AKS test for primes

AKS test for primes
You are encouraged to solve this task according to the task description, using any language you may know.

The AKS algorithm for testing whether a number is prime is a polynomial-time algorithm based on an elementary theorem about Pascal triangles.

The theorem on which the test is based can be stated as follows:

•   a number   ${\displaystyle p}$   is prime   if and only if   all the coefficients of the polynomial expansion of
${\displaystyle (x-1)^{p}-(x^{p}-1)}$

are divisible by   ${\displaystyle p}$.

Example

Using   ${\displaystyle p=3}$:

         (x-1)^3 - (x^3 - 1)
= (x^3 - 3x^2 + 3x - 1) - (x^3 - 1)
= -3x^2 + 3x


And all the coefficients are divisible by 3,   so 3 is prime.

 Note:This task is not the AKS primality test.   It is an inefficient exponential time algorithm discovered in the late 1600s and used as an introductory lemma in the AKS derivation.

1. Create a function/subroutine/method that given   ${\displaystyle p}$   generates the coefficients of the expanded polynomial representation of   ${\displaystyle (x-1)^{p}}$.
2. Use the function to show here the polynomial expansions of   ${\displaystyle (x-1)^{p}}$   for   ${\displaystyle p}$   in the range   0   to at least   7,   inclusive.
3. Use the previous function in creating another function that when given   ${\displaystyle p}$   returns whether   ${\displaystyle p}$   is prime using the theorem.
4. Use your test to generate a list of all primes under   35.
5. As a stretch goal,   generate all primes under   50   (needs integers larger than 31-bit).

References

11l

Translation of: Python
F expand_x_1(p)
V ex = [BigInt(1)]
L(i) 0 .< p
ex.append(ex.last * -(p - i) I/ (i + 1))
R reversed(ex)

F aks_test(p)
I p < 2
R 0B
V ex = expand_x_1(p)
ex[0]++
R !any(ex[0 .< (len)-1].map(mult -> mult % @p != 0))

print(‘# p: (x-1)^p for small p’)
L(p) 12
print(‘#3: #.’.format(p, enumerate(expand_x_1(p)).map((n, e) -> ‘#.#.#.’.format(‘+’ * (e >= 0), e, I n {(‘x^#.’.format(n))} E ‘’)).join(‘ ’)))

print("\n# small primes using the aks test")
print((0..100).filter(p -> aks_test(p)))
Output:
# p: (x-1)^p for small p
0: +1
1: -1 +1x^1
2: +1 -2x^1 +1x^2
3: -1 +3x^1 -3x^2 +1x^3
4: +1 -4x^1 +6x^2 -4x^3 +1x^4
5: -1 +5x^1 -10x^2 +10x^3 -5x^4 +1x^5
6: +1 -6x^1 +15x^2 -20x^3 +15x^4 -6x^5 +1x^6
7: -1 +7x^1 -21x^2 +35x^3 -35x^4 +21x^5 -7x^6 +1x^7
8: +1 -8x^1 +28x^2 -56x^3 +70x^4 -56x^5 +28x^6 -8x^7 +1x^8
9: -1 +9x^1 -36x^2 +84x^3 -126x^4 +126x^5 -84x^6 +36x^7 -9x^8 +1x^9
10: +1 -10x^1 +45x^2 -120x^3 +210x^4 -252x^5 +210x^6 -120x^7 +45x^8 -10x^9 +1x^10
11: -1 +11x^1 -55x^2 +165x^3 -330x^4 +462x^5 -462x^6 +330x^7 -165x^8 +55x^9 -11x^10 +1x^11

# small primes using the aks test
[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]


8th

with: a

: nextrow  \ a -- a
len
[ ( drop [1] ),
( drop [1,1] ),
( ' n:+ y 1 slide 1 push ) ]
swap 2 min caseof ;

;with

with: n

: .x   \ n --
dup
[ ( drop ),
( drop "x" . ),
( "x^" . . ) ]
swap 2 min caseof space ;

