d <= n
If n Mod d = 0 Then Return False
d += 2
If n Mod d = 0 Then Return False
d += 4
Wend
Return True
End Function
Sub getPrimeFactors(factors() As UInteger, n As UInteger)
If n < 2 Then Return
If isPrime(n) Then
Redim factors(0 To 0)
factors(0) = n
Return
End If
Dim factor As UInteger = 2
Do
If n Mod factor = 0 Then
Redim Preserve factors(0 To UBound(factors) + 1)
factors(UBound(factors)) = factor
n \= factor
If n = 1 Then Return
If isPrime(n) Then factor = n
Else
factor += 1
End If
Loop
End Sub
Dim factors() As UInteger
Dim primes(1 To 17) As UInteger = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59}
Dim n As UInteger
For i As UInteger = 1 To 17
Erase factors
n = 1 Shl primes(i) - 1
getPrimeFactors factors(), n
Print "2^";Str(primes(i)); Tab(5); " - 1 = "; Str(n); Tab(30);" => ";
For j As UInteger = LBound(factors) To UBound(factors)
Print factors(j);
If j < UBound(factors) Then Print " x ";
Next j
Print
Next i
Print
Print "Press any key to quit"
Sleep
- Output:
2^2 - 1 = 3 => 3 2^3 - 1 = 7 => 7 2^5 - 1 = 31 => 31 2^7 - 1 = 127 => 127 2^11 - 1 = 2047 => 23 x 89 2^13 - 1 = 8191 => 8191 2^17 - 1 = 131071 => 131071 2^19 - 1 = 524287 => 524287 2^23 - 1 = 8388607 => 47 x 178481 2^29 - 1 = 536870911 => 233 x 1103 x 2089 2^31 - 1 = 2147483647 => 2147483647 2^37 - 1 = 137438953471 => 223 x 616318177 2^41 - 1 = 2199023255551 => 13367 x 164511353 2^43 - 1 = 8796093022207 => 431 x 9719 x 2099863 2^47 - 1 = 140737488355327 => 2351 x 4513 x 13264529 2^53 - 1 = 9007199254740991 => 6361 x 69431 x 20394401 2^59 - 1 = 576460752303423487 => 179951 x 3203431780337
Palo Alto Tiny BASIC
10 REM PRIME DECOMPOSITION
20 INPUT "ENTER A NUMBER"N
30 PRINT "--------------"
40 LET N=ABS(N)
50 IF N<2 STOP
60 LET I=2
70 IF I*I>N GOTO 150
80 LET M=N-(N/I)*I
90 IF M#0 GOTO 130
100 LET N=N/I
110 PRINT I
120 LET I=2
130 IF M#0 LET I=I+1
140 GOTO 70
150 PRINT N
160 STOP
- Output:
3 runs.
ENTER A NUMBER:2520
--------------
2
2
2
3
3
5
7
ENTER A NUMBER:16384
--------------
2
2
2
2
2
2
2
2
2
2
2
2
2
2
ENTER A NUMBER:13
--------------
13
PureBasic
CompilerIf #PB_Compiler_Debugger
CompilerError "Turn off the debugger if you want reasonable speed in this example."
CompilerEndIf
Define.q
Procedure Factor(Number, List Factors())
Protected I = 3
While Number % 2 = 0
AddElement(Factors())
Factors() = 2
Number / 2
Wend
Protected Max = Number
While I <= Max And Number > 1
While Number % I = 0
AddElement(Factors())
Factors() = I
Number/I
Wend
I + 2
Wend
EndProcedure
Number = 9007199254740991
NewList Factors()
time = ElapsedMilliseconds()
Factor(Number, Factors())
time = ElapsedMilliseconds()-time
S.s = "Factored " + Str(Number) + " in " + StrD(time/1000, 2) + " seconds."
ForEach Factors()
S + #CRLF$ + Str(Factors())
Next
MessageRequester("", S)
- Output:
Factored 9007199254740991 in 0.27 seconds. 6361 69431 20394401
S-BASIC
rem - return p mod q
function mod(p, q = integer) = integer
end = p - q * (p/q)
dim integer factors(16) rem log2(maxint) is sufficiently large
comment
Find the prime factors of n and store in global array factors
(arrays cannot be passed as parameters) and return the number
found. If n is prime, it will be stored as the only factor.
end
function primefactors(n = integer) = integer
var p, count = integer
p = 2
count = 1
while n >= (p * p) do
begin
if mod(n, p) = 0 then
begin
factors(count) = p
count = count + 1
n = n / p
end
else
p = p + 1
end
factors(count) = n
end = count
rem -- exercise the routine by checking odd numbers from 77 to 99
var i, k, nfound = integer
for i = 77 to 99 step 2
nfound = primefactors(i)
print i;"; ";
for k = 1 to nfound
print factors(k);
next k
print
next i
end- Output:
77: 7 11 79: 79 81: 3 3 3 3 83: 83 85: 5 17 87: 3 29 89: 89 91: 7 13 93: 3 31 95: 5 19 97: 97 99: 3 3 11
TI-83 BASIC
::prgmPREMIER
Disp "FACTEURS PREMIER"
Prompt N
If N<1:Stop
ClrList L1 ,L2
0→K
iPart(√(N))→L
N→M
For(I,2,L)
0→J
While fPart(M/I)=0
J+1→J
M/I→M
End
If J≠0
Then
K+1→K
I→L 1(K)
J→L2(K)
I→Z:prgmVSTR
" "+Str0→Str1
If J≠1
Then
J→Z:prgmVSTR
Str1+"^"+Str0→Str1
End
Disp Str1
End
If M=1:Stop
End
If M≠1
Then
If M≠N
Then
M→Z:prgmVSTR
" "+Str0→Str1
Disp Str1
Else
Disp "PREMIER"
End
End
::prgmVSTR
{Z,Z}→L5
{1,2}→L6
LinReg(ax+b)L6,L5,Y ₀
Equ►String(Y₀,Str0)
length(Str0)→O
sub(Str0,4,O-3)→Str0
ClrList L5,L6
DelVar Y- Output:
FACTEURS PREMIER
N=?1047552
2^10
3
11
31
Tiny BASIC
10 PRINT "Enter a number."
20 INPUT N
25 PRINT "------"
30 IF N<0 THEN LET N = -N
40 IF N<2 THEN END
50 LET I = 2
60 IF I*I > N THEN GOTO 200
70 IF (N/I)*I = N THEN GOTO 300
80 LET I = I + 1
90 GOTO 60
200 PRINT N
210 END
300 LET N = N / I
310 PRINT I
320 GOTO 50
- Output:
Enter a number. 32 ------ 2 2 2 2 2 Enter a number. 2520 ------ 2 2 2 3 3 5 7 Enter a number. 13 ------ 13
VBScript
Function PrimeFactors(n)
arrP = Split(ListPrimes(n)," ")
divnum = n
Do Until divnum = 1
'The -1 is to account for the null element of arrP
For i = 0 To UBound(arrP)-1
If divnum = 1 Then
Exit For
ElseIf divnum Mod arrP(i) = 0 Then
divnum = divnum/arrP(i)
PrimeFactors = PrimeFactors & arrP(i) & " "
End If
Next
Loop
End Function
Function IsPrime(n)
If n = 2 Then
IsPrime = True
ElseIf n <= 1 Or n Mod 2 = 0 Then
IsPrime = False
Else
IsPrime = True
For i = 3 To Int(Sqr(n)) Step 2
If n Mod i = 0 Then
IsPrime = False
Exit For
End If
Next
End If
End Function
Function ListPrimes(n)
ListPrimes = ""
For i = 1 To n
If IsPrime(i) Then
ListPrimes = ListPrimes & i & " "
End If
Next
End Function
WScript.StdOut.Write PrimeFactors(CInt(WScript.Arguments(0)))
WScript.StdOut.WriteLine
- Output:
C:\>cscript /nologo primefactors.vbs 12 2 3 2 C:\>cscript /nologo primefactors.vbs 50 2 5 5
Batch file
Unfortunately Batch does'nt have a BigNum library so the maximum number that can be decomposed is 2^31-1
@echo off
::usage: cmd /k primefactor.cmd number
setlocal enabledelayedexpansion
set /a compo=%1
if "%compo%"=="" goto:eof
set list=%compo%= (
set /a div=2 & call :loopdiv
set /a div=3 & call :loopdiv
set /a div=5,inc=2
:looptest
call :loopdiv
set /a div+=inc,inc=6-inc,div2=div*div
if %div2% lss %compo% goto looptest
if %compo% neq 1 set list= %list% %compo%
echo %list%) & goto:eof
:loopdiv
set /a "res=compo%%div
if %res% neq 0 goto:eof
set list=%list% %div%,
set/a compo/=div
goto:loopdivBefunge
Handles safely integers only up to 250 (or ones which don't have prime divisors greater than 250).
& 211p > : 1 - #v_ 25*, @ > 11g:. / v
> : 11g %!|
> 11g 1+ 11p v
^ <
BQN
An efficient Factor function using trial division and Pollard's rho algorithm is given in bqn-libs primes.bqn. The following standalone version is based on the trial division there, and builds in the sieve from Extensible prime generator.
Factor ← { 𝕊n:
# Prime sieve
primes ← ↕0
Sieve ← { p 𝕊 a‿b:
p(⍋↑⊣)↩√b ⋄ l←b-a
E ← {↕∘⌈⌾(((𝕩|-a)+𝕩×⊢)⁼)l} # Indices of multiples of 𝕩
a + / (1⥊˜l) E⊸{0¨⌾(𝕨⊸⊏)𝕩}´ p # Primes in segment [a,b)
}
# Factor by trial division
r ← ↕0 # Result list
Try ← {
m ← (1+⌊√n) ⌊ 2×𝕩 # Upper bound for factors this round
𝕩<m ? # Stop if no factors
primes ∾↩ np ← primes Sieve 𝕩‿m # New primes
{0=𝕩|n? r∾↩𝕩 ⋄ n÷↩𝕩 ⋄ 𝕊𝕩 ;@}¨ np # Try each one
𝕊 m # Next segment
;@}
Try 2
r ∾ 1⊸<⊸⥊n
}
- Output:
> ⋈⟜Factor¨ 1232123+↕4 # Some factored numbers
┌─
╵ 1232123 ⟨ 29 42487 ⟩
1232124 ⟨ 2 2 3 102677 ⟩
1232125 ⟨ 5 5 5 9857 ⟩
1232126 ⟨ 2 7 17 31 167 ⟩
┘
Bruijn
:import std/Combinator .
:import std/List .
:import std/Math .
factors \divs primes
divs y [[&[[&[[3 ⋅ 3 >? 4 case-1 (=?0 case-2 case-3)]] (quot-rem 2 1)]]]]
case-1 4 >? (+1) {}4 empty
case-2 3 : (5 1 (3 : 2))
case-3 5 4 2
main [factors <$> ({ (+42) → (+50) })]- Output:
?> {{2t, 3t, 7t}, {43t}, {2t, 2t, 11t}, {3t, 3t, 5t}, {2t, 23t}, {47t}, {2t, 2t, 2t, 2t, 3t}, {7t, 7t}, {2t, 5t, 5t}}
Burlesque
blsq ) 12fC
{2 2 3}C
Version 1
Relatively sophiscated sieve method based on size 30 prime wheel. The code does not pretend to handle prime factors larger than 64 bits. All 32-bit primes are cached with 137MB data. Cache data takes about a minute to compute the first time the program is run, which is also saved to the current directory, and will be loaded in a second if needed again.
#include <inttypes.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <assert.h>
typedef uint32_t pint;
typedef uint64_t xint;
typedef unsigned int uint;
#define PRIuPINT PRIu32 /* printf macro for pint */
#define PRIuXINT PRIu64 /* printf macro for xint */
#define MAX_FACTORS 63 /* because 2^64 is too large for xint */
uint8_t *pbits;
#define MAX_PRIME (~(pint)0)
#define MAX_PRIME_SQ 65535U
#define PBITS (MAX_PRIME / 30 + 1)
pint next_prime(pint);
int is_prime(xint);
void sieve(pint);
uint8_t bit_pos[30] = {
0, 1<<0, 0, 0, 0, 0,
0, 1<<1, 0, 0, 0, 1<<2,
0, 1<<3, 0, 0, 0, 1<<4,
0, 1<<5, 0, 0, 0, 1<<6,
0, 0, 0, 0, 0, 1<<7,
};
uint8_t rem_num[] = { 1, 7, 11, 13, 17, 19, 23, 29 };
void init_primes()
{
FILE *fp;
pint s, tgt = 4;
if (!(pbits = malloc(PBITS))) {
perror("malloc");
exit(1);
}
if ((fp = fopen("primebits", "r"))) {
fread(pbits, 1, PBITS, fp);
fclose(fp);
return;
}
memset(pbits, 255, PBITS);
for (s = 7; s <= MAX_PRIME_SQ; s = next_prime(s)) {
if (s > tgt) {
tgt *= 2;
fprintf(stderr, "sieve %"PRIuPINT"\n", s);
}
sieve(s);
}
fp = fopen("primebits", "w");
fwrite(pbits, 1, PBITS, fp);
fclose(fp);
}
int is_prime(xint x)
{
pint p;
if (x > 5) {
if (x < MAX_PRIME)
return pbits[x/30] & bit_pos[x % 30];
for (p = 2; p && (xint)p * p <= x; p = next_prime(p))
if (x % p == 0) return 0;
return 1;
}
return x == 2 || x == 3 || x == 5;
}
void sieve(pint p)
{
unsigned char b[8];
off_t ofs[8];
int i, q;
for (i = 0; i < 8; i++) {
q = rem_num[i] * p;
b[i] = ~bit_pos[q % 30];
ofs[i] = q / 30;
}
for (q = ofs[1], i = 7; i; i--)
ofs[i] -= ofs[i-1];
for (ofs[0] = p, i = 1; i < 8; i++)
ofs[0] -= ofs[i];
for (i = 1; q < PBITS; q += ofs[i = (i + 1) & 7])
pbits[q] &= b[i];
}
pint next_prime(pint p)
{
off_t addr;
uint8_t bits, rem;
if (p > 5) {
addr = p / 30;
bits = bit_pos[ p % 30 ] << 1;
for (rem = 0; (1 << rem) < bits; rem++);
while (pbits[addr] < bits || !bits) {
if (++addr >= PBITS) return 0;
bits = 1;
rem = 0;
}
if (addr >= PBITS) return 0;
while (!(pbits[addr] & bits)) {
rem++;
bits <<= 1;
}
return p = addr * 30 + rem_num[rem];
}
switch(p) {
case 2: return 3;
case 3: return 5;
case 5: return 7;
}
return 2;
}
int decompose(xint n, xint *f)
{
pint p = 0;
int i = 0;
/* check small primes: not strictly necessary */
if (n <= MAX_PRIME && is_prime(n)) {
f[0] = n;
return 1;
}
while (n >= (xint)p * p) {
if (!(p = next_prime(p))) break;
while (n % p == 0) {
n /= p;
f[i++] = p;
}
}
if (n > 1) f[i++] = n;
return i;
}
int main()
{
int i, len;
pint p = 0;
xint f[MAX_FACTORS], po;
init_primes();
for (p = 1; p < 64; p++) {
po = (1LLU << p) - 1;
printf("2^%"PRIuPINT" - 1 = %"PRIuXINT, p, po);
fflush(stdout);
if ((len = decompose(po, f)) > 1)
for (i = 0; i < len; i++)
printf(" %c %"PRIuXINT, i?'x':'=', f[i]);
putchar('\n');
}
return 0;
}
Using GNU Compiler Collection gcc extensions
Note: The following code sample is experimental as it implements python style iterators for (potentially) infinite sequences. C is not normally written this way, and in the case of this sample it requires the GCC "nested procedure" extension to the C language.
#include <limits.h>
#include <stdio.h>
#include <math.h>
typedef enum{false=0, true=1}bool;
const int max_lint = LONG_MAX;
typedef long long int lint;
#assert sizeof_long_long_int (LONG_MAX>=8) /* XXX */
/* the following line is the only time I have ever required "auto" */
#define FOR(i,iterator) auto bool lambda(i); yield_init = (void *)λ iterator; bool lambda(i)
#define DO {
#define YIELD(x) if(!yield(x))return
#define BREAK return false
#define CONTINUE return true
#define OD CONTINUE; }
/* Warning: _Most_ FOR(,){ } loops _must_ have a CONTINUE as the last statement.
* Otherwise the lambda will return random value from stack, and may terminate early */
typedef void iterator, lint_iterator; /* hint at procedure purpose */
static volatile void *yield_init; /* not thread safe */
#define YIELDS(type) bool (*yield)(type) = yield_init
typedef unsigned int bits;
#define ELEM(shift, bits) ( (bits >> shift) & 0b1 )
bits cache = 0b0, cached = 0b0;
const lint upb_cache = 8 * sizeof(cache);
lint_iterator decompose(lint); /* forward declaration */
bool is_prime(lint n){
bool has_factor = false, out = true;
/* for factor in decompose(n) do */
FOR(lint factor, decompose(n)){
if( has_factor ){ out = false; BREAK; }
has_factor = true;
CONTINUE;
}
return out;
}
bool is_prime_cached (lint n){
lint half_n = n / 2 - 2;
if( half_n <= upb_cache){
/* dont cache the initial four, nor the even numbers */
if (ELEM(half_n,cached)){
return ELEM(half_n,cache);
} else {
bool out = is_prime(n);
cache = cache | out << half_n;
cached = cached | 0b1 << half_n;
return out;
}
} else {
return is_prime(n);
}
}
lint_iterator primes (){
YIELDS(lint);
YIELD(2);
lint n = 3;
while( n < max_lint - 2 ){
YIELD(n);
n += 2;
while( n < max_lint - 2 && ! is_prime_cached(n) ){
n += 2;
}
}
}
lint_iterator decompose (lint in_n){
YIELDS(lint);
lint n = in_n;
/* for p in primes do */
FOR(lint p, primes()){
if( p*p > n ){
BREAK;
} else {
while( n % p == 0 ){
YIELD(p);
n = n / p;
}
}
CONTINUE;
}
if( n > 1 ){
YIELD(n);
}
}
main(){
FOR(lint m, primes()){
lint p = powl(2, m) - 1;
printf("2**%lld-1 = %lld, with factors:",m,p);
FOR(lint factor, decompose(p)){
printf(" %lld",factor);
fflush(stdout);
CONTINUE;
}
printf("\n",m);
if( m >= 59 )BREAK;
CONTINUE;
}
}
- Output:
2**2-1 = 3, with factors: 3 2**3-1 = 7, with factors: 7 2**5-1 = 31, with factors: 31 2**7-1 = 127, with factors: 127 2**11-1 = 2047, with factors: 23 89 2**13-1 = 8191, with factors: 8191 2**17-1 = 131071, with factors: 131071 2**19-1 = 524287, with factors: 524287 2**23-1 = 8388607, with factors: 47 178481 2**29-1 = 536870911, with factors: 233 1103 2089 2**31-1 = 2147483647, with factors: 2147483647 2**37-1 = 137438953471, with factors: 223 616318177 2**41-1 = 2199023255551, with factors: 13367 164511353 2**43-1 = 8796093022207, with factors: 431 9719 2099863 2**47-1 = 140737488355327, with factors: 2351 4513 13264529 2**53-1 = 9007199254740991, with factors: 6361 69431 20394401 2**59-1 = 576460752303423487, with factors: 179951 3203431780337
Note: gcc took 487,719 BogoMI to complete the example.
To understand what was going on with the above code, pass it through cpp and read the outcome. Translated into normal C code sans the function call overhead, it's really this (the following uses a adjustable cache, although setting it beyond a few thousands doesn't gain further benefit):
#include <stdio.h>
#include <stdlib.h>
#include <stdint.h>
typedef uint32_t pint;
typedef uint64_t xint;
typedef unsigned int uint;
int is_prime(xint);
inline int next_prime(pint p)
{
if (p == 2) return 3;
for (p += 2; p > 1 && !is_prime(p); p += 2);
if (p == 1) return 0;
return p;
}
int is_prime(xint n)
{
# define NCACHE 256
# define S (sizeof(uint) * 2)
static uint cache[NCACHE] = {0};
pint p = 2;
int ofs, bit = -1;
if (n < NCACHE * S) {
ofs = n / S;
bit = 1 << ((n & (S - 1)) >> 1);
if (cache[ofs] & bit) return 1;
}
do {
if (n % p == 0) return 0;
if (p * p > n) break;
} while ((p = next_prime(p)));
if (bit != -1) cache[ofs] |= bit;
return 1;
}
int decompose(xint n, pint *out)
{
int i = 0;
pint p = 2;
while (n > p * p) {
while (n % p == 0) {
out[i++] = p;
n /= p;
}
if (!(p = next_prime(p))) break;
}
if (n > 1) out[i++] = n;
return i;
}
int main()
{
int i, j, len;
xint z;
pint out[100];
for (i = 2; i < 64; i = next_prime(i)) {
z = (1ULL << i) - 1;
printf("2^%d - 1 = %llu = ", i, z);
fflush(stdout);
len = decompose(z, out);
for (j = 0; j < len; j++)
printf("%u%s", out[j], j < len - 1 ? " x " : "\n");
}
return 0;
}
Version 2
typedef unsigned long long int ulong; // define a type that represent the limit (64-bit)
ulong mod_mul(ulong a, ulong b, const ulong mod) {
ulong res = 0, c; // return (a * b) % mod, avoiding overflow errors while doing modular multiplication.
for (b %= mod; a; a & 1 ? b >= mod - res ? res -= mod : 0, res += b : 0, a >>= 1, (c = b) >= mod - b ? c -= mod : 0, b += c);
return res % mod;
}
ulong mod_pow(ulong n, ulong exp, const ulong mod) {
ulong res = 1; // return (n ^ exp) % mod
for (n %= mod; exp; exp & 1 ? res = mod_mul(res, n, mod) : 0, n = mod_mul(n, n, mod), exp >>= 1);
return res;
}
ulong square_root(const ulong N) {
ulong res = 0, rem = N, c, d;
for (c = 1 << 62; c; c >>= 2) {
d = res + c;
res >>= 1;
if (rem >= d)
rem -= d, res += c;
} // returns the square root of N.
return res;
}
int is_prime(const ulong N) {
ulong i = 1; // return a truthy value about the primality of N.
if (N > 1) for (; i < 64 && mod_pow(i, N - 1, N) <= 1; ++i);
return i == 64;
}
ulong pollard_rho(const ulong N) {
// Require : N is a composite number, not a square.
// Ensure : res is a non-trivial factor of N.
// Option : change the timeout, change the rand function.
static const int timeout = 18;
static unsigned long long rand_val = 2994439072U;
rand_val = (rand_val * 1025416097U + 286824428U) % 4294967291LLU;
ulong res = 1, a, b, c, i = 0, j = 1, x = 1, y = 1 + rand_val % (N - 1);
for (; res == 1; ++i) {
if (i == j) {
if (j >> timeout)
break;
j <<= 1;
x = y;
}
a = y, b = y; // performs y = (y * y) % N
for (y = 0; a; a & 1 ? b >= N - y ? y -= N : 0, y += b : 0, a >>= 1, (c = b) >= N - b ? c -= N : 0, b += c);
y = (1 + y) % N;
for (a = y > x ? y - x : x - y, b = N; (a %= b) && (b %= a);); // compute the gcd(abs(y - x), N);
res = a | b;
}
return res;
}
void factor(const ulong N, ulong *array) {
// very basic manager that fill the given array (the size of the result is the first array element)
// it does not perform initial trial divisions, which is generally highly recommended.
if (N < 4 || is_prime(N)) {
if (N > 1 || !*array) array[++*array] = N;
return;
}
ulong x = square_root(N);
if (x * x != N) x = pollard_rho(N);
factor(x, array);
factor(N / x, array);
}
#include <stdio.h>
int main(void) {
// simple test.
unsigned long long n = 18446744073709551615U;
ulong fac[65] = {0};
factor(n, fac);
for (ulong i = 1; i <= *fac; ++i)
printf("* %llu\n", fac[i]);
}
C#
using System;
using System.Collections.Generic;
namespace PrimeDecomposition
{
class Program
{
static void Main(string[] args)
{
GetPrimes(12);
}
static List<int> GetPrimes(decimal n)
{
List<int> storage = new List<int>();
while (n > 1)
{
int i = 1;
while (true)
{
if (IsPrime(i))
{
if (((decimal)n / i) == Math.Round((decimal) n / i))
{
n /= i;
storage.Add(i);
break;
}
}
i++;
}
}
return storage;
}
static bool IsPrime(int n)
{
if (n <= 1) return false;
for (int i = 2; i <= Math.Sqrt(n); i++)
if (n % i == 0) return false;
return true;
}
}
}
Simple trial division
This version a translation from Java of the sample presented by Robert C. Martin during a TDD talk at NDC 2011.
Although this three-line algorithm does not mention anything about primes, the fact that factors are taken out of the number n in ascending order garantees the list will only contain primes.
using System.Collections.Generic;
namespace PrimeDecomposition
{
public class Primes
{
public List<int> FactorsOf(int n)
{
var factors = new List<int>();
for (var divisor = 2; n > 1; divisor++)
for (; n % divisor == 0; n /= divisor)
factors.Add(divisor);
return factors;
}
}
C++
#include <iostream>
#include <gmpxx.h>
// This function template works for any type representing integers or
// nonnegative integers, and has the standard operator overloads for
// arithmetic and comparison operators, as well as explicit conversion
// from int.
//
// OutputIterator must be an output iterator with value_type Integer.
// It receives the prime factors.
template<typename Integer, typename OutputIterator>
void decompose(Integer n, OutputIterator out)
{
Integer i(2);
while (n != 1)
{
while (n % i == Integer(0))
{
*out++ = i;
n /= i;
}
++i;
}
}
// this is an output iterator similar to std::ostream_iterator, except
// that it outputs the separation string *before* the value, but not
// before the first value (i.e. it produces an infix notation).
template<typename T> class infix_ostream_iterator:
public std::iterator<T, std::output_iterator_tag>
{
class Proxy;
friend class Proxy;
class Proxy
{
public:
Proxy(infix_ostream_iterator& iter): iterator(iter) {}
Proxy& operator=(T const& value)
{
if (!iterator.first)
{
iterator.stream << iterator.infix;
}
iterator.stream << value;
}
private:
infix_ostream_iterator& iterator;
};
public:
infix_ostream_iterator(std::ostream& os, char const* inf):
stream(os),
first(true),
infix(inf)
{
}
infix_ostream_iterator& operator++() { first = false; return *this; }
infix_ostream_iterator operator++(int)
{
infix_ostream_iterator prev(*this);
++*this;
return prev;
}
Proxy operator*() { return Proxy(*this); }
private:
std::ostream& stream;
bool first;
char const* infix;
};
int main()
{
std::cout << "please enter a positive number: ";
mpz_class number;
std::cin >> number;
if (number <= 0)
std::cout << "this number is not positive!\n;";
else
{
std::cout << "decomposition: ";
decompose(number, infix_ostream_iterator<mpz_class>(std::cout, " * "));
std::cout << "\n";
}
}
Simple trial division
// Factorization by trial division in C++11
#include <iostream>
#include <vector>
using long_pair = std::pair<long,long>;
using lp_vec = std::vector<long_pair>;
lp_vec factorize(long n)
{
lp_vec fs;
int cnt = 0;
for (;n%2==0; n/=2) cnt++; // optimized by compiler
if (cnt > 0)
fs.push_back({2, cnt});
for (long i=3; i*i<=n; i+=2) {
cnt = 0;
for (;n%i==0; n/=i) cnt++;
if (cnt>0)
fs.push_back({i, cnt});
}
if (n>1)
fs.push_back({n, 1});
return fs;
}
int main() {
long n;
std::cin >> n;
auto fs = factorize(n);
for (auto fp : fs) {
std::cout << fp.first << "^" << fp.second << "\n";
}
return 0;
}
Clojure
;;; No stack consuming algorithm
(defn factors
"Return a list of factors of N."
([n]
(factors n 2 ()))
([n k acc]
(if (= 1 n)
acc
(if (= 0 (rem n k))
(recur (quot n k) k (cons k acc))
(recur n (inc k) acc)))))
Common Lisp
;;; Recursive algorithm
(defun factor (n)
"Return a list of factors of N."
(when (> n 1)
(loop with max-d = (isqrt n)
for d = 2 then (if (evenp d) (+ d 1) (+ d 2)) do
(cond ((> d max-d) (return (list n))) ; n is prime
((zerop (rem n d)) (return (cons d (factor (truncate n d)))))))))
;;; Tail-recursive version
(defun factor (n &optional (acc '()))
(when (> n 1) (loop with max-d = (isqrt n)
for d = 2 then (if (evenp d) (1+ d) (+ d 2)) do
(cond ((> d max-d) (return (cons (list n 1) acc)))
((zerop (rem n d))
(return (factor (truncate n d) (if (eq d (caar acc))
(cons
(list (caar acc) (1+ (cadar acc)))
(cdr acc))
(cons (list d 1) acc)))))))))
D
import std.stdio, std.bigint, std.algorithm, std.traits, std.range;
Unqual!T[] decompose(T)(in T number) pure nothrow
in {
assert(number > 1);
} body {
typeof(return) result;
Unqual!T n = number;
for (Unqual!T i = 2; n % i == 0; n /= i)
result ~= i;
for (Unqual!T i = 3; n >= i * i; i += 2)
for (; n % i == 0; n /= i)
result ~= i;
if (n != 1)
result ~= n;
return result;
}
void main() {
writefln("%(%s\n%)", iota(2, 10).map!decompose);
decompose(1023 * 1024).writeln;
BigInt(2 * 3 * 5 * 7 * 11 * 11 * 13 * 17).decompose.writeln;
decompose(16860167264933UL.BigInt * 179951).writeln;
decompose(2.BigInt ^^ 100_000).group.writeln;
}
- Output:
[2] [3] [2, 2] [5] [2, 3] [7] [2, 2, 2] [3, 3] [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 11, 31] [2, 3, 5, 7, 11, 11, 13, 17] [179951, 16860167264933] [Tuple!(BigInt, uint)(2, 100000)]
Delphi
program Prime_decomposition;
{$APPTYPE CONSOLE}
uses
System.SysUtils;
function IsPrime(n: UInt64): Boolean;
var
i: Integer;
begin
if n <= 1 then
exit(False);
i := 2;
while i < Sqrt(n) do
begin
if n mod i = 0 then
exit(False);
inc(i);
end;
Result := True;
end;
function GetPrimes(n: UInt64): TArray<UInt64>;
var
i: Integer;
begin
while n > 1 do
begin
i := 1;
while True do
begin
if IsPrime(i) then
begin
if n / i = (round(n / i)) then
begin
n := n div i;
SetLength(Result, Length(Result) + 1);
Result[High(Result)] := i;
Break;
end;
end;
inc(i);
end;
end;
end;
begin
for var v in GetPrimes(12) do
write(v, ' ');
readln;
end.
E
This example assumes a function isPrime and was tested with this one. It could use a self-referential implementation such as the Python task, but the original author of this example did not like the ordering dependency involved.
def primes := {
var primesCache := [2]
/** A collection of all prime numbers. */
def primes {
to iterate(f) {
primesCache.iterate(f)
for x in (int > primesCache.last()) {
if (isPrime(x)) {
f(primesCache.size(), x)
primesCache with= x
}
}
}
}
}
def primeDecomposition(var x :(int > 0)) {
var factors := []
for p in primes {
while (x % p <=> 0) {
factors with= p
x //= p
}
if (x <=> 1) {
break
}
}
return factors
}EasyLang
func[] decompose num .
t = 2
while t * t <= num
if num mod t = 0
primes[] &= t
num = num / t
else
t += 1
.
.
primes[] &= num
return primes[]
.
for v in [ 27 1232125 9007199254740991 ]
print decompose v
.- Output:
[ 3 3 3 ] [ 5 5 5 9857 ] [ 6361 69431 20394401 ]
EchoLisp
The built-in prime-factors function performs the task.
(prime-factors 1024)
→ (2 2 2 2 2 2 2 2 2 2)
(lib 'bigint)
;; 2^59 - 1
(prime-factors (1- (expt 2 59)))
→ (179951 3203431780337)
(prime-factors 100000000000000000037)
→ (31 821 66590107 59004541)
Eiffel
Uses the feature prime from the Task Primality by Trial Devision in the contract to check if the Result contains only prime numbers.
class
PRIME_DECOMPOSITION
feature
factor (p: INTEGER): ARRAY [INTEGER]
-- Prime decomposition of 'p'.
require
p_positive: p > 0
local
div, i, next, rest: INTEGER
do
create Result.make_empty
if p = 1 then
Result.force (1, 1)
end
div := 2
next := 3
rest := p
from
i := 1
until
rest = 1
loop
from
until
rest \\ div /= 0
loop
Result.force (div, i)
rest := (rest / div).floor
i := i + 1
end
div := next
next := next + 2
end
ensure
is_divisor: across Result as r all p \\ r.item = 0 end
is_prime: across Result as r all prime (r.item) end
end
The test was done in an application class. (Similar as in other Eiffel examples (ex. Selectionsort).)
factor(5000)
- Output:
2x2x2x5x5x5x5
Ela
open integer //arbitrary sized integers
decompose_prime n = loop n 2I
where
loop c p | c < (p * p) = [c]
| c % p == 0I = p :: (loop (c / p) p)
| else = loop c (p + 1I)
decompose_prime 600851475143I- Output:
[71,839,1471,6857]
Elm
module Main exposing (main)
import Html exposing (Html, div, h1, text)
import Html.Attributes exposing (style)
-- See live:
-- <nowiki>https://ellie-app.com/pMYxVPQ4fvca1</nowiki>
accumulator : List Int
accumulator =
[]
compositeNr = 84894624407
ts =
showFactors compositeNr 2 accumulator
main =
div
[ style "margin" "5%"
, style "font-size" "1.5em"
, style "color" "blue"
]
[ h1 [] [ text "Prime Factorizer" ]
, text
("Prime factors: "
++ listAsString ts
++ " from number "
++ String.fromInt (List.product ts)
)
]
showFactors : Int -> Int -> List Int -> List Int
showFactors number factor acc =
if number < 2 then
acc
-- returns the final result if number < 2
else if
modBy factor number == 0
-- modulo used to get prime factors
then let
v2 : List Int
v2 =
factor :: acc
number2 : Int
number2 =
number // factor
in
showFactors number2 factor v2
-- recursive call
-- this modulus function is used
-- in order to output factor !=2
else
let
factor2 : Int
factor2 =
factor + 1
in
showFactors number factor2 acc
listAsString : List Int -> String
listAsString myList =
List.map String.fromInt myList
|> List.map (\el -> " " ++ el)
|> List.foldl (++) " "
- Output:
Prime factors: 3067 4357 6353 from number 84894624407 Prime factors: 2^30 3 from number 3221225472
Elixir
defmodule Prime do
def decomposition(n), do: decomposition(n, 2, [])
defp decomposition(n, k, acc) when n < k*k, do: Enum.reverse(acc, [n])
defp decomposition(n, k, acc) when rem(n, k) == 0, do: decomposition(div(n, k), k, [k | acc])
defp decomposition(n, k, acc), do: decomposition(n, k+1, acc)
end
prime = Stream.iterate(2, &(&1+1)) |>
Stream.filter(fn n-> length(Prime.decomposition(n)) == 1 end) |>
Enum.take(17)
mersenne = Enum.map(prime, fn n -> {n, round(:math.pow(2,n)) - 1} end)
Enum.each(mersenne, fn {n,m} ->
:io.format "~3s :~20w = ~s~n", ["M#{n}", m, Prime.decomposition(m) |> Enum.join(" x ")]
end)
- Output:
M2 : 3 = 3 M3 : 7 = 7 M5 : 31 = 31 M7 : 127 = 127 M11 : 2047 = 23 x 89 M13 : 8191 = 8191 M17 : 131071 = 131071 M19 : 524287 = 524287 M23 : 8388607 = 47 x 178481 M29 : 536870911 = 233 x 1103 x 2089 M31 : 2147483647 = 2147483647 M37 : 137438953471 = 223 x 616318177 M41 : 2199023255551 = 13367 x 164511353 M43 : 8796093022207 = 431 x 9719 x 2099863 M47 : 140737488355327 = 2351 x 4513 x 13264529 M53 : 9007199254740991 = 6361 x 69431 x 20394401 M59 : 576460752303423487 = 179951 x 3203431780337
Erlang
% no stack consuming version
factors(N) ->
factors(N,2,[]).
factors(1,_,Acc) -> Acc;
factors(N,K,Acc) when N < K*K -> [N|Acc];
factors(N,K,Acc) when N rem K == 0 ->
factors(N div K,K, [K|Acc]);
factors(N,K,Acc) ->
factors(N,K+1,Acc).
ERRE
PROGRAM DECOMPOSE
!
! for rosettacode.org
!
!VAR NUM,J
DIM PF[100]
PROCEDURE STORE_FACTOR
PF[0]=PF[0]+1
PF[PF[0]]=CA
I=I/CA
END PROCEDURE
PROCEDURE DECOMP(I)
PF[0]=0 CA=2 ! special case
LOOP
IF I=1 THEN EXIT PROCEDURE END IF
EXIT IF INT(I/CA)*CA<>I
STORE_FACTOR
END LOOP
FOR CA=3 TO INT(SQR(I)) STEP 2 DO
LOOP
IF I=1 THEN EXIT PROCEDURE END IF
EXIT IF INT(I/CA)*CA<>I
STORE_FACTOR
END LOOP
END FOR
IF I>1 THEN CA=I STORE_FACTOR END IF
END PROCEDURE
BEGIN
! ----- function generate
! in ... I ... number
! out ... PF[] ... factors
! PF[0] ... # of factors
! mod ... CA ... pr.fact. candidate
PRINT(CHR$(12);) !CLS
INPUT("Numero ",NUM)
DECOMP(NUM)
PRINT(NUM;"=";)
FOR J=1 TO PF[0] DO
PRINT(PF[J];)
END FOR
PRINT
END PROGRAMEzhil
## இந்த நிரல் தரப்பட்ட எண்ணின் பகாஎண் கூறுகளைக் கண்டறியும்
நிரல்பாகம் பகாஎண்ணா(எண்1)
## இந்த நிரல்பாகம் தரப்பட்ட எண் பகு எண்ணா அல்லது பகா எண்ணா என்று கண்டறிந்து சொல்லும்
## பகுஎண் என்றால் 0 திரும்பத் தரப்படும்
## பகாஎண் என்றால் 1 திரும்பத் தரப்படும்
@(எண்1 < 0) ஆனால்
## எதிர்மறை எண்களை நேராக்குதல்
எண்1 = எண்1 * (-1)
முடி
@(எண்1 < 2) ஆனால்
## பூஜ்ஜியம், ஒன்று ஆகியவை பகா எண்கள் அல்ல
பின்கொடு 0
முடி
@(எண்1 == 2) ஆனால்
## இரண்டு என்ற எண் ஒரு பகா எண்
பின்கொடு 1
முடி
மீதம் = எண்1%2
@(மீதம் == 0) ஆனால்
## இரட்டைப்படை எண், ஆகவே, இது பகா எண் அல்ல
பின்கொடு 0
முடி
எண்1வர்க்கமூலம் = எண்1^0.5
@(எண்2 = 3, எண்2 <= எண்1வர்க்கமூலம், எண்2 = எண்2 + 2) ஆக
மீதம்1 = எண்1%எண்2
@(மீதம்1 == 0) ஆனால்
## ஏதேனும் ஓர் எண்ணால் முழுமையாக வகுபட்டுவிட்டது, ஆகவே அது பகா எண் அல்ல
பின்கொடு 0
முடி
முடி
பின்கொடு 1
முடி
நிரல்பாகம் பகுத்தெடு(எண்1)
## இந்த எண் தரப்பட்ட எண்ணின் பகா எண் கூறுகளைக் கண்டறிந்து பட்டியல் இடும்
கூறுகள் = பட்டியல்()
@(எண்1 < 0) ஆனால்
## எதிர்மறை எண்களை நேராக்குதல்
எண்1 = எண்1 * (-1)
முடி
@(எண்1 <= 1) ஆனால்
## ஒன்று அல்லது அதற்குக் குறைவான எண்களுக்குப் பகா எண் விகிதம் கண்டறியமுடியாது
பின்கொடு கூறுகள்
முடி
@(பகாஎண்ணா(எண்1) == 1) ஆனால்
## தரப்பட்ட எண்ணே பகா எண்ணாக அமைந்துவிட்டால், அதற்கு அதுவே பகாஎண் கூறு ஆகும்
பின்இணை(கூறுகள், எண்1)
பின்கொடு கூறுகள்
முடி
தாற்காலிகஎண் = எண்1
எண்2 = 2
@(எண்2 <= தாற்காலிகஎண்) வரை
விடை1 = பகாஎண்ணா(எண்2)
மீண்டும்தொடங்கு = 0
@(விடை1 == 1) ஆனால்
விடை2 = தாற்காலிகஎண்%எண்2
@(விடை2 == 0) ஆனால்
## பகா எண்ணால் முழுமையாக வகுபட்டுள்ளது, அதனைப் பட்டியலில் இணைக்கிறோம்
பின்இணை(கூறுகள், எண்2)
தாற்காலிகஎண் = தாற்காலிகஎண்/எண்2
## மீண்டும் இரண்டில் தொடங்கி இதே கணக்கிடுதலைத் தொடரவேண்டும்
எண்2 = 2
மீண்டும்தொடங்கு = 1
முடி
முடி
@(மீண்டும்தொடங்கு == 0) ஆனால்
## அடுத்த எண்ணைத் தேர்ந்தெடுத்துக் கணக்கிடுதலைத் தொடரவேண்டும்
எண்2 = எண்2 + 1
முடி
முடி
பின்கொடு கூறுகள்
முடி
அ = int(உள்ளீடு("உங்களுக்குப் பிடித்த ஓர் எண்ணைத் தாருங்கள்: "))
பகாஎண்கூறுகள் = பட்டியல்()
பகாஎண்கூறுகள் = பகுத்தெடு(அ)
பதிப்பி "நீங்கள் தந்த எண்ணின் பகா எண் கூறுகள் இவை: ", பகாஎண்கூறுகள்
F#
let decompose_prime n =
let rec loop c p =
if c < (p * p) then [c]
elif c % p = 0I then p :: (loop (c/p) p)
else loop c (p + 1I)
loop n 2I
printfn "%A" (decompose_prime 600851475143I)
- Output:
[71; 839; 1471; 6857]
Factor
factors from the math.primes.factors vocabulary converts a number into a sequence of its prime divisors; the rest of the code prints this sequence.