: .term  \ coef exp -- ; omit coef for 1x^n when n > 0
over 1 = over 0 > and if  nip .x  else  swap . .x  then ;

: .sgn  \ +/-1 --
[ "-", null, "+" ]
swap 1+ caseof . space ;

: .lhs  \ n --
"(x-1)^" . . ;

: .rhs  \ a -- a
a:len 1- >r
1 swap ( third .sgn r@ rot - .term -1 * ) a:each
nip rdrop ;

: .eqn  \ a -- a
a:len 1- .lhs " = " . .rhs ;

: .binomials  \ --
[] ( nextrow .eqn cr ) 8 times drop ;

: primerow? \ a -- a ?
a:len 3 < if false ;then
1 a:@ >r   \ 2nd position is the number to check for primality
true swap ( nip dup 1 = swap r@ mod 0 = or and ) a:each swap
rdrop ;

: .primes-via-aks \ --
[] ( nextrow primerow? if 1 a:@ . space then ) 50 times drop ;

;with

.binomials cr
"The primes upto 50 are (via AKS): " . .primes-via-aks cr

bye
Output:
(x-1)^0 = + 1
(x-1)^1 = + x - 1
(x-1)^2 = + x^2 - 2x + 1
(x-1)^3 = + x^3 - 3x^2 + 3x - 1
(x-1)^4 = + x^4 - 4x^3 + 6x^2 - 4x + 1
(x-1)^5 = + x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1
(x-1)^6 = + x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1
(x-1)^7 = + x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1

The primes upto 50 are (via AKS): 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47


AArch64 Assembly

Works with: as version Raspberry Pi 3B version Buster 64 bits
or android 64 bits with application Termux
/* ARM assembly AARCH64 Raspberry PI 3B or android 64 bits */
/*  program AKS64.s   */

/*******************************************/
/* Constantes file                         */
/*******************************************/
/* for this file see task include a file in language AArch64 assembly*/
.include "../includeConstantesARM64.inc"
.equ MAXI,       64
.equ NUMBERLOOP, 10

/*********************************/
/* Initialized data              */
/*********************************/
.data
szMessResult:        .asciz " (x-1)^@ = "
szMessResult1:       .asciz " @ x^@   "
szMessResPrime:      .asciz "Number @ is prime. \n"
szCarriageReturn:    .asciz "\n"

/*********************************/
/* UnInitialized data            */
/*********************************/
.bss
sZoneConv:        .skip 24
qTabCoef:         .skip 8 * MAXI
/*********************************/
/*  code section                 */
/*********************************/
.text
.global main
main:                               // entry of program

mov x4,#1
1:                                  // loop
mov x0,x4
bl computeCoef                  // compute coefficient
mov x0,x4
bl displayCoef                  // display coefficient
cmp x4,NUMBERLOOP
blt 1b

mov x4,1
2:
mov x0,x4
bl isPrime                      // is prime ?
cmp x0,1
bne 3f
mov x0,x4
bl conversion10                  // call decimal conversion
strb wzr,[x1]
ldr x1,qAdrsZoneConv             // insert value conversion in message
bl strInsertAtCharInc
bl affichageMess

3:
cmp x4,MAXI
blt 2b

100:                                 // standard end of the program
mov x0,0                         // return code
mov x8,EXIT                      // request to exit program
svc 0                            // perform the system call