USING: io kernel math math.parser math.primes.factors sequences ;
27720 factors
[ number>string ] map
" " join print ;
FALSE
[2[\$@$$*@>~][\$@$@$@$@\/*=$[%$." "$@\/\0~]?~[1+1|]?]#%.]d:
27720d;! {2 2 2 3 3 5 7 11}Forth
: decomp ( n -- )
2
begin 2dup dup * >=
while 2dup /mod swap
if drop 1+ 1 or \ next odd number
else -rot nip dup .
then
repeat
drop . ;
Fortran
module PrimeDecompose
implicit none
integer, parameter :: huge = selected_int_kind(18)
! => integer(8) ... more fails on my 32 bit machine with gfortran(gcc) 4.3.2
contains
subroutine find_factors(n, d)
integer(huge), intent(in) :: n
integer, dimension(:), intent(out) :: d
integer(huge) :: div, next, rest
integer :: i
i = 1
div = 2; next = 3; rest = n
do while ( rest /= 1 )
do while ( mod(rest, div) == 0 )
d(i) = div
i = i + 1
rest = rest / div
end do
div = next
next = next + 2
end do
end subroutine find_factors
end module PrimeDecompose
program Primes
use PrimeDecompose
implicit none
integer, dimension(100) :: outprimes
integer i
outprimes = 0
call find_factors(12345649494449_huge, outprimes)
do i = 1, 100
if ( outprimes(i) == 0 ) exit
print *, outprimes(i)
end do
end program Primes
Frink
Frink has a built-in factoring function which uses wheel factoring, trial division, Pollard p-1 factoring, and Pollard rho factoring. It also recognizes some special forms (e.g. Mersenne numbers) and handles them efficiently.
println[factor[2^508-1]]- Output:
(total process time including JVM startup = 1.515 s)
[[3, 1], [5, 1], [509, 1], [18797, 1], [26417, 1], [72118729, 1], [140385293, 1], [2792688414613, 1], [8988357880501, 1], [90133566917913517709497, 1], [56713727820156410577229101238628035243, 1], [170141183460469231731687303715884105727, 1]]
Note that this means 31 * 51 * ...
GAP
Built-in function :
FactorsInt(2^67-1);
# [ 193707721, 761838257287 ]
Or using the FactInt package :
FactInt(2^67-1);
# [ [ 193707721, 761838257287 ], [ ] ]
Go
package main
import (
"fmt"
"math/big"
)
var (
ZERO = big.NewInt(0)
ONE = big.NewInt(1)
)
func Primes(n *big.Int) []*big.Int {
res := []*big.Int{}
mod, div := new(big.Int), new(big.Int)
for i := big.NewInt(2); i.Cmp(n) != 1; {
div.DivMod(n, i, mod)
for mod.Cmp(ZERO) == 0 {
res = append(res, new(big.Int).Set(i))
n.Set(div)
div.DivMod(n, i, mod)
}
i.Add(i, ONE)
}
return res
}
func main() {
vals := []int64{
1 << 31,
1234567,
333333,
987653,
2 * 3 * 5 * 7 * 11 * 13 * 17,
}
for _, v := range vals {
fmt.Println(v, "->", Primes(big.NewInt(v)))
}
}
- Output:
2147483648 -> [2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2] 1234567 -> [127 9721] 333333 -> [3 3 7 11 13 37] 987653 -> [29 34057] 510510 -> [2 3 5 7 11 13 17]
Groovy
This solution uses the fact that a given factor must be prime if no smaller factor divides it evenly, so it does not require an "isPrime-like function", assumed or otherwise.
def factorize = { long target ->
if (target == 1) return [1L]
if (target < 4) return [1L, target]
def targetSqrt = Math.sqrt(target)
def lowfactors = (2L..targetSqrt).findAll { (target % it) == 0 }
if (lowfactors == []) return [1L, target]
def nhalf = lowfactors.size() - ((lowfactors[-1]**2 == target) ? 1 : 0)
[1] + lowfactors + (0..<nhalf).collect { target.intdiv(lowfactors[it]) }.reverse() + [target]
}
def decomposePrimes = { target ->
def factors = factorize(target) - [1]
def primeFactors = []
factors.eachWithIndex { f, i ->
if (i==0 || factors[0..<i].every {f % it != 0}) {
primeFactors << f
def pfPower = f*f
while (target % pfPower == 0) {
primeFactors << f
pfPower *= f
}
}
}
primeFactors
}
- Test #1:
((1..30) + [97*4, 1000, 1024, 333333]).each { println ([number:it, primes:decomposePrimes(it)]) }
- Output #1:
[number:1, primes:[]] [number:2, primes:[2]] [number:3, primes:[3]] [number:4, primes:[2, 2]] [number:5, primes:[5]] [number:6, primes:[2, 3]] [number:7, primes:[7]] [number:8, primes:[2, 2, 2]] [number:9, primes:[3, 3]] [number:10, primes:[2, 5]] [number:11, primes:[11]] [number:12, primes:[2, 2, 3]] [number:13, primes:[13]] [number:14, primes:[2, 7]] [number:15, primes:[3, 5]] [number:16, primes:[2, 2, 2, 2]] [number:17, primes:[17]] [number:18, primes:[2, 3, 3]] [number:19, primes:[19]] [number:20, primes:[2, 2, 5]] [number:21, primes:[3, 7]] [number:22, primes:[2, 11]] [number:23, primes:[23]] [number:24, primes:[2, 2, 2, 3]] [number:25, primes:[5, 5]] [number:26, primes:[2, 13]] [number:27, primes:[3, 3, 3]] [number:28, primes:[2, 2, 7]] [number:29, primes:[29]] [number:30, primes:[2, 3, 5]] [number:388, primes:[2, 2, 97]] [number:1000, primes:[2, 2, 2, 5, 5, 5]] [number:1024, primes:[2, 2, 2, 2, 2, 2, 2, 2, 2, 2]] [number:333333, primes:[3, 3, 7, 11, 13, 37]]
- Test #2:
def isPrime = {factorize(it).size() == 2}
(1..60).step(2).findAll(isPrime).each { println ([number:"2**${it}-1", value:2**it-1, primes:decomposePrimes(2**it-1)]) }
- Output #2:
[number:2**3-1, value:7, primes:[7]] [number:2**5-1, value:31, primes:[31]] [number:2**7-1, value:127, primes:[127]] [number:2**11-1, value:2047, primes:[23, 89]] [number:2**13-1, value:8191, primes:[8191]] [number:2**17-1, value:131071, primes:[131071]] [number:2**19-1, value:524287, primes:[524287]] [number:2**23-1, value:8388607, primes:[47, 178481]] [number:2**29-1, value:536870911, primes:[233, 1103, 2089]] [number:2**31-1, value:2147483647, primes:[2147483647]] [number:2**37-1, value:137438953471, primes:[223, 616318177]] [number:2**41-1, value:2199023255551, primes:[13367, 164511353]] [number:2**43-1, value:8796093022207, primes:[431, 9719, 2099863]] [number:2**47-1, value:140737488355327, primes:[2351, 4513, 13264529]] [number:2**53-1, value:9007199254740991, primes:[6361, 69431, 20394401]] [number:2**59-1, value:576460752303423487, primes:[179951, 3203431780337]]
Perhaps a more sophisticated algorithm is in order. It took well over 1 hour to calculate the last three decompositions using this solution.
Haskell
The task description hints at using the isPrime function from the trial division task:
factorize n = [ d | p <- [2..n], isPrime p, d <- divs n p ]
-- [2..n] >>= (\p-> [p|isPrime p]) >>= divs n
where
divs n p | rem n p == 0 = p : divs (quot n p) p
| otherwise = []
but it is not very efficient, to put it mildly. Inlining and fusing gets us the progressively more optimized
import Data.Maybe (listToMaybe)
import Data.List (unfoldr)
factorize :: Integer -> [Integer]
factorize n
= unfoldr (\n -> listToMaybe [(x, div n x) | x <- [2..n], mod n x==0]) n
= unfoldr (\(d,n) -> listToMaybe [(x, (x, div n x)) | x <- [d..n], mod n x==0]) (2,n)
= unfoldr (\(d,n) -> listToMaybe [(x, (x, div n x)) | x <-
takeWhile ((<=n).(^2)) [d..] ++ [n|n>1], mod n x==0]) (2,n)
= unfoldr (\(ds,n) -> listToMaybe [(x, (dropWhile (< x) ds, div n x)) | n>1, x <-
takeWhile ((<=n).(^2)) ds ++ [n|n>1], mod n x==0]) (primesList,n)
The library function listToMaybe gets at most one element from its list argument. The last variant can be written as the optimal
factorize n = divs n primesList
where
divs n ds@(d:t) | d*d > n = [n | n > 1]
| r == 0 = d : divs q ds
| otherwise = divs n t
where (q,r) = quotRem n d
See Sieve of Eratosthenes or Primality by trial division for a source of primes to use with this function.
Actually as some other entries notice, with any ascending order list containing all primes (e.g. 2:[3,5..]) used in place of primesList, the factors found by this function are guaranteed to be prime, so no separate testing for primality is strictly needed; however using just primes is more efficient, if we already have them.
- Output:
λ> mapM_ (print . factorize) $ take 11 [123123451..] [11,41,273001] [2,2,17,53,127,269] [3,229,277,647] [2,61561727] [5,7,13,270601] [2,2,2,2,2,2,2,2,3,3,3,47,379] [37,109,30529] [2,19,97,33403] [3,3167,12959] [2,2,5,6156173] [123123461]
Icon and Unicon
Uses genfactors and prime from factors
Sample Output showing factors of a large integer:
8796093022207=431*9719*2099863
J
q:
- Example use:
q: 3684
2 2 3 307
and, more elaborately:
_1+2^128x
340282366920938463463374607431768211455
q: _1+2^128x
3 5 17 257 641 65537 274177 6700417 67280421310721
*/ q: _1+2^128x
340282366920938463463374607431768211455
Java
This is a version for arbitrary-precision integers which assumes the existence of a function with the signature:
public boolean prime(BigInteger i);
You will need to import java.util.List, java.util.LinkedList, and java.math.BigInteger.
public static List<BigInteger> primeFactorBig(BigInteger a){
List<BigInteger> ans = new LinkedList<BigInteger>();
//loop until we test the number itself or the number is 1
for (BigInteger i = BigInteger.valueOf(2); i.compareTo(a) <= 0 && !a.equals(BigInteger.ONE);
i = i.add(BigInteger.ONE)){
while (a.remainder(i).equals(BigInteger.ZERO) && prime(i)) { //if we have a prime factor
ans.add(i); //put it in the list
a = a.divide(i); //factor it out of the number
}
}
return ans;
}
Alternate version, optimised to be faster.
private static final BigInteger two = BigInteger.valueOf(2);
public List<BigInteger> primeDecomp(BigInteger a) {
// impossible for values lower than 2
if (a.compareTo(two) < 0) {
return null;
}
//quickly handle even values
List<BigInteger> result = new ArrayList<BigInteger>();
while (a.and(BigInteger.ONE).equals(BigInteger.ZERO)) {
a = a.shiftRight(1);
result.add(two);
}
//left with odd values
if (!a.equals(BigInteger.ONE)) {
BigInteger b = BigInteger.valueOf(3);
while (b.compareTo(a) < 0) {
if (b.isProbablePrime(10)) {
BigInteger[] dr = a.divideAndRemainder(b);
if (dr[1].equals(BigInteger.ZERO)) {
result.add(b);
a = dr[0];
}
}
b = b.add(two);
}
result.add(b); //b will always be prime here...
}
return result;
}
Another alternate version designed to make fewer modular calculations:
private static final BigInteger TWO = BigInteger.valueOf(2);
private static final BigInteger THREE = BigInteger.valueOf(3);
private static final BigInteger FIVE = BigInteger.valueOf(5);
public static ArrayList<BigInteger> primeDecomp(BigInteger n){
if(n.compareTo(TWO) < 0) return null;
ArrayList<BigInteger> factors = new ArrayList<BigInteger>();
// handle even values
while(n.and(BigInteger.ONE).equals(BigInteger.ZERO)){
n = n.shiftRight(1);
factors.add(TWO);
}
// handle values divisible by three
while(n.mod(THREE).equals(BigInteger.ZERO)){
factors.add(THREE);
n = n.divide(THREE);
}
// handle values divisible by five
while(n.mod(FIVE).equals(BigInteger.ZERO)){
factors.add(FIVE);
n = n.divide(FIVE);
}
// much like how we can skip multiples of two, we can also skip
// multiples of three and multiples of five. This increment array
// helps us to accomplish that
int[] pattern = {4,2,4,2,4,6,2,6};
int pattern_index = 0;
BigInteger current_test = BigInteger.valueOf(7);
while(!n.equals(BigInteger.ONE)){
while(n.mod(current_test).equals(BigInteger.ZERO)){
factors.add(current_test);
n = n.divide(current_test);
}
current_test = current_test.add(BigInteger.valueOf(pattern[pattern_index]));
pattern_index = (pattern_index + 1) & 7;
}
return factors;
}
Simple but very inefficient method, because it will test divisibility of all numbers from 2 to max prime factor. When decomposing a large prime number this will take O(n) trial divisions instead of more common O(log n).
public static List<BigInteger> primeFactorBig(BigInteger a){
List<BigInteger> ans = new LinkedList<BigInteger>();
for(BigInteger divisor = BigInteger.valueOf(2);
a.compareTo(ONE) > 0; divisor = divisor.add(ONE))
while(a.mod(divisor).equals(ZERO)){
ans.add(divisor);
a = a.divide(divisor);
}
return ans;
}
JavaScript
This code uses the BigInteger Library jsbn and jsbn2
function run_factorize(input, output) {
var n = new BigInteger(input.value, 10);
var TWO = new BigInteger("2", 10);
var divisor = new BigInteger("3", 10);
var prod = false;
if (n.compareTo(TWO) < 0)
return;
output.value = "";
while (true) {
var qr = n.divideAndRemainder(TWO);
if (qr[1].equals(BigInteger.ZERO)) {
if (prod)
output.value += "*";
else
prod = true;
output.value += "2";
n = qr[0];
}
else
break;
}
while (!n.equals(BigInteger.ONE)) {
var qr = n.divideAndRemainder(divisor);
if (qr[1].equals(BigInteger.ZERO)) {
if (prod)
output.value += "*";
else
prod = true;
output.value += divisor;
n = qr[0];
}
else
divisor = divisor.add(TWO);
}
}
Without any library.
function run_factorize(n) {
if (n <= 3)
return [n];
var ans = [];
var done = false;
while (!done) {
if (n % 2 === 0) {
ans.push(2);
n /= 2;
continue;
}
if (n % 3 === 0) {
ans.push(3);
n /= 3;
continue;
}
if (n === 1)
return ans;
var sr = Math.sqrt(n);
done = true;
// try to divide the checked number by all numbers till its square root.
for (var i = 6; i <= (sr + 6); i += 6) {
if (n % (i - 1) === 0) { // is n divisible by i-1?
ans.push((i - 1));
n /= (i - 1);
done = false;
break;
}
if (n % (i + 1) === 0) { // is n divisible by i+1?
ans.push((i + 1));
n /= (i + 1);
done = false;
break;
}
}
}
ans.push(n);
return ans;
}
TDD using Jasmine
PrimeFactors.js
function factors(n) {
if (!n || n < 2)
return [];
var f = [];
for (var i = 2; i <= n; i++){
while (n % i === 0){
f.push(i);
n /= i;
}
}
return f;
};
SpecPrimeFactors.js (with tag for Chutzpah)
/// <reference path="PrimeFactors.js" />
describe("Prime Factors", function() {
it("Given nothing, empty is returned", function() {
expect(factors()).toEqual([]);
});
it("Given 1, empty is returned", function() {
expect(factors(1)).toEqual([]);
});
it("Given 2, 2 is returned", function() {
expect(factors(2)).toEqual([2]);
});
it("Given 3, 3 is returned", function() {
expect(factors(3)).toEqual([3]);
});
it("Given 4, 2 and 2 is returned", function() {
expect(factors(4)).toEqual([2, 2]);
});
it("Given 5, 5 is returned", function() {
expect(factors(5)).toEqual([5]);
});
it("Given 6, 2 and 3 is returned", function() {
expect(factors(6)).toEqual([2, 3]);
});
it("Given 7, 7 is returned", function() {
expect(factors(7)).toEqual([7]);
});
it("Given 8; 2, 2, and 2 is returned", function() {
expect(factors(8)).toEqual([2, 2, 2]);
});
it("Given a large number, many primes factors are returned", function() {
expect(factors(2*2*2*3*3*7*11*17))
.toEqual([2, 2, 2, 3, 3, 7, 11, 17]);
});
it("Given a large prime number, that number is returned", function() {
expect(factors(997)).toEqual([997]);
});
});
jq
Works with gojq, the Go implementation of jq
`factors` as defined below emits a stream of all the prime factors of the input integer. The implementation is compact, fast and space-efficient: no space is required to store the primes or factors already computed, there is no reliance on an "is_prime" function, and square roots are only computed if needed.
The economy comes about through the use of the builtin filter recurse/1, and the use of the state vector: [p, n, valid, sqrt], where p is the candidate factor, n is the number still to be factored, valid is a flag, and sqrt is either null or the square root of n.
gojq supports unlimited-precision integer arithmetic, but the C implementation of jq currently uses IEEE 754 64-bit numbers, so using the latter, the following program will only be reliable for integers up to and including 9,007,199,254,740,992 (2^53). However, "factors" could be easily modified to work with a "BigInt" library for jq, such as BigInt.jq.
def factors:
. as $in
| [2, $in, false]
| recurse(
. as [$p, $q, $valid, $s]
| if $q == 1 then empty
elif $q % $p == 0 then [$p, $q/$p, true]
elif $p == 2 then [3, $q, false, $s]
else ($s // ($q | sqrt)) as $s
| if $p + 2 <= $s then [$p + 2, $q, false, $s]
else [$q, 1, true]
end
end )
| if .[2] then .[0] else empty end ;Examples:
24 | factors
#=> 2 2 2 3
[9007199254740992 | factors] | length
#=> 53
# 2**29-1 is 536870911
[ 536870911 | factors ]
#=> [233,1103,2089]Julia
using package Primes.jl:
julia> Pkg.add("Primes")
julia> factor(8796093022207)
[9719=>1,431=>1,2099863=>1]
(The factor function returns a dictionary
whose keys are the factors and whose values are the multiplicity of each factor.)
Kotlin
// version 1.0.6
import java.math.BigInteger
val bigTwo = BigInteger.valueOf(2L)
val bigThree = BigInteger.valueOf(3L)
fun getPrimeFactors(n: BigInteger): MutableList<BigInteger> {
val factors = mutableListOf<BigInteger>()
if (n < bigTwo) return factors
if (n.isProbablePrime(20)) {
factors.add(n)
return factors
}
var factor = bigTwo
var nn = n
while (true) {
if (nn % factor == BigInteger.ZERO) {
factors.add(factor)
nn /= factor
if (nn == BigInteger.ONE) return factors
if (nn.isProbablePrime(20)) factor = nn
}
else if (factor >= bigThree) factor += bigTwo
else factor = bigThree
}
}
fun main(args: Array<String>) {
val primes = intArrayOf(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97)
for (prime in primes) {
val bigPow2 = bigTwo.pow(prime) - BigInteger.ONE
println("2^${"%2d".format(prime)} - 1 = ${bigPow2.toString().padEnd(30)} => ${getPrimeFactors(bigPow2)}")
}
}
- Output:
2^ 2 - 1 = 3 => [3] 2^ 3 - 1 = 7 => [7] 2^ 5 - 1 = 31 => [31] 2^ 7 - 1 = 127 => [127] 2^11 - 1 = 2047 => [23, 89] 2^13 - 1 = 8191 => [8191] 2^17 - 1 = 131071 => [131071] 2^19 - 1 = 524287 => [524287] 2^23 - 1 = 8388607 => [47, 178481] 2^29 - 1 = 536870911 => [233, 1103, 2089] 2^31 - 1 = 2147483647 => [2147483647] 2^37 - 1 = 137438953471 => [223, 616318177] 2^41 - 1 = 2199023255551 => [13367, 164511353] 2^43 - 1 = 8796093022207 => [431, 9719, 2099863] 2^47 - 1 = 140737488355327 => [2351, 4513, 13264529] 2^53 - 1 = 9007199254740991 => [6361, 69431, 20394401] 2^59 - 1 = 576460752303423487 => [179951, 3203431780337] 2^61 - 1 = 2305843009213693951 => [2305843009213693951] 2^67 - 1 = 147573952589676412927 => [193707721, 761838257287] 2^71 - 1 = 2361183241434822606847 => [228479, 48544121, 212885833] 2^73 - 1 = 9444732965739290427391 => [439, 2298041, 9361973132609] 2^79 - 1 = 604462909807314587353087 => [2687, 202029703, 1113491139767] 2^83 - 1 = 9671406556917033397649407 => [167, 57912614113275649087721] 2^89 - 1 = 618970019642690137449562111 => [618970019642690137449562111] 2^97 - 1 = 158456325028528675187087900671 => [11447, 13842607235828485645766393]
Lambdatalk
{def prime_fact.smallest
{def prime_fact.smallest.r
{lambda {:q :r :i}
{if {and {> :r 0} {< :i :q}}
then {prime_fact.smallest.r :q {% :q {+ :i 1}} {+ :i 1}}
else :i}}}
{lambda {:q} {prime_fact.smallest.r :q {% :q 2} 2}}}
{def prime_fact
{def prime_fact.r
{lambda {:q :d}
{if {> :q 1}
then {let { {:q :q} {:d :d}
{:i {prime_fact.smallest :q}}}
{prime_fact.r {floor {/ :q :i}} {#.push! :d :i}} }
else {if {= {#.length :d} 1} then {b :d} else :d}}}}
{lambda {:n} :n:{prime_fact.r :n {#.new}}}}
{prime_fact {* 2 3 3 3 31 47 173}}
-> 13611294:[2,3,3,3,31,47,173]
{map prime_fact {serie 2 101}}
-> 2:[2] 3:[3] 4:[2,2] 5:[5] 6:[2,3] 7:[7] 8:[2,2,2] 9:[3,3] 10:[2,5] 11:[11] 12:[2,2,3] 13:[13] 14:[2,7] 15:[3,5]
16:[2,2,2,2] 17:[17] 18:[2,3,3] 19:[19] 20:[2,2,5] 21:[3,7] 22:[2,11] 23:[23] 24:[2,2,2,3] 25:[5,5] 26:[2,13] 27:[3,3,3]
28:[2,2,7] 29:[29] 30:[2,3,5] 31:[31] 32:[2,2,2,2,2] 33:[3,11] 34:[2,17] 35:[5,7] 36:[2,2,3,3] 37:[37] 38:[2,19] 39:[3,13]
40:[2,2,2,5] 41:[41] 42:[2,3,7] 43:[43] 44:[2,2,11] 45:[3,3,5] 46:[2,23] 47:[47] 48:[2,2,2,2,3] 49:[7,7] 50:[2,5,5] 51:[3,17]
52:[2,2,13] 53:[53] 54:[2,3,3,3] 55:[5,11] 56:[2,2,2,7] 57:[3,19] 58:[2,29] 59:[59] 60:[2,2,3,5] 61:[61] 62:[2,31] 63:[3,3,7]
64:[2,2,2,2,2,2] 65:[5,13] 66:[2,3,11] 67:[67] 68:[2,2,17] 69:[3,23] 70:[2,5,7] 71:[71] 72:[2,2,2,3,3] 73:[73] 74:[2,37]
75:[3,5,5] 76:[2,2,19] 77:[7,11] 78:[2,3,13] 79:[79] 80:[2,2,2,2,5] 81:[3,3,3,3] 82:[2,41] 83:[83] 84:[2,2,3,7] 85:[5,17]
86:[2,43] 87:[3,29] 88:[2,2,2,11] 89:[89] 90:[2,3,3,5] 91:[7,13] 92:[2,2,23] 93:[3,31] 94:[2,47] 95:[5,19] 96:[2,2,2,2,2,3]
97:[97] 98:[2,7,7] 99:[3,3,11] 100:[2,2,5,5] 101:[101]
LFE
(defun factors (n)
(factors n 2 '()))
(defun factors
((1 _ acc)
acc)
((n k acc) (when (== 0 (rem n k)))
(factors (div n k) k (cons k acc)))
((n k acc)
(factors n (+ k 1) acc)))
Lingo
-- Returns list of prime factors for given number.
-- To overcome the limits of integers (signed 32-bit in Lingo),
-- the number can be specified as float (which works up to 2^53).
-- For the same reason, values in returned list are floats, not integers.
on getPrimeFactors (n)
f = []
f.sort()
c = sqrt(n)
i = 1.0
repeat while TRUE
i=i+1
if i>c then exit repeat
check = n/i
if bitOr(check,0)=check then
f.add(i)
n = check
c = sqrt(n)
i = 1.0
end if
end repeat
f.add(n)
return f
endput getPrimeFactors(12)
-- [2.0000, 2.0000, 3.0000]
-- print floats without fractional digits
the floatPrecision=0
put getPrimeFactors(12)
-- [2, 2, 3]
put getPrimeFactors(1125899906842623.0)
-- [3, 251, 601, 4051, 614141]Logo
to decompose :n [:p 2]
if :p*:p > :n [output (list :n)]
if less? 0 modulo :n :p [output (decompose :n bitor 1 :p+1)]
output fput :p (decompose :n/:p :p)
endLua
The code of the used auxiliary function "IsPrime(n)" is located at Primality by trial division#Lua
function PrimeDecomposition( n )
local f = {}
if IsPrime( n ) then
f[1] = n
return f
end
local i = 2
repeat
while n % i == 0 do
f[#f+1] = i
n = n / i
end
repeat
i = i + 1
until IsPrime( i )
until n == 1
return f
end
M2000 Interpreter
Module Prime_decomposition {
Inventory Known1=2@, 3@
IsPrime=lambda Known1 (x as decimal) -> {
=0=1
if exist(Known1, x) then =1=1 : exit
if x<=5 OR frac(x) then {if x == 2 OR x == 3 OR x == 5 then Append Known1, x : =1=1
Break}
if frac(x/2) else exit
if frac(x/3) else exit
x1=sqrt(x):d = 5@
{if frac(x/d ) else exit
d += 2: if d>x1 then Append Known1, x : =1=1 : exit
if frac(x/d) else exit
d += 4: if d<= x1 else Append Known1, x : =1=1: exit
loop}
}
decompose=lambda IsPrime (n as decimal) -> {
Inventory queue Factors
{
k=2@
While frac(n/k)=0 {
n/=k
Append Factors, k
}
if n=1 then exit
k++
While frac(n/k)=0 {
n/=k
Append Factors, k
}
if n=1 then exit
{
k+=2
while not isprime(k) {k+=2}
While frac(n/k)=0 {
n/=k
Append Factors, k
}
if n=1 then exit
loop
}
}
=Factors
}
Data 10, 100, 12, 144, 496, 1212454
while not empty {
Print Decompose(Number)
}
}
Prime_decompositionMaple
Maple has two commands for integer factorization: ifactor, which returns results in a form resembling textbook presentation and ifactors, which returns a list of two-element lists of prime factors and their multiplicities:
> ifactor(1337);
(7) (191)> ifactors(1337);
[1, [[7, 1], [191, 1]]]Mathematica /Wolfram Language
Bare built-in function does:
FactorInteger[2016] => {{2, 5}, {3, 2}, {7, 1}}
Read as: 2 to the power 5 times 3 squared times 7 (to the power 1). To show them nicely we could use the following functions:
supscript[x_,y_]:=If[y==1,x,Superscript[x,y]]
ShowPrimeDecomposition[input_Integer]:=Print@@{input," = ",Sequence@@Riffle[supscript@@@FactorInteger[input]," "]}
Example for small prime:
ShowPrimeDecomposition[1337]
gives:
1337 = 7 191
Examples for large primes:
Table[AbsoluteTiming[ShowPrimeDecomposition[2^a-1]]//Print[#[[1]]," sec"]&,{a,50,150,10}];
gives back:
1125899906842623 = 3 11 31 251 601 1801 4051
0.000231 sec
1152921504606846975 = 3^2 5^2 7 11 13 31 41 61 151 331 1321
0.000146 sec
1180591620717411303423 = 3 11 31 43 71 127 281 86171 122921
0.001008 sec
1208925819614629174706175 = 3 5^2 11 17 31 41 257 61681 4278255361
0.000340 sec
1237940039285380274899124223 = 3^3 7 11 19 31 73 151 331 631 23311 18837001
0.000192 sec
1267650600228229401496703205375 = 3 5^3 11 31 41 101 251 601 1801 4051 8101 268501
0.000156 sec
1298074214633706907132624082305023 = 3 11^2 23 31 89 683 881 2971 3191 201961 48912491
0.001389 sec
1329227995784915872903807060280344575 = 3^2 5^2 7 11 13 17 31 41 61 151 241 331 1321 61681 4562284561
0.000374 sec
1361129467683753853853498429727072845823 = 3 11 31 131 2731 8191 409891 7623851 145295143558111
0.024249 sec
1393796574908163946345982392040522594123775 = 3 5^2 11 29 31 41 43 71 113 127 281 86171 122921 7416361 47392381
0.009419 sec
1427247692705959881058285969449495136382746623 = 3^2 7 11 31 151 251 331 601 1801 4051 100801 10567201 1133836730401
0.007705 sec
MATLAB
function [outputPrimeDecomposition] = primedecomposition(inputValue)
outputPrimeDecomposition = factor(inputValue);
Maxima
Using the built-in function:
(%i1) display2d: false$ /* disable rendering exponents as superscripts */
(%i2) factor(2016);
(%o2) 2^5*3^2*7
Using the underlying language:
prime_dec(n) := flatten(create_list(makelist(first(a), second(a)), a, ifactors(n)))$
/* or, slighlty more "functional" */
prime_dec(n) := flatten(map(lambda([a], apply(makelist, a)), ifactors(n)))$
prime_dec(2^4*3^5*5*7^2);
/* [2, 2, 2, 2, 3, 3, 3, 3, 3, 5, 7, 7] */
Modula-2
MODULE PrimeDecomposition;
FROM STextIO IMPORT
SkipLine, WriteLn, WriteString;
FROM SWholeIO IMPORT
ReadCard, WriteInt;
CONST
MaxFacIndex = 31;
(* 2^31 has most prime factors (31 twos) than other 32-bit unsigned integer. *)
TYPE
TFacs = ARRAY [0 .. MaxFacIndex] OF CARDINAL;
VAR
Facs: TFacs;
I, N, FacsCnt: CARDINAL;
PROCEDURE CalcFacs(N: CARDINAL; VAR Facs: TFacs; VAR FacsCnt: CARDINAL);
VAR
I: CARDINAL;
BEGIN
FacsCnt := 0;
IF N >= 2 THEN
I := 2;
WHILE I * I <= N DO
IF N MOD I = 0 THEN
N := N DIV I;
Facs[FacsCnt] := I;
FacsCnt := FacsCnt + 1;
I := 2
ELSE
I := I + 1
END
END;
Facs[FacsCnt] := N;
FacsCnt := FacsCnt + 1
END;
END CalcFacs;
BEGIN
WriteString("Enter a number: ");
ReadCard(N);
SkipLine;
CalcFacs(N, Facs, FacsCnt);
(* There is at least one factor *)
IF FacsCnt > 1 THEN
FOR I := 0 TO FacsCnt - 2 DO
WriteInt(Facs[I], 1);
WriteString(" ")
END;
END;
WriteInt(Facs[FacsCnt - 1], 1);
WriteLn
END PrimeDecomposition.
- Output:
3 runs.
Enter a number: 32 2 2 2 2 2
Enter a number: 2520 2 2 2 3 3 5 7
Enter a number: 13 13
MUMPS
ERATO1(HI)
SET HI=HI\1
KILL ERATO1 ;Don't make it new - we want it to remain after the quit
NEW I,J,P
FOR I=2:1:(HI**.5)\1 DO
.FOR J=I*I:I:HI DO
..SET P(J)=1 ;$SELECT($DATA(P(J))#10:P(J)+1,1:1)
;WRITE !,"Prime numbers between 2 and ",HI,": "
FOR I=2:1:HI DO
.S:'$DATA(P(I)) ERATO1(I)=I ;WRITE $SELECT((I<3):"",1:", "),I
KILL I,J,P
QUIT
PRIMDECO(N)
;Returns its results in the string PRIMDECO
;Kill that before the first call to this recursive function
QUIT:N<=1
IF $D(PRIMDECO)=1 SET PRIMDECO="" D ERATO1(N)
SET N=N\1,I=0
FOR SET I=$O(ERATO1(I)) Q:+I<1 Q:'(N#I)
IF I>1 SET PRIMDECO=$S($L(PRIMDECO)>0:PRIMDECO_"^",1:"")_I D PRIMDECO(N/I)
;that is, if I is a factor of N, add it to the string
QUIT- Usage:
USER>K ERATO1,PRIMDECO D PRIMDECO^ROSETTA(31415) W PRIMDECO 5^61^103 USER>K ERATO,PRIMDECO D PRIMDECO^ROSETTA(31318) W PRIMDECO 2^7^2237 USER>K ERATO,PRIMDECO D PRIMDECO^ROSETTA(34) W PRIMDECO 2^17 USER>K ERATO,PRIMDECO D PRIMDECO^ROSETTA(68) W PRIMDECO 2^2^17 USER>K ERATO,PRIMDECO D PRIMDECO^ROSETTA(7) W PRIMDECO 7 USER>K ERATO,PRIMDECO D PRIMDECO^ROSETTA(777) W PRIMDECO 3^7^37
Nim
Based on python floating point solution, but using integers rather than floats.
import math, sequtils, strformat, strutils, times
proc getStep(n: int64): int64 {.inline.} =
result = 1 + n shl 2 - n shr 1 shl 1
proc primeFac(n: int64): seq[int64] =
var maxq = int64(sqrt(float(n)))
var d = 1
var q: int64 = 2 + (n and 1) # Start with 2 or 3 according to oddity.
while q <= maxq and n %% q != 0:
q = getStep(d)
inc d
if q <= maxq:
let q1 = primeFac(n /% q)
let q2 = primeFac(q)
result = concat(q2, q1, result)
else:
result.add(n)
iterator primes(limit: int): int =
var isPrime = newSeq[bool](limit + 1)
for n in 2..limit: isPrime[n] = true
for n in 2..limit:
if isPrime[n]:
yield n
for i in countup(n *% n, limit, n):
isPrime[i] = false
when isMainModule:
# Example: calculate factors of Mersenne numbers from M2 to M59.
for m in primes(59):
let p = 2i64^m - 1
let s = &"2^{m}-1"
stdout.write &"{s:<6} = {p} with factors: "
let start = cpuTime()
stdout.write primeFac(p).join(", ")
echo &" => {(1000 * (cpuTime() - start)).toInt} ms"
- Output:
Compiled with option -d:release
2^2-1 = 3 with factors: 3 => 0 ms 2^3-1 = 7 with factors: 7 => 0 ms 2^5-1 = 31 with factors: 31 => 0 ms 2^7-1 = 127 with factors: 127 => 0 ms 2^11-1 = 2047 with factors: 23, 89 => 0 ms 2^13-1 = 8191 with factors: 8191 => 0 ms 2^17-1 = 131071 with factors: 131071 => 0 ms 2^19-1 = 524287 with factors: 524287 => 0 ms 2^23-1 = 8388607 with factors: 47, 178481 => 0 ms 2^29-1 = 536870911 with factors: 233, 1103, 2089 => 0 ms 2^31-1 = 2147483647 with factors: 2147483647 => 1 ms 2^37-1 = 137438953471 with factors: 223, 616318177 => 0 ms 2^41-1 = 2199023255551 with factors: 13367, 164511353 => 0 ms 2^43-1 = 8796093022207 with factors: 431, 9719, 2099863 => 0 ms 2^47-1 = 140737488355327 with factors: 2351, 4513, 13264529 => 0 ms 2^53-1 = 9007199254740991 with factors: 6361, 69431, 20394401 => 1 ms 2^59-1 = 576460752303423487 with factors: 179951, 3203431780337 => 6 ms
OCaml
open Big_int;;
let prime_decomposition x =
let rec inner c p =
if lt_big_int p (square_big_int c) then
[p]
else if eq_big_int (mod_big_int p c) zero_big_int then
c :: inner c (div_big_int p c)
else
inner (succ_big_int c) p
in
inner (succ_big_int (succ_big_int zero_big_int)) x;;
Octave
r = factor(120202039393)
Oforth
Oforth handles aribitrary precision integers.
: factors(n) // ( aInteger -- aList )
| k p |
ListBuffer new
2 ->k
n nsqrt ->p
while( k p <= ) [
n k /mod swap ifZero: [
dup ->n nsqrt ->p
k over add continue
]
drop k 1+ ->k
]
n 1 > ifTrue: [ n over add ]
dup freeze ;- Output:
>2 128 pow 1 - dup println factors println 340282366920938463463374607431768211455 [3, 5, 17, 257, 641, 65537, 274177, 6700417, 67280421310721] ok
PARI/GP
GP normally returns factored integers as a matrix
with the first column representing the primes
and the second their exponents.
Thus factor(12)==[2,2;3,1] is true.
But it's simple enough to convert this to a vector with repetition:
pd(n)={
my(f=factor(n),v=f[,1]~);
for(i=1,#v,
while(f[i,2]--,
v=concat(v,f[i,1])
)
);
vecsort(v)
};Pascal
Program PrimeDecomposition(output);
type
DynArray = array of integer;
procedure findFactors(n: Int64; var d: DynArray);
var
divisor, next, rest: Int64;
i: integer;
begin
i := 0;
divisor := 2;
next := 3;
rest := n;
while (rest <> 1) do
begin
while (rest mod divisor = 0) do
begin
setlength(d, i+1);
d[i] := divisor;
inc(i);
rest := rest div divisor;
end;
divisor := next;
next := next + 2;
end;
end;
var
factors: DynArray;
j: integer;
begin
setlength(factors, 1);
findFactors(1023*1024, factors);
for j := low(factors) to high(factors) do
writeln (factors[j]);
end.
- Output:
% ./PrimeDecomposition 2 2 2 2 2 2 2 2 2 2 3 11 31
Optimization:
Program PrimeDecomposition(output);
type
DynArray = array of integer;
procedure findFactors(n: Int64; var d: DynArray);
var
divisor, next, rest: Int64;
i: integer;
begin
i := 0;
divisor := 2;
next := 3;
rest := n;
while (rest <> 1) do
begin
while (rest mod divisor = 0) do
begin
setlength(d, i+1);
d[i] := divisor;
inc(i);
rest := rest div divisor;
end;
divisor := next;
next := next + 2; // try only odd numbers
// cut condition: avoid many useless iterations
if (rest < divisor * divisor) then
begin
setlength(d, i+1);
d[i] := rest;
rest := 1;
end;
end;
end;
var
factors: DynArray;
j: integer;
begin
setlength(factors, 1);
findFactors(1023*1024, factors);
for j := low(factors) to high(factors) do
writeln (factors[j]);
readln;
end.
PascalABC.NET
function Factors(N: BigInteger): List<BigInteger>;
begin
var lst := new List<BigInteger>;
if N = 1 then
lst.Add(N);
var i := 2bi;
while i * i <= N do
begin
while N mod i = 0 do
begin
lst.Add(i);
N := N div i;
end;
i += 1;
end;
if N >= 2 then
lst.Add(N);
Result := lst;
end;
- Output:
[71,839,1471,6857]
Perl
These will work for large integers by adding the use bigint; clause.