/***************************************************/
/*     display coefficients                        */
/***************************************************/
// x0 contains a number
displayCoef:
stp x1,lr,[sp,-16]!         // save  registres
stp x2,x3,[sp,-16]!         // save  registres
stp x4,x5,[sp,-16]!         // save  registres
stp x6,x7,[sp,-16]!         // save  registres
mov x2,x0
bl conversion10             // call decimal conversion
strb wzr,[x1]
ldr x1,qAdrsZoneConv        // insert value conversion in message
bl strInsertAtCharInc
bl affichageMess
1:
ldr x0,[x3,x2,lsl #3]
bl conversion10S            // call decimal conversion
2:                              // removing spaces
ldrb w6,[x1]
cmp x6,' '
cinc x1,x1,eq
beq 2b

bl strInsertAtCharInc
mov x4,x0
mov x0,x2
ldr x1,qAdrsZoneConv        // else display odd message
bl conversion10             // call decimal conversion
strb wzr,[x1]
mov x0,x4
ldr x1,qAdrsZoneConv        // insert value conversion in message
bl strInsertAtCharInc
bl affichageMess
subs x2,x2,#1
bge 1b

bl affichageMess
100:
ldp x6,x7,[sp],16         // restaur des  2 registres
ldp x4,x5,[sp],16         // restaur des  2 registres
ldp x2,x3,[sp],16         // restaur des  2 registres
ldp x1,lr,[sp],16         // restaur des  2 registres
ret
/***************************************************/
/*     compute coefficient               */
/***************************************************/
// x0 contains a number
computeCoef:
stp x1,lr,[sp,-16]!         // save  registres
stp x2,x3,[sp,-16]!         // save  registres
stp x4,x5,[sp,-16]!         // save  registres
stp x6,x7,[sp,-16]!         // save  registres
mov x2,1
str x2,[x1]                 // store 1 to coeff [0]
mov x3,0                    // indice 1
1:
mov x5,1
str x5,[x1,x4,lsl #3]
mov x6,x3                   // indice 2 = indice 1
2:
cmp x6,0                    // zero ? -> end loop
ble 3f
sub x4,x6,1
ldr x5,[x1,x4,lsl 3]
ldr x4,[x1,x6,lsl 3]
sub x5,x5,x4
str x5,[x1,x6,lsl 3]
sub x6,x6,1
b 2b
3:
ldr x2,[x1]                 // inversion coeff [0]
neg x2,x2
str x2,[x1]
cmp x3,x0
blt 1b

100:
ldp x6,x7,[sp],16         // restaur des  2 registres
ldp x4,x5,[sp],16         // restaur des  2 registres
ldp x2,x3,[sp],16         // restaur des  2 registres
ldp x1,lr,[sp],16         // restaur des  2 registres
ret
/***************************************************/
/*     verify number is prime              */
/***************************************************/
// x0 contains a number
isPrime:
stp x1,lr,[sp,-16]!         // save  registres
stp x2,x3,[sp,-16]!         // save  registres
stp x4,x5,[sp,-16]!         // save  registres
bl computeCoef
ldr x2,[x4]
str x2,[x4]
ldr x2,[x4,x0,lsl 3]
sub x2,x2,#1
str x2,[x4,x0,lsl 3]
mov x5,x0                  // number start
1:
ldr x1,[x4,x5,lsl 3]       // load one coeff
sdiv x2,x1,x0
msub x3,x2,x0,x1           // compute remainder
cmp x3,#0                  // remainder = zéro ?
bne 99f                    // if <> no prime
subs x5,x5,#1              // next coef
bgt 1b                     // and loop
mov x0,#1                  // prime
b 100f
99:
mov x0,0                   // no prime
100:
ldp x4,x5,[sp],16         // restaur des  2 registres
ldp x2,x3,[sp],16         // restaur des  2 registres
ldp x1,lr,[sp],16         // restaur des  2 registres
ret
/********************************************************/
/*        File Include fonctions                        */
/********************************************************/
/* for this file see task include a file in language AArch64 assembly */
.include "../includeARM64.inc"
Output:
 (x-1)^1 =  +1 x^1    -1 x^0
(x-1)^2 =  +1 x^2    -2 x^1    +1 x^0
(x-1)^3 =  +1 x^3    -3 x^2    +3 x^1    -1 x^0
(x-1)^4 =  +1 x^4    -4 x^3    +6 x^2    -4 x^1    +1 x^0
(x-1)^5 =  +1 x^5    -5 x^4    +10 x^3    -10 x^2    +5 x^1    -1 x^0
(x-1)^6 =  +1 x^6    -6 x^5    +15 x^4    -20 x^3    +15 x^2    -6 x^1    +1 x^0
(x-1)^7 =  +1 x^7    -7 x^6    +21 x^5    -35 x^4    +35 x^3    -21 x^2    +7 x^1    -1 x^0
(x-1)^8 =  +1 x^8    -8 x^7    +28 x^6    -56 x^5    +70 x^4    -56 x^3    +28 x^2    -8 x^1    +1 x^0
(x-1)^9 =  +1 x^9    -9 x^8    +36 x^7    -84 x^6    +126 x^5    -126 x^4    +84 x^3    -36 x^2    +9 x^1    -1 x^0
Number 1 is prime.
Number 2 is prime.
Number 3 is prime.
Number 5 is prime.
Number 7 is prime.
Number 11 is prime.
Number 13 is prime.
Number 17 is prime.
Number 19 is prime.
Number 23 is prime.
Number 29 is prime.
Number 31 is prime.
Number 37 is prime.
Number 41 is prime.
Number 43 is prime.
Number 47 is prime.
Number 53 is prime.
Number 59 is prime.
Number 61 is prime.