Trivial trial division (very slow)
sub prime_factors {
my ($n, $d, @out) = (shift, 1);
while ($n > 1 && $d++) {
$n /= $d, push @out, $d until $n % $d;
}
@out
}
print "@{[prime_factors(1001)]}\n";
Better trial division
This is much faster than the trivial version above.
sub prime_factors {
my($n, $p, @out) = (shift, 3);
return if $n < 1;
while (!($n&1)) { $n >>= 1; push @out, 2; }
while ($n > 1 && $p*$p <= $n) {
while ( ($n % $p) == 0) {
$n /= $p;
push @out, $p;
}
$p += 2;
}
push @out, $n if $n > 1;
@out;
}
Modules
As usual, there are CPAN modules for this that will be much faster. These both take about 1 second to factor all Mersenne numbers from M_1 to M_150.
use ntheory qw/factor forprimes/;
use bigint;
forprimes {
my $p = 2 ** $_ - 1;
print "2**$_-1: ", join(" ", factor($p)), "\n";
} 100, 150;
- Output:
2^101-1: 7432339208719 341117531003194129 2^103-1: 2550183799 3976656429941438590393 2^107-1: 162259276829213363391578010288127 2^109-1: 745988807 870035986098720987332873 2^113-1: 3391 23279 65993 1868569 1066818132868207 2^127-1: 170141183460469231731687303715884105727 2^131-1: 263 10350794431055162386718619237468234569 2^137-1: 32032215596496435569 5439042183600204290159 2^139-1: 5625767248687 123876132205208335762278423601 2^149-1: 86656268566282183151 8235109336690846723986161
use Math::Pari qw/:int factorint isprime/;
# Convert Math::Pari's format into simple vector
sub factor {
my ($pn,$pc) = @{Math::Pari::factorint(shift)};
map { ($pn->[$_]) x $pc->[$_] } 0 .. $#$pn;
}
for (100 .. 150) {
next unless isprime($_);
my $p = 2 ** $_ - 1;
print "2^$_-1: ", join(" ", factor($p)), "\n";
}
With the same output.
Phix
For small numbers less than 253 on 32bit and 264 on 64bit just use prime_factors().
with javascript_semantics requires("1.0.0") include mpfr.e atom t0 = time() mpz z = mpz_init() for i=1 to 25 do integer pi = get_prime(i) mpz_ui_pow_ui(z,2,pi) mpz_sub_ui(z,z,1) string zs = mpz_get_str(z), fs = mpz_factorstring(mpz_pollard_rho(z)) if fs!=zs then zs &= " = "&fs end if printf(1,"2^%d-1 = %s\n",{pi,zs}) end for for s in {"600851475143","100000000000000000037"} do mpz_set_str(z,s) printf(1,"%s = %s\n",{s,mpz_factorstring(mpz_pollard_rho(z))}) end for ?elapsed(time()-t0)
- Output:
2^2-1 = 3 2^3-1 = 7 2^5-1 = 31 2^7-1 = 127 2^11-1 = 2047 = 23*89 2^13-1 = 8191 2^17-1 = 131071 2^19-1 = 524287 2^23-1 = 8388607 = 47*178481 2^29-1 = 536870911 = 233*1103*2089 2^31-1 = 2147483647 2^37-1 = 137438953471 = 223*616318177 2^41-1 = 2199023255551 = 13367*164511353 2^43-1 = 8796093022207 = 431*9719*2099863 2^47-1 = 140737488355327 = 2351*4513*13264529 2^53-1 = 9007199254740991 = 6361*69431*20394401 2^59-1 = 576460752303423487 = 179951*3203431780337 2^61-1 = 2305843009213693951 2^67-1 = 147573952589676412927 = 193707721*761838257287 2^71-1 = 2361183241434822606847 = 228479*48544121*212885833 2^73-1 = 9444732965739290427391 = 439*2298041*9361973132609 2^79-1 = 604462909807314587353087 = 2687*202029703*1113491139767 2^83-1 = 9671406556917033397649407 = 167*57912614113275649087721 2^89-1 = 618970019642690137449562111 2^97-1 = 158456325028528675187087900671 = 11447*13842607235828485645766393 600851475143 = 71*839*1471*6857 100000000000000000037 = 31*821*59004541*66590107 "0.9s"
Picat
go =>
% Checking 2**prime-1
foreach(P in primes(60))
Factors = factors(2**P-1),
println([n=2**P-1,factors=Factors])
end,
nl,
% Testing a larger number
println(factors(1361129467683753853853498429727072845823)),
nl.
%
% factors of N
%
factors(N) = Factors =>
Factors = [],
M = N,
while (M mod 2 == 0)
Factors := Factors ++ [2],
M := M div 2
end,
T = 3,
while (M > 1, T < 1+(sqrt(M)))
if M mod T == 0 then
[Divisors, NewM] = alldivisorsM(M, T),
Factors := Factors ++ Divisors,
M := NewM
end,
T := T + 2
end,
if M > 1 then Factors := Factors ++ [M] end.
alldivisorsM(N,Div) = [Divisors,M] =>
M = N,
Divisors = [],
while (M mod Div == 0)
Divisors := Divisors ++ [Div],
M := M div Div
end.- Output:
[n = 3,factors = [3]] [n = 7,factors = [7]] [n = 31,factors = [31]] [n = 127,factors = [127]] [n = 2047,factors = [23,89]] [n = 8191,factors = [8191]] [n = 131071,factors = [131071]] [n = 524287,factors = [524287]] [n = 8388607,factors = [47,178481]] [n = 536870911,factors = [233,1103,2089]] [n = 2147483647,factors = [2147483647]] [n = 137438953471,factors = [223,616318177]] [n = 2199023255551,factors = [13367,164511353]] [n = 8796093022207,factors = [431,9719,2099863]] [n = 140737488355327,factors = [2351,4513,13264529]] [n = 9007199254740991,factors = [6361,69431,20394401]] [n = 576460752303423487,factors = [179951,3203431780337]] [3,11,31,131,2731,8191,409891,7623851,145295143558111]
PicoLisp
The following solution generates a sequence of "trial divisors" (2 3 5 7 11 13 17 19 23 29 31 37 ..), as described by Donald E. Knuth, "The Art of Computer Programming", Vol.2, p.365.
(de factor (N)
(make
(let (D 2 L (1 2 2 . (4 2 4 2 4 6 2 6 .)) M (sqrt N))
(while (>= M D)
(if (=0 (% N D))
(setq M (sqrt (setq N (/ N (link D)))))
(inc 'D (pop 'L)) ) )
(link N) ) ) )
(factor 1361129467683753853853498429727072845823)- Output:
-> (3 11 31 131 2731 8191 409891 7623851 145295143558111)
PL/0
The overlapping loops, created with GOTOs in the original version, have been replaced with structures with single entries and single exits (like in structured programming).
The program waits for a number, and then displays the prime factors of the number.
var i, n, nmodi;
begin
? n;
if n < 0 then n := -n;
if n >= 2 then
begin
i := 2;
while i * i <= n do
begin
nmodi := n - (n / i) * i;
if nmodi = 0 then
begin
n := n / i;
! i;
i := 2
end;
if nmodi <> 0 then
i := i + 1
end;
! n
end;
end.
3 runs.
- Input:
2520
- Output:
2
2
2
3
3
5
7
- Input:
16384
- Output:
2
2
2
2
2
2
2
2
2
2
2
2
2
2
- Input:
13
- Output:
13
PL/I
test: procedure options (main, reorder);
declare (n, i) fixed binary (31);
get list (n);
put edit ( n, '[' ) (x(1), a);
restart:
if is_prime(n) then
do;
put edit (trim(n), ']' ) (x(1), a);
stop;
end;
do i = n/2 to 2 by -1;
if is_prime(i) then
if (mod(n, i) = 0) then
do;
put edit ( trim(i) ) (x(1), a);
n = n / i;
go to restart;
end;
end;
put edit ( ' ]' ) (a);
is_prime: procedure (n) options (reorder) returns (bit(1));
declare n fixed binary (31);
declare i fixed binary (31);
if n < 2 then return ('0'b);
if n = 2 then return ('1'b);
if mod(n, 2) = 0 then return ('0'b);
do i = 3 to sqrt(n) by 2;
if mod(n, i) = 0 then return ('0'b);
end;
return ('1'b);
end is_prime;
end test;- Results from various runs:
1234567 [ 9721 127 ]
32768 [ 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ]
99 [ 11 3 3 ]
9876543 [ 14503 227 3 ]
100 [ 5 5 2 2 ]
9999999 [ 4649 239 3 3 ]
5040 [ 7 5 3 3 2 2 2 2 ]
PowerShell
function eratosthenes ($n) {
if($n -gt 1){
$prime = @(1..($n+1) | foreach{$true})
$prime[1] = $false
$m = [Math]::Floor([Math]::Sqrt($n))
function multiple($i) {
for($j = $i*$i; $j -le $n; $j += $i) {
$prime[$j] = $false
}
}
multiple 2
for($i = 3; $i -le $m; $i += 2) {
if($prime[$i]) {multiple $i}
}
1..$n | where{$prime[$_]}
} else {
Write-Error "$n is not greater than 1"
}
}
function prime-decomposition ($n) {
$array = eratosthenes $n
$prime = @()
foreach($p in $array) {
while($n%$p -eq 0) {
$n /= $p
$prime += @($p)
}
}
$prime
}
"$(prime-decomposition 12)"
"$(prime-decomposition 100)"
Output:
2 2 3 2 2 5 5
Alternative version, significantly faster with big numbers
function prime-decomposition ($n) {
$values = [System.Collections.Generic.List[string]]::new()
while ((($n % 2) -eq 0) -and ($n -gt 2)) {
$values.Add(2)
$n /= 2
}
for ($i = 3; $n -ge ($i * $i); $i += 2) {
if (($n % $i) -eq 0){
$values.Add($i)
$n /= $i
$i -= 2
}
}
$values.Add($n)
return $values
}
"$(prime-decomposition 1000000)"
Prolog
prime_decomp(N, L) :-
SN is sqrt(N),
prime_decomp_1(N, SN, 2, [], L).
prime_decomp_1(1, _, _, L, L) :- !.
% Special case for 2, increment 1
prime_decomp_1(N, SN, D, L, LF) :-
( 0 is N mod D ->
Q is N / D,
SQ is sqrt(Q),
prime_decomp_1(Q, SQ, D, [D |L], LF)
;
D1 is D+1,
( D1 > SN ->
LF = [N |L]
;
prime_decomp_2(N, SN, D1, L, LF)
)
).
% General case, increment 2
prime_decomp_2(1, _, _, L, L) :- !.
prime_decomp_2(N, SN, D, L, LF) :-
( 0 is N mod D ->
Q is N / D,
SQ is sqrt(Q),
prime_decomp_2(Q, SQ, D, [D |L], LF);
D1 is D+2,
( D1 > SN ->
LF = [N |L]
;
prime_decomp_2(N, SN, D1, L, LF)
)
).
- Output:
?- time(prime_decomp(9007199254740991, L)).
% 138,882 inferences, 0.344 CPU in 0.357 seconds (96% CPU, 404020 Lips)
L = [20394401,69431,6361].
?- time(prime_decomp(576460752303423487, L)).
% 2,684,734 inferences, 0.672 CPU in 0.671 seconds (100% CPU, 3995883 Lips)
L = [3203431780337,179951].
?- time(prime_decomp(1361129467683753853853498429727072845823, L)).
% 18,080,807 inferences, 7.953 CPU in 7.973 seconds (100% CPU, 2273422 Lips)
L = [145295143558111,7623851,409891,8191,2731,131,31,11,3].
Simple version
Optimized to stop on square root, and count by +2 on odds, above 2.
factors( N, FS):-
factors2( N, FS).
factors2( N, FS):-
( N < 2 -> FS = []
; 4 > N -> FS = [N]
; 0 is N rem 2 -> FS = [K|FS2], N2 is N div 2, factors2( N2, FS2)
; factors( N, 3, FS)
).
factors( N, K, FS):-
( N < 2 -> FS = []
; K*K > N -> FS = [N]
; 0 is N rem K -> FS = [K|FS2], N2 is N div K, factors( N2, K, FS2)
; K2 is K+2, factors( N, K2, FS)
).
Expression Tree version
Uses a 2*3*5*7 factor wheel, but the main feature is that it returns the decomposition as a fully simplified expression tree.
wheel2357(L) :-
W = [2, 4, 2, 4, 6, 2, 6, 4,
2, 4, 6, 6, 2, 6, 4, 2,
6, 4, 6, 8, 4, 2, 4, 2,
4, 8, 6, 4, 6, 2, 4, 6,
2, 6, 6, 4, 2, 4, 6, 2,
6, 4, 2, 4, 2, 10, 2, 10 | W],
L = [1, 2, 2, 4 | W].
factor(1, 1) :- !.
factor(N, Fac) :-
N > 1,
wheel2357(W),
factor(N, 2, W, 1, Fac0),
reverse_factors(Fac0, Fac).
factor(N, F, _, Fac1, Fac2) :- F*F > N, !, add_factor(N, Fac1, Fac2).
factor(N, F, W, Fac1, Fac) :-
divmod(N, F, Q, 0), !,
add_factor(F, Fac1, Fac2),
factor(Q, F, W, Fac2, Fac).
factor(N, F1, [A|As], Fac1, Fac) :-
F2 is F1 + A,
factor(N, F2, As, Fac1, Fac).
add_factor(F, 1, F) :- !.
add_factor(F, F, F**2) :- !.
add_factor(F, F**Ex1, F**Ex2) :- succ(Ex1, Ex2), !.
add_factor(F, F*A, F**2*A) :- !.
add_factor(F, F**Ex1*Rest, F**Ex2*Rest) :- succ(Ex1, Ex2), !.
add_factor(F, Fac, F*Fac).
reverse_factors(A*B, C*A) :- reverse_factors(B, C), !.
reverse_factors(A, A).- Output:
?- factor(277,X). X = 277. ?- factor(1003,X). X = 17*59. ?- factor(1024,X). X = 2**10. ?- factor(768,X). X = 2**8*3. ?- factor(1361129467683753853853498429727072845823,X). X = 3*11*31*131*2731*8191*409891*7623851*145295143558111. ?- factor(360,X). X = 2**3*3**2*5.
Pure
factor n = factor 2 n with
factor k n = k : factor k (n div k) if n mod k == 0;
= if n>1 then [n] else [] if k*k>n;
= factor (k+1) n if k==2;
= factor (k+2) n otherwise;
end;Python
Python: Using Croft Spiral sieve
Note: the program below is saved to file prime_decomposition.py and imported as a library here, here, here, here and here.
from __future__ import print_function
import sys
from itertools import cycle
def is_prime(n):
return list(zip((True, False), decompose(n)))[-1][0]
class IsPrimeCached(dict):
def __missing__(self, n):
r = is_prime(n)
self[n] = r
return r
is_prime_cached = IsPrimeCached()
def croft():
"""Yield prime integers using the Croft Spiral sieve.
This is a variant of wheel factorisation modulo 30.
"""
# Copied from:
# https://code.google.com/p/pyprimes/source/browse/src/pyprimes.py
# Implementation is based on erat3 from here:
# http://stackoverflow.com/q/2211990
# and this website:
# http://www.primesdemystified.com/
# Memory usage increases roughly linearly with the number of primes seen.
# dict ``roots`` stores an entry x:p for every prime p.
for p in (2, 3, 5):
yield p
roots = {} # Map x*d -> 2*d.
not_primeroot = tuple(x not in {1,7,11,13,17,19,23,29} for x in range(30))
q = 1
for x in cycle((6, 4, 2, 4, 2, 4, 6, 2)):
# Iterate over prime candidates 7, 11, 13, 17, ...
q += x
# Using dict membership testing instead of pop gives a
# 5-10% speedup over the first three million primes.
if q in roots:
p = roots.pop(q)
x = q + p
while not_primeroot[x % 30] or x in roots:
x += p
roots[x] = p
else:
roots[q * q] = q + q
yield q
primes = croft
def decompose(n):
for p in primes():
if p*p > n: break
while n % p == 0:
yield p
n //=p
if n > 1:
yield n
if __name__ == '__main__':
# Example: calculate factors of Mersenne numbers to M59 #
import time
for m in primes():
p = 2 ** m - 1
print( "2**{0:d}-1 = {1:d}, with factors:".format(m, p) )
start = time.time()
for factor in decompose(p):
print(factor, end=' ')
sys.stdout.flush()
print( "=> {0:.2f}s".format( time.time()-start ) )
if m >= 59:
break- Output:
2**2-1 = 3, with factors: 3 => 0.00s 2**3-1 = 7, with factors: 7 => 0.01s 2**5-1 = 31, with factors: 31 => 0.00s 2**7-1 = 127, with factors: 127 => 0.00s 2**11-1 = 2047, with factors: 23 89 => 0.00s 2**13-1 = 8191, with factors: 8191 => 0.00s 2**17-1 = 131071, with factors: 131071 => 0.00s 2**19-1 = 524287, with factors: 524287 => 0.00s 2**23-1 = 8388607, with factors: 47 178481 => 0.01s 2**29-1 = 536870911, with factors: 233 1103 2089 => 0.01s 2**31-1 = 2147483647, with factors: 2147483647 => 0.03s 2**37-1 = 137438953471, with factors: 223 616318177 => 0.02s 2**41-1 = 2199023255551, with factors: 13367 164511353 => 0.01s 2**43-1 = 8796093022207, with factors: 431 9719 2099863 => 0.01s 2**47-1 = 140737488355327, with factors: 2351 4513 13264529 => 0.01s 2**53-1 = 9007199254740991, with factors: 6361 69431 20394401 => 0.04s 2**59-1 = 576460752303423487, with factors: 179951 3203431780337 => 1.22s
Python: Using floating point
Here a shorter and marginally faster algorithm:
from math import floor, sqrt
try:
long
except NameError:
long = int
def fac(n):
step = lambda x: 1 + (x<<2) - ((x>>1)<<1)
maxq = long(floor(sqrt(n)))
d = 1
q = 2 if n % 2 == 0 else 3
while q <= maxq and n % q != 0:
q = step(d)
d += 1
return [q] + fac(n // q) if q <= maxq else [n]
if __name__ == '__main__':
import time
start = time.time()
tocalc = 2**59-1
print("%s = %s" % (tocalc, fac(tocalc)))
print("Needed %ss" % (time.time() - start))- Output:
576460752303423487 = [3203431780337, 179951] Needed 0.9240529537200928s
Quackery
prime is defined at Miller-Rabin primality test#Quackery.
[ dup prime iff
nested done
[] swap
dup times
[ i^ 2 + prime
not if done
[ dup i^ 2 + /mod
0 = while
nip dip
[ i^ 2 + join ]
again ]
drop
dup 1 = if conclude ]
drop ] is primefactors ( n --> [ )- Output:
[ 2 2 2 2 2 2 2 2 2 2 3 11 31 ]
R
findfactors <- function(num) {
x <- NULL
firstprime<- 2; secondprime <- 3; everyprime <- num
while( everyprime != 1 ) {
while( everyprime%%firstprime == 0 ) {
x <- c(x, firstprime)
everyprime <- floor(everyprime/ firstprime)
}
firstprime <- secondprime
secondprime <- secondprime + 2
}
x
}
print(findfactors(1027*4))Or a more explicit (but less efficient) recursive approach:
Recursive Approach (Less efficient for large numbers)
primes <- as.integer(c())
max_prime_checker <- function(n){
divisor <<- NULL
primes <- primes[primes <= n]
for(i in 1:length(primes)){
if((n/primes[i]) %% 1 == 0){
divisor[i]<<-1
} else {
divisor[i]<<-0
}
}
num_find <<- primes*as.integer(divisor)
return(max(num_find))
}
#recursive prime finder
prime_factors <- function(n){
factors <- NULL
large <- max_prime_checker(n)
n1 <- n/large
if(max_prime_checker(n1) == n1){
factors <- c(large,n1)
return(factors)
} else {
factors <- c(large, prime_factors(n1))
return(factors)
}
}Alternate solution
findfactors <- function(n) {
a <- NULL
if (n > 1) {
while (n %% 2 == 0) {
a <- c(a, 2)
n <- n %/% 2
}
k <- 3
while (k * k <= n) {
while (n %% k == 0) {
a <- c(a, k)
n <- n %/% k
}
k <- k + 2
}
if (n > 1) a <- c(a, n)
}
a
}Racket
#lang racket
(require math)
(define (factors n)
(append-map (λ (x) (make-list (cadr x) (car x))) (factorize n)))Or, an explicit (and less efficient) computation:
#lang racket
(define (factors number)
(let loop ([n number] [i 2])
(if (= n 1)
'()
(let-values ([(q r) (quotient/remainder n i)])
(if (zero? r) (cons i (loop q i)) (loop n (add1 i)))))))Raku
(formerly Perl 6)
Pure Raku
This is a pure Raku version that uses no outside libraries. It uses a variant of Pollard's rho factoring algorithm and is fairly performent when factoring numbers < 2⁸⁰; typically taking well under a second on an i7. It starts to slow down with larger numbers, but really bogs down factoring numbers that have more than 1 factor larger than about 2⁴⁰.
sub prime-factors ( Int $n where * > 0 ) {
return $n if $n.is-prime;
return () if $n == 1;
my $factor = find-factor( $n );
sort flat ( $factor, $n div $factor ).map: &prime-factors;
}
sub find-factor ( Int $n, $constant = 1 ) {
return 2 unless $n +& 1;
if (my $gcd = $n gcd 6541380665835015) > 1 { # magic number: [*] primes 3 .. 43
return $gcd if $gcd != $n
}
my $x = 2;
my $rho = 1;
my $factor = 1;
while $factor == 1 {
$rho = $rho +< 1;
my $fixed = $x;
my int $i = 0;
while $i < $rho {
$x = ( $x * $x + $constant ) % $n;
$factor = ( $x - $fixed ) gcd $n;
last if 1 < $factor;
$i = $i + 1;
}
}
$factor = find-factor( $n, $constant + 1 ) if $n == $factor;
$factor;
}
.put for (2²⁹-1, 2⁴¹-1, 2⁵⁹-1, 2⁷¹-1, 2⁷⁹-1, 2⁹⁷-1, 2¹¹⁷-1, 2²⁴¹-1,
5465610891074107968111136514192945634873647594456118359804135903459867604844945580205745718497)\
.hyper(:1batch).map: -> $n {
my $start = now;
"factors of $n: ",
prime-factors($n).join(' × '), " \t in ", (now - $start).fmt("%0.3f"), " sec."
}- Output:
factors of 536870911: 233 × 1103 × 2089 in 0.004 sec. factors of 2199023255551: 13367 × 164511353 in 0.011 sec. factors of 576460752303423487: 179951 × 3203431780337 in 0.023 sec. factors of 2361183241434822606847: 228479 × 48544121 × 212885833 in 0.190 sec. factors of 604462909807314587353087: 2687 × 202029703 × 1113491139767 in 0.294 sec. factors of 158456325028528675187087900671: 11447 × 13842607235828485645766393 in 0.005 sec. factors of 166153499473114484112975882535043071: 7 × 73 × 79 × 937 × 6553 × 8191 × 86113 × 121369 × 7830118297 in 0.022 sec. factors of 3533694129556768659166595001485837031654967793751237916243212402585239551: 22000409 × 160619474372352289412737508720216839225805656328990879953332340439 in 0.085 sec. factors of 5465610891074107968111136514192945634873647594456118359804135903459867604844945580205745718497: 165901 × 10424087 × 18830281 × 53204737 × 56402249 × 59663291 × 91931221 × 95174413 × 305293727939 × 444161842339 × 790130065009 in 28.427 sec.
There is a Raku module available: Prime::Factor, that uses essentially this algorithm with some minor performance tweaks.
External library
If you really need a speed boost, load the highly optimized Perl 5 ntheory module. It needs a little extra plumbing to deal with the lack of built-in big integer support, but for large number factoring the interface overhead is worth it.
use Inline::Perl5;
my $p5 = Inline::Perl5.new();
$p5.use( 'ntheory' );
sub prime-factors ($i) {
my &primes = $p5.run('sub { map { ntheory::todigitstring $_ } sort {$a <=> $b} ntheory::factor $_[0] }');
primes("$i");
}
for 2²⁹-1, 2⁴¹-1, 2⁵⁹-1, 2⁷¹-1, 2⁷⁹-1, 2⁹⁷-1, 2¹¹⁷-1,
5465610891074107968111136514192945634873647594456118359804135903459867604844945580205745718497
-> $n {
my $start = now;
say "factors of $n: ",
prime-factors($n).join(' × '), " \t in ", (now - $start).fmt("%0.3f"), " sec."
}- Output:
factors of 536870911: 233 × 1103 × 2089 in 0.001 sec. factors of 2199023255551: 13367 × 164511353 in 0.001 sec. factors of 576460752303423487: 179951 × 3203431780337 in 0.001 sec. factors of 2361183241434822606847: 228479 × 48544121 × 212885833 in 0.012 sec. factors of 604462909807314587353087: 2687 × 202029703 × 1113491139767 in 0.003 sec. factors of 158456325028528675187087900671: 11447 × 13842607235828485645766393 in 0.001 sec. factors of 166153499473114484112975882535043071: 7 × 73 × 79 × 937 × 6553 × 8191 × 86113 × 121369 × 7830118297 in 0.001 sec. factors of 5465610891074107968111136514192945634873647594456118359804135903459867604844945580205745718497: 165901 × 10424087 × 18830281 × 53204737 × 56402249 × 59663291 × 91931221 × 95174413 × 305293727939 × 444161842339 × 790130065009 in 0.064 sec.
REXX
Version 1
No (error) checking was done for the input arguments to test their validity.
The number of decimal digits is adjusted to match the size of the top-of-the-range (top).
Also, a count of primes found is shown.
If the top number is negative, only the number of primes up to abs(top) is shown.
A method exists in this REXX program to also test Mersenne-type numbers (2n - 1).
Since the majority of computing time is spent looking for primes, that part of the program was
optimized somewhat (but could be extended if more optimization is wanted).
/*REXX pgm does prime decomposition of a range of positive integers (with a prime count)*/
Numeric Digits 1000 /*handle thousand digits For the powers*/
Parse Arg bot top step base add /*get optional arguments from the C.L. */
If bot='?' Then Do
Say 'rexx pfoo bot top step base add'
Exit
End
If bot=='' Then /* no arguments given */
Parse Value 1 100 With bot top /* set default range .*/
If top=='' Then top=bot /* process one number */
If step=='' Then step=1 /* step=2 to process only odd numbers */
If add =='' Then add=-1 /* for Mersenne tests */
tell=top>0 /*If TOP is negative, suppress displays*/
top=abs(top)
w=length(top) /*get maximum width For aligned display*/
If base\=='' Then
w=length(base**top) /*will be testing powers of two later? */
tag.=left('', 7) /*some literals: pad; prime (or not).*/
tag.0='{unity}'
tag.1='[prime]'
Numeric Digits max(9,w+1) /*maybe increase the digits precision. */
np=0 /*np: is the number of primes found.*/
Do n=bot To top by step /*process a single number or a range. */
?=n
If base\=='' Then /*should we perform a 'Mersenne' test? */
?=base**n+add
pf=factr(?) /* get prime factors */
f=words(pf) /* number of prime factors */
If f=1 Then /* If the number is prime */
np=np+1 /* Then bump prime counter */
If tell Then
Say right(?,w) right('('f')',9) 'prime factors: ' tag.f pf
End /*n*/
Say ''
ps='prime'
If f>1 Then ps=ps's' /*setup For proper English in sentence.*/
Say right(np, w+9+1) ps 'found.' /*display the number of primes found. */
Exit /*stick a fork in it, we're all done. */
/*---------------------------------------------------------------------------*/
factr: Procedure
Parse Arg x 1 d,pl /*set X, D to argument 1, pl to null */
If x==1 Then Return '' /*handle the special case of X=1. */
primes=2 3 5 7
Do While primes>'' /* first check the small primes */
Parse Var primes prime primes
Do While x//prime==0
pl=pl prime
x=x%prime
End
End
r=isqrt(x)
Do j=11 by 6 To r /*insure that J isn't divisible by 3. */
Parse var j '' -1 _ /*obtain the last decimal digit of J. */
If _\==5 Then Do
Do While x//j==0
pl=pl j
x=x%j
End /*maybe reduce by J. */
End
If _ ==3 Then Iterate /*If next Y is divisible by 5? Skip. */
y=j+2
Do While x//y==0
pl=pl y
x=x%y
End /*maybe reduce by y. */
End /*j*/
If x==1 Then Return pl /*Is residual=unity? Then don't append.*/
Return pl x /*return pl with appended residual.*/
isqrt: Procedure
Parse Arg x
x=abs(x)
Parse Value 0 x with lo hi
Do While lo<=hi
t=(lo+hi)%2
If t**2>x Then
hi=t-1
Else
lo=t+1
End
Return toutput when using the default input of: 1 100 All timings were done using ooRexx. (Shown at three-quarter size.)
1 (0) prime factors: {unity}
2 (1) prime factors: [prime] 2
3 (1) prime factors: [prime] 3
4 (2) prime factors: 2 2
5 (1) prime factors: [prime] 5
6 (2) prime factors: 2 3
7 (1) prime factors: [prime] 7
8 (3) prime factors: 2 2 2
9 (2) prime factors: 3 3
10 (2) prime factors: 2 5
11 (1) prime factors: [prime] 11
12 (3) prime factors: 2 2 3
13 (1) prime factors: [prime] 13
14 (2) prime factors: 2 7
15 (2) prime factors: 3 5
16 (4) prime factors: 2 2 2 2
17 (1) prime factors: [prime] 17
18 (3) prime factors: 2 3 3
19 (1) prime factors: [prime] 19
20 (3) prime factors: 2 2 5
21 (2) prime factors: 3 7
22 (2) prime factors: 2 11
23 (1) prime factors: [prime] 23
24 (4) prime factors: 2 2 2 3
25 (2) prime factors: 5 5
26 (2) prime factors: 2 13
27 (3) prime factors: 3 3 3
28 (3) prime factors: 2 2 7
29 (1) prime factors: [prime] 29
30 (3) prime factors: 2 3 5
31 (1) prime factors: [prime] 31
32 (5) prime factors: 2 2 2 2 2
33 (2) prime factors: 3 11
34 (2) prime factors: 2 17
35 (2) prime factors: 5 7
36 (4) prime factors: 2 2 3 3
37 (1) prime factors: [prime] 37
38 (2) prime factors: 2 19
39 (2) prime factors: 3 13
40 (4) prime factors: 2 2 2 5
41 (1) prime factors: [prime] 41
42 (3) prime factors: 2 3 7
43 (1) prime factors: [prime] 43
44 (3) prime factors: 2 2 11
45 (3) prime factors: 3 3 5
46 (2) prime factors: 2 23
47 (1) prime factors: [prime] 47
48 (5) prime factors: 2 2 2 2 3
49 (2) prime factors: 7 7
50 (3) prime factors: 2 5 5
51 (2) prime factors: 3 17
52 (3) prime factors: 2 2 13
53 (1) prime factors: [prime] 53
54 (4) prime factors: 2 3 3 3
55 (2) prime factors: 5 11
56 (4) prime factors: 2 2 2 7
57 (2) prime factors: 3 19
58 (2) prime factors: 2 29
59 (1) prime factors: [prime] 59
60 (4) prime factors: 2 2 3 5
61 (1) prime factors: [prime] 61
62 (2) prime factors: 2 31
63 (3) prime factors: 3 3 7
64 (6) prime factors: 2 2 2 2 2 2
65 (2) prime factors: 5 13
66 (3) prime factors: 2 3 11
67 (1) prime factors: [prime] 67
68 (3) prime factors: 2 2 17
69 (2) prime factors: 3 23
70 (3) prime factors: 2 5 7
71 (1) prime factors: [prime] 71
72 (5) prime factors: 2 2 2 3 3
73 (1) prime factors: [prime] 73
74 (2) prime factors: 2 37
75 (3) prime factors: 3 5 5
76 (3) prime factors: 2 2 19
77 (2) prime factors: 7 11
78 (3) prime factors: 2 3 13
79 (1) prime factors: [prime] 79
80 (5) prime factors: 2 2 2 2 5
81 (4) prime factors: 3 3 3 3
82 (2) prime factors: 2 41
83 (1) prime factors: [prime] 83
84 (4) prime factors: 2 2 3 7
85 (2) prime factors: 5 17
86 (2) prime factors: 2 43
87 (2) prime factors: 3 29
88 (4) prime factors: 2 2 2 11
89 (1) prime factors: [prime] 89
90 (4) prime factors: 2 3 3 5
91 (2) prime factors: 7 13
92 (3) prime factors: 2 2 23
93 (2) prime factors: 3 31
94 (2) prime factors: 2 47
95 (2) prime factors: 5 19
96 (6) prime factors: 2 2 2 2 2 3
97 (1) prime factors: [prime] 97
98 (3) prime factors: 2 7 7
99 (3) prime factors: 3 3 11
100 (4) prime factors: 2 2 5 5
25 primes found.<br>
0.046 seconds
output when using the input of: 9007199254740991
9007199254740991 (3) prime factors: 6361 69431 20394401
0 primes found.
5.201 seconds
output when using the input of: 2543821448263974486045199
2543821448263974486045199 (6) prime factors: 701 1123 1123 2411 1092461 1092461
0 primes found.<br>
> 1000 seconds
output when using the input of: 1 -1000000
78498 primes found. 32.251 seconds
output when using the input of: 2 50 1 2 -1
(essentially testing for Mersenne primes: 2n -1)
(Shown at three-quarter size.)
3 (1) prime factors: [prime] 3
7 (1) prime factors: [prime] 7
15 (2) prime factors: 3 5
31 (1) prime factors: [prime] 31
63 (3) prime factors: 3 3 7
127 (1) prime factors: [prime] 127
255 (3) prime factors: 3 5 17
511 (2) prime factors: 7 73
1023 (3) prime factors: 3 11 31
2047 (2) prime factors: 23 89
4095 (5) prime factors: 3 3 5 7 13
8191 (1) prime factors: [prime] 8191
16383 (3) prime factors: 3 43 127
32767 (3) prime factors: 7 31 151
65535 (4) prime factors: 3 5 17 257
131071 (1) prime factors: [prime] 131071
262143 (6) prime factors: 3 3 3 7 19 73
524287 (1) prime factors: [prime] 524287
1048575 (6) prime factors: 3 5 5 11 31 41
2097151 (4) prime factors: 7 7 127 337
4194303 (4) prime factors: 3 23 89 683
8388607 (2) prime factors: 47 178481
16777215 (7) prime factors: 3 3 5 7 13 17 241
33554431 (3) prime factors: 31 601 1801
67108863 (3) prime factors: 3 2731 8191
134217727 (3) prime factors: 7 73 262657
268435455 (6) prime factors: 3 5 29 43 113 127
536870911 (3) prime factors: 233 1103 2089
1073741823 (7) prime factors: 3 3 7 11 31 151 331
2147483647 (1) prime factors: [prime] 2147483647
4294967295 (5) prime factors: 3 5 17 257 65537
8589934591 (4) prime factors: 7 23 89 599479
17179869183 (3) prime factors: 3 43691 131071
34359738367 (4) prime factors: 31 71 127 122921
68719476735 (10) prime factors: 3 3 3 5 7 13 19 37 73 109
137438953471 (2) prime factors: 223 616318177
274877906943 (3) prime factors: 3 174763 524287
549755813887 (4) prime factors: 7 79 8191 121369
1099511627775 (8) prime factors: 3 5 5 11 17 31 41 61681
2199023255551 (2) prime factors: 13367 164511353
4398046511103 (8) prime factors: 3 3 7 7 43 127 337 5419
8796093022207 (3) prime factors: 431 9719 2099863
17592186044415 (7) prime factors: 3 5 23 89 397 683 2113
35184372088831 (6) prime factors: 7 31 73 151 631 23311
70368744177663 (4) prime factors: 3 47 178481 2796203
140737488355327 (3) prime factors: 2351 4513 13264529
281474976710655 (10) prime factors: 3 3 5 7 13 17 97 241 257 673
562949953421311 (2) prime factors: 127 4432676798593
1125899906842623 (7) prime factors: 3 11 31 251 601 1801 4051
8 primes found.
3.942 seconds
output when using the input of: 1 50 1 2 +1
(essentially testing for 2n +1)
(Shown at three-quarter size.)
3 (1) prime factors: [prime] 3
5 (1) prime factors: [prime] 5
9 (2) prime factors: 3 3
17 (1) prime factors: [prime] 17
33 (2) prime factors: 3 11
65 (2) prime factors: 5 13
129 (2) prime factors: 3 43
257 (1) prime factors: [prime] 257
513 (4) prime factors: 3 3 3 19
1025 (3) prime factors: 5 5 41
2049 (2) prime factors: 3 683
4097 (2) prime factors: 17 241
8193 (2) prime factors: 3 2731
16385 (3) prime factors: 5 29 113
32769 (4) prime factors: 3 3 11 331
65537 (1) prime factors: [prime] 65537
131073 (2) prime factors: 3 43691
262145 (4) prime factors: 5 13 37 109
524289 (2) prime factors: 3 174763
1048577 (2) prime factors: 17 61681
2097153 (4) prime factors: 3 3 43 5419
4194305 (3) prime factors: 5 397 2113
8388609 (2) prime factors: 3 2796203
16777217 (3) prime factors: 97 257 673
33554433 (4) prime factors: 3 11 251 4051
67108865 (4) prime factors: 5 53 157 1613
134217729 (6) prime factors: 3 3 3 3 19 87211
268435457 (2) prime factors: 17 15790321
536870913 (3) prime factors: 3 59 3033169
1073741825 (6) prime factors: 5 5 13 41 61 1321
2147483649 (2) prime factors: 3 715827883
4294967297 (2) prime factors: 641 6700417
8589934593 (5) prime factors: 3 3 67 683 20857
17179869185 (4) prime factors: 5 137 953 26317
34359738369 (5) prime factors: 3 11 43 281 86171
68719476737 (4) prime factors: 17 241 433 38737
137438953473 (3) prime factors: 3 1777 25781083
274877906945 (4) prime factors: 5 229 457 525313
549755813889 (4) prime factors: 3 3 2731 22366891
1099511627777 (2) prime factors: 257 4278255361
2199023255553 (3) prime factors: 3 83 8831418697
4398046511105 (6) prime factors: 5 13 29 113 1429 14449
8796093022209 (2) prime factors: 3 2932031007403
17592186044417 (3) prime factors: 17 353 2931542417
35184372088833 (7) prime factors: 3 3 3 11 19 331 18837001
70368744177665 (5) prime factors: 5 277 1013 1657 30269
140737488355329 (3) prime factors: 3 283 165768537521
281474976710657 (3) prime factors: 193 65537 22253377
562949953421313 (3) prime factors: 3 43 4363953127297
1125899906842625 (7) prime factors: 5 5 5 41 101 8101 268501
5 primes found.
3.164 seconds
Version 2
This REXX version is about 20% faster than the 1st REXX version when factoring one million numbers.
/*REXX pgm does prime decomposition of a range of positive integers (with a prime count)*/
Numeric Digits 1000 /*handle thousand digits For the powers*/
Parse Arg bot top step base add /*get optional arguments from the C.L. */
If bot='?' Then Do
Say 'rexx pfoo bot top step base add'
Exit
End
If bot=='' Then /* no arguments given */
Parse Value 1 100 With bot top /* set default range .*/
If top=='' Then top=bot /* process one number */
If step=='' Then step=1 /* step=2 to process only odd numbers */
If add =='' Then add=-1 /* for Mersenne tests */
tell=top>0 /*If TOP is negative, suppress displays*/
top=abs(top)
w=length(top) /*get maximum width For aligned display*/
If base\=='' Then
w=length(base**top) /*will be testing powers of two later? */
tag.=left('', 7) /*some literals: pad; prime (or not).*/
tag.0='{unity}'
tag.1='[prime]'
Numeric Digits max(9,w+1) /*maybe increase the digits precision. */
np=0 /*np: is the number of primes found.*/
Do n=bot To top by step /*process a single number or a range. */
?=n
If base\=='' Then /*should we perform a 'Mersenne' test? */
?=base**n+add
pf=factr(?) /* get prime factors */
f=words(pf) /* number of prime factors */
If f=1 Then /* If the number is prime */
np=np+1 /* Then bump prime counter */
If tell Then
Say right(?,w) right('('f')',9) 'prime factors: ' tag.f pf
End /*n*/
Say ''
ps='prime'
If f>1 Then ps=ps's' /*setup For proper English in sentence.*/
Say right(np, w+9+1) ps 'found.' /*display the number of primes found. */
Exit /*stick a fork in it, we're all done. */
/*---------------------------------------------------------------------------*/
factr: Procedure
Parse Arg x 1 d,pl /*set X, D to argument 1, pl to null */
If x==1 Then Return '' /*handle the special case of X=1. */
primes=2 3 5 7 11 13 17 19 23
Do While primes>'' /* first check the small primes */
Parse Var primes prime primes
Do While x//prime==0
pl=pl prime
x=x%prime
End
End
r=isqrt(x)
Do j=29 by 6 To r /*insure that J isn't divisible by 3.*/
Parse var j '' -1 _ /*obtain the last decimal digit of J. */
If _\==5 Then Do
Do While x//j==0
pl=pl j
x=x%j
End /*maybe reduce by J. */
End
If _ ==3 Then Iterate /*If next Y is divisible by 5? Skip. */
y=j+2
Do While x//y==0
pl=pl y
x=x%y
End /*maybe reduce by y. */
End /*j*/
If x==1 Then Return pl /*Is residual=unity? Then don't append.*/
Return pl x /*return pl with appended residual.*/
isqrt: Procedure
Parse Arg x
x=abs(x)
Parse Value 0 x with lo hi
Do While lo<=hi
t=(lo+hi)%2
If t**2>x Then
hi=t-1
Else
lo=t+1
End
Return toutput is identical to the 1st REXX version Timings slightly faster.