with Ada.Text_IO;

procedure Test_For_Primes is

type Pascal_Triangle_Type is array (Natural range <>) of Long_Long_Integer;

function Calculate_Pascal_Triangle (N : in Natural) return Pascal_Triangle_Type is
Pascal_Triangle : Pascal_Triangle_Type (0 .. N);
begin
Pascal_Triangle (0) := 1;
for I in Pascal_Triangle'First .. Pascal_Triangle'Last - 1 loop
Pascal_Triangle (1 + I) := 1;
for J in reverse 1 .. I loop
Pascal_Triangle (J) := Pascal_Triangle (J - 1) - Pascal_Triangle (J);
end loop;
Pascal_Triangle (0) := -Pascal_Triangle (0);
end loop;
return Pascal_Triangle;
end Calculate_Pascal_Triangle;

function Is_Prime (N : Integer) return Boolean is
I      : Integer;
Result : Boolean := True;
Pascal_Triangle : constant Pascal_Triangle_Type := Calculate_Pascal_Triangle (N);
begin
I := N / 2;
while Result and I > 1 loop
Result := Result and Pascal_Triangle (I) mod Long_Long_Integer (N) = 0;
I := I - 1;
end loop;
return Result;
end Is_Prime;

function Image (N    : in Long_Long_Integer;
Sign : in Boolean := False) return String is
Image : constant String := N'Image;
begin
if N < 0 then
return Image;
else
if Sign then
return "+" & Image (Image'First + 1 .. Image'Last);
else
return Image (Image'First + 1 .. Image'Last);
end if;
end if;
end Image;

procedure Show (Triangle : in Pascal_Triangle_Type) is
Begin
for I in reverse Triangle'Range loop
Put (Image (Triangle (I), Sign => True));
Put ("x^");
Put (Image (Long_Long_Integer (I)));
Put (" ");
end loop;
end Show;

procedure Show_Pascal_Triangles is
begin
for N in 0 .. 9 loop
declare
Pascal_Triangle : constant Pascal_Triangle_Type := Calculate_Pascal_Triangle (N);
begin
Put ("(x-1)^" & Image (Long_Long_Integer (N)) & " = ");
Show (Pascal_Triangle);
New_Line;
end;
end loop;
end Show_Pascal_Triangles;

procedure Show_Primes is
begin
for N in 2 .. 63 loop
if Is_Prime (N) then
Put (N'Image);
end if;
end loop;
New_Line;
end Show_Primes;

begin
Show_Pascal_Triangles;
Show_Primes;
end Test_For_Primes;