Version 3
In this version
r=isqrt(x)
Do j=29 by 6 To r
is replaced by
do j = 29 by 6 while j*j <= x
together with some other minor improvements.
/*REXX pgm does prime decomposition of a range of positive integers (with a prime count)*/
call time('r')
Numeric Digits 1000 /*handle thousand digits For the powers*/
Parse Arg bot top step base add /*get optional arguments from the C.L. */
If bot='?' Then Do
Say 'rexx pfoo bot top step base add'
Exit
End
If bot=='' Then /* no arguments given */
Parse Value 1 100 With bot top /* set default range .*/
If top=='' Then top=bot /* process one number */
If step=='' Then step=1 /* step=2 to process only odd numbers */
If add =='' Then add=-1 /* for Mersenne tests */
tell=top>0 /*If TOP is negative, suppress displays*/
top=abs(top)
w=length(top) /*get maximum width For aligned display*/
If base\=='' Then
w=length(base**top) /*will be testing powers of two later? */
tag.=left('', 7) /*some literals: pad; prime (or not).*/
tag.0='{unity}'
tag.1='[prime]'
Numeric Digits max(9,w+1) /*maybe increase the digits precision. */
np=0 /*np: is the number of primes found.*/
Do n=bot To top by step /*process a single number or a range. */
?=n
If base\=='' Then /*should we perform a 'Mersenne' test? */
?=base**n+add
pf=factr(?) /* get prime factors */
f=words(pf) /* number of prime factors */
If f=1 Then /* If the number is prime */
np=np+1 /* Then bump prime counter */
If tell Then
Say right(?,w) right('('f')',9) 'prime factors: ' tag.f pf
End /*n*/
Say ''
ps='prime'
If f>1 Then ps=ps's' /*setup For proper English in sentence.*/
Say right(np, w+9+1) ps 'found.' /*display the number of primes found. */
say format(time('e'),,3) 'seconds'
Exit /*stick a fork in it, we're all done. */
/*---------------------------------------------------------------------------*/
Factr:
procedure expose glob.
arg x
primes = '2 3 5 7 11 13 17 19 23'; pl = ''
do n = 1 to words(primes)
prime = word(primes,n)
do while x//prime = 0
pl = pl prime; x = x%prime
end
end
do j = 29 by 6 while j*j <= x
d = right(j,1)
if d <> 5 then do
do while x//j = 0
pl = pl j; x = x%j
end
end
if d = 3 then
iterate
y = j+2
do while x//y = 0
pl = pl y; x = x%y
end
end
if x = 1 then
return pl
else
return pl xThe output is identical to Version 1, except for the timings. Version 3 is much, much faster on big numbers!
Input 9007199254740991 0.011 seconds Input 2543821448263974486045199 0.082 seconds Input 340282366920938463463374607431768211455 340282366920938463463374607431768211455 (9) prime factors: 3 5 17 257 641 65537 274177 6700417 67280421310721 2.093 seconds This big number only succeeded because it has some small factors first. Dividing by these factors quickly reduces the size of the number. Looks like a 'lucky shot'. If a big number is the product of 2 not so small primes (i.e. semiprimes), then also Version 3 will not find the factors within a reasonable time. Input 1 -1000000 19.369 seconds Input 2 50 1 2 -1 0.238 seconds Input 2 50 1 2 +1 0.370 seconds
Version 4
Libraries: How to use
Library: Numbers
Library: Functions
The factorization routine in below version is almost the same as the one in Version 3, but is also finds (number of) factors and stores all results in global variables. Thus it has some overhead for this specific task.
/*REXX pgm does prime decomposition of a range of positive integers (with a prime count)*/
call time('r')
Numeric Digits 1000 /*handle thousand digits For the powers*/
Parse Arg bot top step base add /*get optional arguments from the C.L. */
If bot='qq' Then Do
Say 'rexx pfoo bot top step base add'
Exit
End
If bot=='' Then /* no arguments given */
Parse Value 1 100 With bot top /* set default range .*/
If top=='' Then top=bot /* process one number */
If step=='' Then step=1 /* step=2 to process only odd numbers */
If add =='' Then add=-1 /* for Mersenne tests */
tell=top>0 /*If TOP is negative, suppress displays*/
top=abs(top)
w=length(top) /*get maximum width For aligned display*/
If base\=='' Then
w=length(base**top) /*will be testing powers of two laterqq */
tag.=left('', 7) /*some literals: pad; prime (or not).*/
tag.0='{unity}'
tag.1='[prime]'
Numeric Digits max(9,w+1) /*maybe increase the digits precision. */
np=0 /*np: is the number of primes found.*/
Do n=bot To top by step /*process a single number or a range. */
qq=n
If base\=='' Then /*should we perform a 'Mersenne' testqq */
qq=base**n+add
f=factors(qq) /* number of prime factors */
If f=1 Then /* If the number is prime */
np=np+1 /* Then bump prime counter */
If tell Then do
pf = ''
do i = 1 to f
pf = pf glob.factor.i
end
Say right(qq,w) right('('f')',9) 'prime factors: ' tag.f pf
end
End /*n*/
Say ''
ps='prime'
If f>1 Then ps=ps's' /*setup For proper English in sentence.*/
Say right(np, w+9+1) ps 'found.' /*display the number of primes found. */
say format(time('e'),,3) 'seconds'
Exit /*stick a fork in it, we're all done. */
Factors:
/* Prime factors of an integer */
procedure expose glob.
arg x
/* Validity */
if \ Whole(x) then
return 'X'
if x < 1 then
return 'X'
/* Fast values */
if x = 1 then do
glob.factor.0 = 0
return 0
end
if x < 4 then do
glob.factor.1 = x; glob.factor.0 = 1
return 1
end
if Prime(x) then do
glob.factor.1 = x; glob.factor.0 = 1
return 1
end
/* Check low factors */
n = 0
pr = '2 3 5 7 11 13 17 19 23'
do i = 1 to Words(pr)
p = Word(pr,i)
do while x//p = 0
n = n+1; glob.factor.n = p
x = x%p
end
end
/* Check higher factors */
do j = 29 by 6 while j*j <= x
p = Right(j,1)
if p <> 5 then do
do while x//j = 0
n = n+1; glob.factor.n = j
x = x%j
end
end
if p = 3 then
iterate
y = j+2
do while x//y = 0
n = n+1; glob.factor.n = y
x = x%y
end
end
/* Last factor */
if x > 1 then do
n = n+1; glob.factor.n = x
end
glob.factor.0 = n
/* Return number of factors */
return n
include Functions
include NumbersThe same run as given for Version 3 give:
- Output:
Input 9007199254740991 0.015 seconds Input 2543821448263974486045199 0.078 seconds Input 340282366920938463463374607431768211455 340282366920938463463374607431768211455 (9) prime factors: 3 5 17 257 641 65537 274177 6700417 67280421310721 2.077 seconds Input 1 -1000000 21.391 seconds Input 2 50 1 2 -1 0.203 seconds Input 2 50 1 2 +1 0.359 seconds
More or less the same performance.
Ring
prime = 18705
decomp(prime)
func decomp nr
x = ""
for i = 1 to nr
if isPrime(i) and nr % i = 0
x = x + string(i) + " * " ok
if i = nr
x2 = substr(x,1,(len(x)-2))
see string(nr) + " = " + x2 + nl ok
next
func isPrime num
if (num <= 1) return 0 ok
if (num % 2 = 0) and num != 2 return 0 ok
for i = 3 to floor(num / 2) -1 step 2
if (num % i = 0) return 0 ok
next
return 1RPL
≪ { } SWAP DUP √ CEIL → lim
≪ 2
WHILE OVER 1 > OVER lim ≤ AND REPEAT
DUP2 /
IF DUP FP
THEN DROP DUP 2 ≠ + 1 +
ELSE SWAP ROT DROP ROT OVER + ROT ROT END
END DROP
IF DUP 1 ≠ THEN + ELSE DROP END
≫ ≫ ‘PDIV’ STO
1048 PDIV
- Output:
1: { 2 2 2 131 }
Version for binary integers
≪ { } → pdiv
≪ #2
WHILE DUP2 DUP * ≥ REPEAT
IF DUP2 / LAST 3 PICK * -
THEN DROP 1 + #1 OR
ELSE ROT DROP SWAP pdiv OVER + 'pdiv' STO END
END DROP pdiv SWAP +
≫ ≫ ‘PDIVB’ STO
#1048 PDIVB
- Output:
1: { #2d #2d #2d #131d }
Ruby
Built in
irb(main):001:0> require 'prime'
=> true
irb(main):003:0> 2543821448263974486045199.prime_division
=> [[701, 1], [1123, 2], [2411, 1], [1092461, 2]]Simple algorithm
# Get prime decomposition of integer _i_.
# This routine is terribly inefficient, but elegance rules.
def prime_factors(i)
v = (2..i-1).detect{|j| i % j == 0}
v ? ([v] + prime_factors(i/v)) : [i]
end
# Example: Decompose all possible Mersenne primes up to 2**31-1.
# This may take several minutes to show that 2**31-1 is prime.
(2..31).each do |i|
factors = prime_factors(2**i-1)
puts "2**#{i}-1 = #{2**i-1} = #{factors.join(' * ')}"
end- Output:
... 2**28-1 = 268435455 = 3 * 5 * 29 * 43 * 113 * 127 2**29-1 = 536870911 = 233 * 1103 * 2089 2**30-1 = 1073741823 = 3 * 3 * 7 * 11 * 31 * 151 * 331 2**31-1 = 2147483647 = 2147483647
Faster algorithm
# Get prime decomposition of integer _i_.
# This routine is more efficient than prime_factors,
# and quite similar to Integer#prime_division of MRI 1.9.
def prime_factors_faster(i)
factors = []
check = proc do |p|
while(q, r = i.divmod(p)
r.zero?)
factors << p
i = q
end
end
check[2]
check[3]
p = 5
while p * p <= i
check[p]
p += 2
check[p]
p += 4 # skip multiples of 2 and 3
end
factors << i if i > 1
factors
end
# Example: Decompose all possible Mersenne primes up to 2**70-1.
# This may take several minutes to show that 2**61-1 is prime,
# but 2**62-1 and 2**67-1 are not prime.
(2..70).each do |i|
factors = prime_factors_faster(2**i-1)
puts "2**#{i}-1 = #{2**i-1} = #{factors.join(' * ')}"
end- Output:
... 2**67-1 = 147573952589676412927 = 193707721 * 761838257287 2**68-1 = 295147905179352825855 = 3 * 5 * 137 * 953 * 26317 * 43691 * 131071 2**69-1 = 590295810358705651711 = 7 * 47 * 178481 * 10052678938039 2**70-1 = 1180591620717411303423 = 3 * 11 * 31 * 43 * 71 * 127 * 281 * 86171 * 122921
This benchmark compares the different implementations.
require 'benchmark'
require 'mathn'
Benchmark.bm(24) do |x|
[2**25 - 6, 2**35 - 7].each do |i|
puts "#{i} = #{prime_factors_faster(i).join(' * ')}"
x.report(" prime_factors") { prime_factors(i) }
x.report(" prime_factors_faster") { prime_factors_faster(i) }
x.report(" Integer#prime_division") { i.prime_division }
end
endWith MRI 1.8, prime_factors is slow, Integer#prime_division is fast, and prime_factors_faster is very fast. With MRI 1.9, Integer#prime_division is also very fast.
Rust
Rust's largest built-in integer type is u128 (128-bit unsigned integer) which is pretty large, but not unlimited.
The solution therefore uses external crates for big integers.
Add the dependencies in Cargo.toml:
[package] name = "prime_decomposition" version = "0.1.1" edition = "2018" [dependencies] num-bigint = "0.3.0" num-traits = "0.2.12"
The implementation:
use num_bigint::BigUint;
use num_traits::{One, Zero};
use std::fmt::{Display, Formatter};
#[derive(Clone, Debug)]
pub struct Factors {
pub number: BigUint,
pub result: Vec<BigUint>,
}
impl Factors {
pub fn of(number: BigUint) -> Factors {
let mut factors = Self {
number: number.clone(),
result: Vec::new(),
};
let big_2 = BigUint::from(2u8);
let big_4 = BigUint::from(4u8);
factors.check(&big_2);
factors.check(&BigUint::from(3u8));
let mut divisor = BigUint::from(5u8);
while &divisor * &divisor <= factors.number {
factors.check(&divisor);
divisor += &big_2;
factors.check(&divisor);
divisor += &big_4;
}
if factors.number > BigUint::one() {
factors.result.push(factors.number);
}
factors.number = number; // Restore the number
factors
}
pub fn is_prime(&self) -> bool {
self.result.len() == 1
}
fn check(&mut self, divisor: &BigUint) {
while (&self.number % divisor).is_zero() {
self.result.push(divisor.clone());
self.number /= divisor;
}
}
}
impl Display for Factors {
fn fmt(&self, f: &mut Formatter) -> std::fmt::Result {
let mut iter = self.result.iter();
match iter.next() {
None => write!(f, "[]"),
Some(first) => {
write!(f, "[{}", first)?;
for next in iter {
write!(f, ", {}", next)?;
}
write!(f, "]")
}
}
}
}
fn print_factors(number: BigUint) {
let factors = Factors::of(number);
if factors.is_prime() {
println!("{} -> {} (prime)", factors.number, factors);
} else {
println!("{} -> {}", factors.number, factors);
}
}
fn main() {
print_factors(24u32.into());
print_factors(32u32.into());
print_factors(37u32.into());
// Find Mersenne primes
for n in 2..70 {
print!("2**{} - 1: ", n);
print_factors((BigUint::from(2u8) << n) - BigUint::one());
}
}Scala
import annotation.tailrec
import collection.parallel.mutable.ParSeq
object PrimeFactors extends App {
def factorize(n: Long): List[Long] = {
@tailrec
def factors(tuple: (Long, Long, List[Long], Int)): List[Long] = {
tuple match {
case (1, _, acc, _) => acc
case (n, k, acc, _) if (n % k == 0) => factors((n / k, k, acc ++ ParSeq(k), Math.sqrt(n / k).toInt))
case (n, k, acc, sqr) if (k < sqr) => factors(n, k + 1, acc, sqr)
case (n, k, acc, sqr) if (k >= sqr) => factors((1, k, acc ++ ParSeq(n), 0))
}
}
factors((n, 2, List[Long](), Math.sqrt(n).toInt))
}
def mersenne(p: Int): BigInt = (BigInt(2) pow p) - 1
def sieve(nums: Stream[Int]): Stream[Int] =
Stream.cons(nums.head, sieve((nums.tail) filter (_ % nums.head != 0)))
// An infinite stream of primes, lazy evaluation and memo-ized
val oddPrimes = sieve(Stream.from(3, 2))
def primes = sieve(2 #:: oddPrimes)
oddPrimes takeWhile (_ <= 59) foreach { p =>
{ // Needs some intermediate results for nice formatting
val numM = s"M${p}"
val nMersenne = mersenne(p).toLong
val lit = f"${nMersenne}%30d"
val datum = System.nanoTime
val result = factorize(nMersenne)
val mSec = ((System.nanoTime - datum) / 1.0e+6).round
def decStr = { if (lit.length > 30) f"(M has ${lit.length}%3d dec)" else "" }
def sPrime = { if (result.isEmpty) " is a prime number." else "" }
println(
f"$numM%4s = 2^$p%03d - 1 = ${lit}%s${sPrime} ($mSec%,4d msec) composed of ${result.mkString(" × ")}")
}
}
}- Output:
M3 = 2^003 - 1 = 7 ( 23 msec) composed of 7 M5 = 2^005 - 1 = 31 ( 0 msec) composed of 31 M7 = 2^007 - 1 = 127 ( 0 msec) composed of 127 M11 = 2^011 - 1 = 2047 ( 0 msec) composed of 23 × 89 M13 = 2^013 - 1 = 8191 ( 0 msec) composed of 8191 M17 = 2^017 - 1 = 131071 ( 1 msec) composed of 131071 M19 = 2^019 - 1 = 524287 ( 1 msec) composed of 524287 M23 = 2^023 - 1 = 8388607 ( 1 msec) composed of 47 × 178481 M29 = 2^029 - 1 = 536870911 ( 2 msec) composed of 233 × 1103 × 2089 M31 = 2^031 - 1 = 2147483647 ( 39 msec) composed of 2147483647 M37 = 2^037 - 1 = 137438953471 ( 8 msec) composed of 223 × 616318177 M41 = 2^041 - 1 = 2199023255551 ( 2 msec) composed of 13367 × 164511353 M43 = 2^043 - 1 = 8796093022207 ( 2 msec) composed of 431 × 9719 × 2099863 M47 = 2^047 - 1 = 140737488355327 ( 2 msec) composed of 2351 × 4513 × 13264529 M53 = 2^053 - 1 = 9007199254740991 ( 7 msec) composed of 6361 × 69431 × 20394401 M59 = 2^059 - 1 = 576460752303423487 ( 152 msec) composed of 179951 × 3203431780337
Getting the prime factors does not require identifying prime numbers. Since the problems seems to ask for it, here is one version that does it:
class PrimeFactors(n: BigInt) extends Iterator[BigInt] {
val zero = BigInt(0)
val one = BigInt(1)
val two = BigInt(2)
def isPrime(n: BigInt) = n.isProbablePrime(10)
var currentN = n
var prime = two
def nextPrime =
if (prime == two) {
prime += one
} else {
prime += two
while (!isPrime(prime)) {
prime += two
if (prime * prime > currentN)
prime = currentN
}
}
def next = {
if (!hasNext)
throw new NoSuchElementException("next on empty iterator")
while(currentN % prime != zero) {
nextPrime
}
currentN /= prime
prime
}
def hasNext = currentN != one && currentN > zero
}The method isProbablePrime(n) has a chance of 1 - 1/(2^n) of correctly identifying a prime. Next is a version that does not depend on identifying primes, and works with arbitrary integral numbers:
class PrimeFactors[N](n: N)(implicit num: Integral[N]) extends Iterator[N] {
import num._
val two = one + one
var currentN = n
var divisor = two
def next = {
if (!hasNext)
throw new NoSuchElementException("next on empty iterator")
while(currentN % divisor != zero) {
if (divisor == two)
divisor += one
else
divisor += two
if (divisor * divisor > currentN)
divisor = currentN
}
currentN /= divisor
divisor
}
def hasNext = currentN != one && currentN > zero
}- Output:
Both versions can be rather slow, as they accept arbitrarily big numbers, as requested.
- Test:
scala> BigInt(2) to BigInt(30) filter (_ isProbablePrime 10) map (p => (p, BigInt(2).pow(p.toInt) - 1)) foreach {
| case (prime, n) => println("2**"+prime+"-1 = "+n+", with factors: "+new PrimeFactors(n).mkString(", "))
| }
2**2-1 = 3, with factors: 3
2**3-1 = 7, with factors: 7
2**5-1 = 31, with factors: 31
2**7-1 = 127, with factors: 127
2**11-1 = 2047, with factors: 23, 89
2**13-1 = 8191, with factors: 8191
2**17-1 = 131071, with factors: 131071
2**19-1 = 524287, with factors: 524287
2**23-1 = 8388607, with factors: 47, 178481
2**29-1 = 536870911, with factors: 233, 1103, 2089
2**31-1 = 2147483647, with factors: 2147483647
2**37-1 = 137438953471, with factors: 223, 616318177
2**41-1 = 2199023255551, with factors: 13367, 164511353
2**43-1 = 8796093022207, with factors: 431, 9719, 2099863
2**47-1 = 140737488355327, with factors: 2351, 4513, 13264529
2**53-1 = 9007199254740991, with factors: 6361, 69431, 20394401
2**59-1 = 576460752303423487, with factors: 179951, 3203431780337
Alternatively, Scala LazyLists and Iterators support quite elegant one-line encodings of iterative/recursive algorithms, allowing us to to define the prime factorization like so:
import spire.math.SafeLong
import spire.implicits._
def pFactors(num: SafeLong): Vector[SafeLong] = Iterator.iterate((Vector[SafeLong](), num, SafeLong(2))){case (ac, n, f) => if(n%f == 0) (ac :+ f, n/f, f) else (ac, n, f + 1)}.dropWhile(_._2 != 1).next._1Scheme
(define (factor number)
(define (*factor divisor number)
(if (> (* divisor divisor) number)
(list number)
(if (= (modulo number divisor) 0)
(cons divisor (*factor divisor (/ number divisor)))
(*factor (+ divisor 1) number))))
(*factor 2 number))
(display (factor 111111111111))
(newline)- Output:
(3 7 11 13 37 101 9901)
Seed7
const func array integer: factorise (in var integer: number) is func
result
var array integer: result is 0 times 0;
local
var integer: checker is 2;
begin
while checker * checker <= number do
if number rem checker = 0 then
result &:= [](checker);
number := number div checker;
else
incr(checker);
end if;
end while;
if number <> 1 then
result &:= [](number);
end if;
end func;Original source: [1]
SequenceL
Recursive Using isPrime
isPrime(n) := n = 2 or (n > 1 and none(n mod ([2]++((1...floor(sqrt(n)/2))*2+1)) = 0));
primeFactorization(num) := primeFactorizationHelp(num, []);
primeFactorizationHelp(num, current(1)) :=
let
primeFactors[i] := i when num mod i = 0 and isPrime(i) foreach i within 2 ... num;
in
current when size(primeFactors) = 0
else
primeFactorizationHelp(num / product(primeFactors), current ++ primeFactors);Using isPrime Based On: [2]
Recursive Trial Division
primeFactorization(num) := primeFactorizationHelp(num, 2, []);
primeFactorizationHelp(num, divisor, factors(1)) :=
factors when num <= 1
else
primeFactorizationHelp(num, divisor + 1, factors) when num mod divisor /= 0
else
primeFactorizationHelp(num / divisor, divisor, factors ++ [divisor]);Sidef
Built-in:
say factor(536870911) #=> [233, 1103, 2089]
say factor_exp(536870911) #=> [[233, 1], [1103, 1], [2089, 1]]Trial division:
func prime_factors(n) {
return [] if (n < 1)
gather {
while (!(n & 1)) {
n >>= 1
take(2)
}
var p = 3
while ((n > 1) && (p*p <= n)) {
while (n %% p) {
n //= p
take(p)
}
p += 2
}
take(n) if (n > 1)
}
}Calling the function:
say prime_factors(536870911) #=> [233, 1103, 2089]Simula
Simula has no built-in function to test for prime numbers.
Code for class bignum can be found here: https://rosettacode.org/wiki/Pi#Simula
EXTERNAL CLASS BIGNUM;
BIGNUM
BEGIN
CLASS TEXTLIST;
BEGIN
CLASS TEXTARRAY(N); INTEGER N;
BEGIN
TEXT ARRAY DATA(1:N);
END TEXTARRAY;
PROCEDURE EXPAND(N); INTEGER N;
BEGIN
REF(TEXTARRAY) NEWARR;
INTEGER I;
NEWARR :- NEW TEXTARRAY(N);
FOR I := 1 STEP 1 UNTIL SIZE DO BEGIN
NEWARR.DATA(I) :- ARR.DATA(I);
END;
ARR :- NEWARR;
END EXPAND;
PROCEDURE APPEND(T); TEXT T;
BEGIN
IF SIZE = ARR.N THEN
EXPAND(2*ARR.N);
SIZE := SIZE+1;
ARR.DATA(SIZE) :- T;
END EXPAND;
TEXT PROCEDURE GET(I); INTEGER I;
GET :- ARR.DATA(I);
REF(TEXTARRAY) ARR;
INTEGER SIZE;
EXPAND(20);
END TEXTLIST;
REF(TEXTLIST) PROCEDURE PRIME_FACTORS(N); TEXT N;
BEGIN
REF(TEXTLIST) FACTORS;
REF(DIVMOD) DM;
TEXT P;
FACTORS :- NEW TEXTLIST;
IF TCMP(N, "1") < 0 THEN
GOTO RETURN;
P :- "2";
FOR DM :- TDIVMOD(N,P) WHILE TISZERO(DM.MOD) DO BEGIN
N :- DM.DIV;
FACTORS.APPEND(P);
END;
P :- "3";
WHILE TCMP(N,"1") > 0 AND THEN TCMP(TMUL(P,P),N) <= 0 DO BEGIN
FOR DM :- TDIVMOD(N, P) WHILE TISZERO(DM.MOD) DO BEGIN
N :- DM.DIV;
FACTORS.APPEND(P);
END;
P :- TADD(P,"2");
END;
IF TCMP(N,"1") > 0 THEN
FACTORS.APPEND(N);
RETURN:
PRIME_FACTORS :- FACTORS;
END PRIME_FACTORS;
REF(TEXTLIST) FACTORS;
TEXT INP;
INTEGER I;
FOR INP :- "536870911", "6768768", "1957", "64865899369365843" DO BEGIN
FACTORS :- PRIME_FACTORS(INP);
OUTTEXT("PRIME FACTORS OF ");
OUTTEXT(INP);
OUTTEXT(" => [");
FOR I := 1 STEP 1 UNTIL FACTORS.SIZE DO BEGIN
IF I > 1 THEN
OUTTEXT(", ");
OUTTEXT(FACTORS.GET(I));
END;
OUTTEXT("]");
OUTIMAGE;
END;
END;- Output:
PRIME FACTORS OF 536870911 => [233, 1103, 2089] PRIME FACTORS OF 6768768 => [2, 2, 2, 2, 2, 2, 2, 3, 17627] PRIME FACTORS OF 1957 => [19, 103] PRIME FACTORS OF 64865899369365843 => [3, 7, 397, 276229, 28166791] 5320 garbage collection(s) in 1.9 seconds.
Slate
Admittedly, this is just based on the Smalltalk entry below:
n@(Integer traits) primesDo: block
"Decomposes the Integer into primes, applying the block to each (in increasing
order)."
[| div next remaining |
div: 2.
next: 3.
remaining: n.
[[(remaining \\ div) isZero]
whileTrue:
[block applyTo: {div}.
remaining: remaining // div].
remaining = 1] whileFalse:
[div: next.
next: next + 2] "Just look at the next odd integer."
].Smalltalk
Integer extend [
primesDo: aBlock [
| div next rest |
div := 2. next := 3.
rest := self.
[ [ rest \\ div == 0 ]
whileTrue: [
aBlock value: div.
rest := rest // div ].
rest = 1] whileFalse: [
div := next. next := next + 2 ]
]
]
123456 primesDo: [ :each | each printNl ]SPAD
(1) -> factor 102400
12 2
(1) 2 5
Type: Factored(Integer)
(2) -> factor 23193931893819371
(2) 83 3469 71341 1129153
Type: Factored(Integer)Domain:Factored(R)
Standard ML
Trial division
val factor = fn n :IntInf.int =>
let
val unfactored = fn (u,_,_) => u
val factors = fn (_,f,_) => f
val try = fn (_,_,i) => i
fun getresult t = unfactored t::(factors t)
fun until done change x =
if done x
then getresult x
else until done change (change x); (* iteration *)
fun lastprime t = unfactored t < (try t)*(try t)
fun trymore t = if unfactored t mod (try t) = 0
then (unfactored t div (try t) , try t::(factors t) , try t )
else (unfactored t , factors t , try t + 1)
in
until lastprime trymore (n,[],2)
end;- Array.fromList(factor 122489234920000001278234798233);; val it = fromList[658601127263, 41259943, 34942753, 43, 3]: IntInf.int array
Stata
The following Mata function will factor any representable positive integer (that is, between 1 and 2^53).
function factor(n_) {
n = n_
a = J(0,2,.)
if (n<2) {
return(a)
}
else if (n<4) {
return((n,1))
}
else {
if (mod(n,2)==0) {
for (i=0; mod(n,2)==0; i++) n = floor(n/2)
a = a\(2,i)
}
for (k=3; k*k<=n; k=k+2) {
if (mod(n,k)==0) {
for (i=0; mod(n,k)==0; i++) n = floor(n/k)
a = a\(k,i)
}
}
if (n>1) a = a\(n,1)
return(a)
}
}Swift
Uses the sieve of Eratosthenes. This is generic on any type that conforms to BinaryInteger. So in theory any BigInteger library should work with it.
func primeDecomposition<T: BinaryInteger>(of n: T) -> [T] {
guard n > 2 else { return [] }
func step(_ x: T) -> T {
return 1 + (x << 2) - ((x >> 1) << 1)
}
let maxQ = T(Double(n).squareRoot())
var d: T = 1
var q: T = n % 2 == 0 ? 2 : 3
while q <= maxQ && n % q != 0 {
q = step(d)
d += 1
}
return q <= maxQ ? [q] + primeDecomposition(of: n / q) : [n]
}
for prime in Eratosthenes(upTo: 60) {
let m = Int(pow(2, Double(prime))) - 1
let decom = primeDecomposition(of: m)
print("2^\(prime) - 1 = \(m) => \(decom)")
}- Output:
2^2 - 1 = 3 => [3] 2^3 - 1 = 7 => [7] 2^5 - 1 = 31 => [31] 2^7 - 1 = 127 => [127] 2^11 - 1 = 2047 => [23, 89] 2^13 - 1 = 8191 => [8191] 2^17 - 1 = 131071 => [131071] 2^19 - 1 = 524287 => [524287] 2^23 - 1 = 8388607 => [47, 178481] 2^29 - 1 = 536870911 => [233, 1103, 2089] 2^31 - 1 = 2147483647 => [2147483647] 2^37 - 1 = 137438953471 => [223, 616318177] 2^41 - 1 = 2199023255551 => [13367, 164511353] 2^43 - 1 = 8796093022207 => [431, 9719, 2099863] 2^47 - 1 = 140737488355327 => [2351, 4513, 13264529] 2^53 - 1 = 9007199254740991 => [6361, 69431, 20394401] 2^59 - 1 = 576460752303423487 => [179951, 3203431780337]
Tcl
proc factors {x} {
# list the prime factors of x in ascending order
set result [list]
while {$x % 2 == 0} {
lappend result 2
set x [expr {$x / 2}]
}
for {set i 3} {$i*$i <= $x} {incr i 2} {
while {$x % $i == 0} {
lappend result $i
set x [expr {$x / $i}]
}
}
if {$x != 1} {lappend result $x}
return $result
}Testing
foreach m {2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59} {
set n [expr {2**$m - 1}]
catch {time {set primes [factors $n]} 1} tm
puts [format "2**%02d-1 = %-18s = %-22s => %s" $m $n [join $primes *] $tm]
}- Output:
2**02-1 = 3 = 3 => 184 microseconds per iteration 2**03-1 = 7 = 7 => 8 microseconds per iteration 2**05-1 = 31 = 31 => 8 microseconds per iteration 2**07-1 = 127 = 127 => 23 microseconds per iteration 2**11-1 = 2047 = 23*89 => 12 microseconds per iteration 2**13-1 = 8191 = 8191 => 22 microseconds per iteration 2**17-1 = 131071 = 131071 => 69 microseconds per iteration 2**19-1 = 524287 = 524287 => 131 microseconds per iteration 2**23-1 = 8388607 = 47*178481 => 81 microseconds per iteration 2**29-1 = 536870911 = 233*1103*2089 => 199 microseconds per iteration 2**31-1 = 2147483647 = 2147483647 => 9509 microseconds per iteration 2**37-1 = 137438953471 = 223*616318177 => 4377 microseconds per iteration 2**41-1 = 2199023255551 = 13367*164511353 => 2389 microseconds per iteration 2**43-1 = 8796093022207 = 431*9719*2099863 => 1711 microseconds per iteration 2**47-1 = 140737488355327 = 2351*4513*13264529 => 802 microseconds per iteration 2**53-1 = 9007199254740991 = 6361*69431*20394401 => 13109 microseconds per iteration 2**59-1 = 576460752303423487 = 179951*3203431780337 => 316009 microseconds per iteration
TXR
@(next :args)
@(do
(defun factor (n)
(if (> n 1)
(for ((max-d (isqrt n))
(d 2))
()
((inc d (if (evenp d) 1 2)))
(cond ((> d max-d) (return (list n)))
((zerop (mod n d))
(return (cons d (factor (trunc n d))))))))))
@{num /[0-9]+/}
@(bind factors @(factor (int-str num 10)))
@(output)
@num -> {@(rep)@factors, @(last)@factors@(end)}
@(end)- Output:
$ txr factor.txr 1139423842450982345
1139423842450982345 -> {5, 19, 37, 12782467, 25359769}
$ txr factor.txr 1
1 -> {}
$ txr factor.txr 2
2 -> {2}
$ txr factor.txr 3
3 -> {3}
$ txr factor.txr 2
2 -> {2}
$ txr factor.txr 3
3 -> {3}
$ txr factor.txr 4
4 -> {2, 2}
$ txr factor.txr 5
5 -> {5}
$ txr factor.txr 6
6 -> {2, 3}
V
like in scheme (using variables)
[prime-decomposition
[inner [c p] let
[c c * p >]
[p unit]
[ [p c % zero?]
[c c p c / inner cons]
[c 1 + p inner]
ifte]
ifte].
2 swap inner].(mostly) the same thing using stack (with out variables)
[prime-decomposition
[inner
[dup * <]
[pop unit]
[ [% zero?]
[ [p c : [c p c / c]] view i inner cons]
[succ inner]
ifte]
ifte].
2 inner].Using it
|1221 prime-decomposition puts=[3 11 37]
Wren
The examples are borrowed from the Go solution.
import "./big" for BigInt
import "./fmt" for Fmt
var vals = [1 << 31, 1234567, 333333, 987653, 2 * 3 * 5 * 7 * 11 * 13 * 17]
for (val in vals) {
Fmt.print("$10d -> $n", val, BigInt.primeFactors(val))
}- Output:
2147483648 -> [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]
1234567 -> [127, 9721]
333333 -> [3, 3, 7, 11, 13, 37]
987653 -> [29, 34057]
510510 -> [2, 3, 5, 7, 11, 13, 17]
XPL0
\ Prime decomposition
code Abs=0, Rem=2, CrLf=9, IntIn=10, IntOut=11, Text=12;
define MaxFacIndex = 30;
\ -(2^31) has most prime factors (31 twos) than other 32-bit signed integer.
integer I, N, Facs(MaxFacIndex), FacsCnt;
procedure CalcFacs(N, Facs, FacsCnt);
integer N, Facs, FacsCnt;
integer I, Cnt;
begin
N:= Abs(N);
Cnt:=0;
if N >= 2 then
begin
I:= 2;
while I * I <= N do
begin
if Rem(N / I) = 0 then
begin
N:= N / I;
Facs(Cnt):= I; Cnt:= Cnt + 1;
I:= 2
end
else
I:= I + 1
end;
Facs(Cnt):= N; Cnt:= Cnt + 1
end;
FacsCnt(0):= Cnt
end;
begin
Text(0, "Enter a number: "); N:= IntIn(0);
CalcFacs(N, Facs, addr FacsCnt);
for I:= 0 to FacsCnt - 2 do
begin
IntOut(0, Facs(I)); Text(0, " ")
end;
IntOut(0, Facs(FacsCnt - 1)); CrLf(0)
end- Output:
3 runs.
Enter a number: 32 2 2 2 2 2
Enter a number: 2520 2 2 2 3 3 5 7
Enter a number: 13 13
XSLT
Let's assume that in XSLT the application of a template is similar to the invocation of a function. So when the following template
<xsl:stylesheet xmlns:xsl="http://www.w3.org/1999/XSL/Transform" version="1.0">
<xsl:template match="/numbers">
<html>
<body>
<ul>
<xsl:apply-templates />
</ul>
</body>
</html>
</xsl:template>
<xsl:template match="number">
<li>
Number:
<xsl:apply-templates mode="value" />
Factors:
<xsl:apply-templates mode="factors" />
</li>
</xsl:template>
<xsl:template match="value" mode="value">
<xsl:apply-templates />
</xsl:template>
<xsl:template match="value" mode="factors">
<xsl:call-template name="generate">
<xsl:with-param name="number" select="number(current())" />
<xsl:with-param name="candidate" select="number(2)" />
</xsl:call-template>
</xsl:template>
<xsl:template name="generate">
<xsl:param name="number" />
<xsl:param name="candidate" />
<xsl:choose>
<!-- 1 is no prime and does not have any factors -->
<xsl:when test="$number = 1"></xsl:when>
<!-- if the candidate is larger than the sqrt of the number, it's prime and the last factor -->
<xsl:when test="$candidate * $candidate > $number">
<xsl:value-of select="$number" />
</xsl:when>
<!-- if the number is factored by the candidate, add the factor and try again with the same factor -->
<xsl:when test="$number mod $candidate = 0">
<xsl:value-of select="$candidate" />
<xsl:text> </xsl:text>
<xsl:call-template name="generate">
<xsl:with-param name="number" select="$number div $candidate" />
<xsl:with-param name="candidate" select="$candidate" />
</xsl:call-template>
</xsl:when>
<!-- else try again with the next factor -->
<xsl:otherwise>
<!-- increment by 2 to save stack depth -->
<xsl:choose>
<xsl:when test="$candidate = 2">
<xsl:call-template name="generate">
<xsl:with-param name="number" select="$number" />
<xsl:with-param name="candidate" select="$candidate + 1" />
</xsl:call-template>
</xsl:when>
<xsl:otherwise>
<xsl:call-template name="generate">
<xsl:with-param name="number" select="$number" />
<xsl:with-param name="candidate" select="$candidate + 2" />
</xsl:call-template>
</xsl:otherwise>
</xsl:choose>
</xsl:otherwise>
</xsl:choose>
</xsl:template>
</xsl:stylesheet>is applied against the document
<numbers>
<number><value>1</value></number>
<number><value>2</value></number>
<number><value>4</value></number>
<number><value>8</value></number>
<number><value>9</value></number>
<number><value>255</value></number>
</numbers>then the output contains the prime decomposition of each number:
<html>
<body>
<ul>
<li>
Number:
1
Factors:
</li>
<li>
Number:
2
Factors:
2</li>
<li>
Number:
4
Factors:
2 2</li>
<li>
Number:
8
Factors:
2 2 2</li>
<li>
Number:
9
Factors:
3 3</li>
<li>
Number:
255
Factors:
3 5 17</li>
</ul>
</body>
</html>zkl
With 64 bit ints:
fcn primeFactors(n){ // Return a list of factors of n
acc:=fcn(n,k,acc,maxD){ // k is 2,3,5,7,9,... not optimum
if(n==1 or k>maxD) acc.close();
else{
q,r:=n.divr(k); // divr-->(quotient,remainder)
if(r==0) return(self.fcn(q,k,acc.write(k),q.toFloat().sqrt()));
return(self.fcn(n,k+1+k.isOdd,acc,maxD))
}
}(n,2,Sink(List),n.toFloat().sqrt());
m:=acc.reduce('*,1); // mulitply factors
if(n!=m) acc.append(n/m); // opps, missed last factor
else acc;
}foreach n in (T(5,12, 2147483648, 2199023255551, 8796093022207,
9007199254740991, 576460752303423487)){
println(n,": ",primeFactors(n).concat(", "))
}- Output:
5: 5 12: 2, 2, 3 2147483648: 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 2199023255551: 13367, 164511353 8796093022207: 431, 9719, 2099863 9007199254740991: 6361, 69431, 20394401 576460752303423487: 179951, 3203431780337
Unfortunately, big ints (GMP) don't have (quite) the same interface as ints (since there is no big float, BI.toFloat() truncates to a double so BI.toFloat().sqrt() is wrong). So mostly duplicate code is needed:
fcn factorsBI(n){ // Return a list of factors of n
acc:=fcn(n,k,acc,maxD){ // k is 2,3,5,7,9,... not optimum
if(n==1 or k>maxD) acc.close();
else{
q,r:=n.div2(k); // divr-->(quotient,remainder)
if(r==0) return(self.fcn(q,k,acc.write(k),q.root(2)));
return(self.fcn(n,k+1+k.isOdd,acc,maxD))
}
}(n,2,Sink(List),n.root(2));
m:=acc.reduce('*,BN(1)); // mulitply factors
if(n!=m) acc.append(n/m); // opps, missed last factor
else acc;
}var BN=Import("zklBigNum");
foreach n in (T(BN("12"),
BN("340282366920938463463374607431768211455"))){
println(n,": ",factorsBI(n).concat(", "))
}- Output:
12: 2, 2, 3 340282366920938463463374607431768211455: 3, 5, 17, 257, 641, 65537, 274177, 6700417, 67280421310721