Output:
(x-1)^0 = +1x^0
(x-1)^1 = +1x^1 -1x^0
(x-1)^2 = +1x^2 -2x^1 +1x^0
(x-1)^3 = +1x^3 -3x^2 +3x^1 -1x^0
(x-1)^4 = +1x^4 -4x^3 +6x^2 -4x^1 +1x^0
(x-1)^5 = +1x^5 -5x^4 +10x^3 -10x^2 +5x^1 -1x^0
(x-1)^6 = +1x^6 -6x^5 +15x^4 -20x^3 +15x^2 -6x^1 +1x^0
(x-1)^7 = +1x^7 -7x^6 +21x^5 -35x^4 +35x^3 -21x^2 +7x^1 -1x^0
(x-1)^8 = +1x^8 -8x^7 +28x^6 -56x^5 +70x^4 -56x^3 +28x^2 -8x^1 +1x^0
(x-1)^9 = +1x^9 -9x^8 +36x^7 -84x^6 +126x^5 -126x^4 +84x^3 -36x^2 +9x^1 -1x^0
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61


ALGOL 68

The code below uses Algol 68 Genie which provides arbitrary precision arithmetic for LONG LONG modes.

BEGIN
COMMENT
Mathematical preliminaries.

First note that the homogeneous polynomial (a+b)^n is symmetrical
(to see this just swap the variables a and b).  Therefore its
coefficients need be calculated only to that of (ab)^{n/2} for even
n or (ab)^{(n-1)/2} for odd n.

Second, the coefficients are the binomial coefficients C(n,k) where
the coefficient of a^k b^(n-k) is C(n,k) = n! / k!  (k-1)!.  This
leads to an immediate and relatively efficient implementation for
which we do not need to compute n! before dividing by k! and (k-1)!
but, rather cancel common factors as we go along.  Further, the
well-known symmetry identity C(n,k) = C(n, n-k) allows a
significant reduction in computational effort.

Third, (x-1)^n is the value of (a + b)^n when a=x and b = -1.  The
powers of -1 alternate between +1 and -1 so we may as well compute
(x+1)^n and negate every other coefficient when printing.
COMMENT
PR precision=300 PR
MODE LLI = LONG LONG INT;	CO For brevity CO
PROC choose = (INT n, k) LLI :
BEGIN
LLI result := 1;
INT sym k := (k >= n%2 | n-k | k);	CO Use symmetry CO
IF sym k > 0 THEN
FOR i FROM 0 TO sym k-1
DO
result TIMESAB (n-i);
result OVERAB (i+1)
OD
FI;
result
END;
PROC coefficients = (INT n) [] LLI :
BEGIN
[0:n] LLI a;
FOR i FROM 0 TO n%2
DO
a[i] := a[n-i] := choose (n, i)		CO Use symmetry CO
OD;
a
END;
COMMENT
First print the polynomials (x-1)^n, remembering to alternate signs
and to tidy up the constant term, the x^1 term and the x^n term.
This means we must treat (x-1)^0 and (x-1)^1 specially
COMMENT
FOR n FROM 0 TO 7
DO
[0:n] LLI a := coefficients (n);
printf (($"(x-1)^", g(0), " = "$, n));
CASE n+1 IN
printf (($g(0)l$, a[0])),
printf (($"x - ", g(0)l$, a[1]))
OUT
printf (($"x^", g(0)$, n));
FOR i TO n-2
DO
printf (($xax, g(0), "x^", g(0)$, (ODD i | "-" | "+"), a[i], n-i))
OD;
printf (($xax, g(0), "x"$, (ODD (n-1) | "-" | "+"), a[n-1]));
printf (($xaxg(0)l$, (ODD n | "-" | "+"), a[n]))
ESAC
OD;
COMMENT
Finally, for the "AKS" portion of the task, the sign of the
coefficient has no effect on its divisibility by p so, once again,
we may as well use the positive coefficients.  Symmetry clearly
reduces the necessary number of tests by a factor of two.
COMMENT
PROC is prime = (INT n) BOOL :
BEGIN
BOOL prime := TRUE;
FOR i FROM 1 TO n%2 WHILE prime DO prime := choose (n, i) MOD n = 0 OD;
prime
END;
print ("Primes < 50 are ");
FOR n FROM 2 TO 50 DO (is prime (n) | printf (($g(0)x$, n)) ) OD;
print (newline);
print ("And just to show off, the primes between 900 and 1000 are ");
FOR n FROM 900 TO 1000 DO IF is prime (n) THEN printf (($g(0)x$, n)) FI OD;
print (newline)
END
Output:
(x-1)^0 = 1
(x-1)^1 = x - 1
(x-1)^2 = x^2 - 2x + 1
(x-1)^3 = x^3 - 3x^2 + 3x - 1
(x-1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1
(x-1)^5 = x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1
(x-1)^6 = x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1
(x-1)^7 = x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1
Primes < 50 are 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
And just to show off, the primes between 900 and 1000 are 907 911 919 929 937 941 947 953 967 971 977 983 991 997


ARM Assembly

Works with: as version Raspberry Pi
or android 32 bits with application Termux
/* ARM assembly Raspberry PI  or android 32 bits */
/*  program AKS.s   */

/* REMARK 1 : this program use routines in a include file
see task Include a file language arm assembly
for the routine affichageMess conversion10
see at end of this program the instruction include */
/* for constantes see task include a file in arm assembly */
/************************************/
/* Constantes                       */
/************************************/
.include "../constantes.inc"
.equ MAXI,       32
.equ NUMBERLOOP, 10

/*********************************/
/* Initialized data              */
/*********************************/
.data
szMessResult:        .asciz " (x-1)^@ = "
szMessResult1:       .asciz " @ x^@   "
szMessResPrime:      .asciz "Number @ is prime. \n"
szCarriageReturn:    .asciz "\n"

/*********************************/
/* UnInitialized data            */
/*********************************/
.bss
sZoneConv:        .skip 24
iTabCoef:         .skip 4 * MAXI
/*********************************/
/*  code section                 */
/*********************************/
.text
.global main
main:                               @ entry of program

mov r4,#1
1:                                  @ loop
mov r0,r4
bl computeCoef                  @ compute coefficient
mov r0,r4
bl displayCoef                  @ display coefficient
cmp r4,#NUMBERLOOP
blt 1b

mov r4,#1
2:
mov r0,r4
bl isPrime                      @ is prime ?
cmp r0,#1
bne 3f
mov r0,r4
bl conversion10                  @ call decimal conversion
mov r5,#0
strb r5,[r1]
ldr r1,iAdrsZoneConv             @ insert value conversion in message
bl strInsertAtCharInc
bl affichageMess

3:
cmp r4,#MAXI
blt 2b

100:                                  @ standard end of the program
mov r0, #0                        @ return code
mov r7, #EXIT                     @ request to exit program
svc #0                            @ perform the system call

/***************************************************/
/*     display coefficients                        */
/***************************************************/
// r0 contains a number
displayCoef:
push {r1-r6,lr}             @ save  registers
mov r2,r0
bl conversion10             @ call decimal conversion
mov r5,#0
strb r5,[r1]
ldr r1,iAdrsZoneConv        @ insert value conversion in message
bl strInsertAtCharInc
bl affichageMess
1:
ldr r0,[r3,r2,lsl #2]
bl conversion10S            @ call decimal conversion
2:                              @ removing spaces
ldrb r6,[r1]
cmp r6,#' '
beq 2b

bl strInsertAtCharInc
mov r4,r0
mov r0,r2
ldr r1,iAdrsZoneConv        @ else display odd message
bl conversion10             @ call decimal conversion
mov r5,#0
strb r5,[r1]
mov r0,r4
ldr r1,iAdrsZoneConv        @ insert value conversion in message
bl strInsertAtCharInc
bl affichageMess
subs r2,r2,#1
bge 1b

bl affichageMess
100:
pop {r1-r6,lr}             @ restaur registers
bx lr                      @ return
/***************************************************/
/*     compute coefficient               */
/***************************************************/
// r0 contains a number
computeCoef:
push {r1-r6,lr}             @ save  registers
mov r2,#1
str r2,[r1]                 @ store 1 to coeff [0]
mov r3,#0                   @ indice 1
1:
mov r5,#1
str r5,[r1,r4,lsl #2]
mov r6,r3                   @ indice 2 = indice 1
2:
cmp r6,#0                   @ zero ? -> end loop
ble 3f
sub r4,r6,#1
ldr r5,[r1,r4,lsl #2]
ldr r4,[r1,r6,lsl #2]
sub r5,r5,r4
str r5,[r1,r6,lsl #2]
sub r6,r6,#1
b 2b
3:
ldr r2,[r1]                 @ inversion coeff [0]
neg r2,r2
str r2,[r1]
cmp r3,r0
blt 1b

100:
pop {r1-r6,lr}             @ restaur registers
bx lr                      @ return
/***************************************************/
/*     verify number is prime              */
/***************************************************/
// r0 contains a number
isPrime:
push {r1-r5,lr}             @ save  registers
bl computeCoef
ldr r2,[r4]
str r2,[r4]
ldr r2,[r4,r0,lsl #2]
sub r2,r2,#1
str r2,[r4,r0,lsl #2]
mov r5,r0                  @ number start
mov r1,r0                  @ divisor
1:
ldr r0,[r4,r5,lsl #2]      @ load one coeff
cmp r0,#0                  @ if negative inversion
neglt r0,r0
bl division                @ because this routine is number positive only
cmp r3,#0                  @ remainder = zéro ?
movne r0,#0                @  if <> no prime
bne 100f
subs r5,r5,#1              @ next coef
bgt 1b
mov r0,#1                  @ prime

100:
pop {r1-r5,lr}             @ restaur registers
bx lr                      @ return
/***************************************************/
/*      ROUTINES INCLUDE                           */
/***************************************************/
.include "../affichage.inc"
Output:
 (x-1)^1 =  +1 x^1    -1 x^0
(x-1)^2 =  +1 x^2    -2 x^1    +1 x^0
(x-1)^3 =  +1 x^3    -3 x^2    +3 x^1    -1 x^0
(x-1)^4 =  +1 x^4    -4 x^3    +6 x^2    -4 x^1    +1 x^0
(x-1)^5 =  +1 x^5    -5 x^4    +10 x^3    -10 x^2    +5 x^1    -1 x^0
(x-1)^6 =  +1 x^6    -6 x^5    +15 x^4    -20 x^3    +15 x^2    -6 x^1    +1 x^0
(x-1)^7 =  +1 x^7    -7 x^6    +21 x^5    -35 x^4    +35 x^3    -21 x^2    +7 x^1    -1 x^0
(x-1)^8 =  +1 x^8    -8 x^7    +28 x^6    -56 x^5    +70 x^4    -56 x^3    +28 x^2    -8 x^1    +1 x^0
(x-1)^9 =  +1 x^9    -9 x^8    +36 x^7    -84 x^6    +126 x^5    -126 x^4    +84 x^3    -36 x^2    +9 x^1    -1 x^0
Number 1 is prime.
Number 2 is prime.
Number 3 is prime.
Number 5 is prime.
Number 7 is prime.
Number 11 is prime.
Number 13 is prime.
Number 17 is prime.
Number 19 is prime.
Number 23 is prime.
Number 29 is prime.
Number 31 is prime.


AutoHotkey

Works with: AutoHotkey L
; 1. Create a function/subroutine/method that given p generates the coefficients of the expanded polynomial representation of (x-1)^p.
; Function modified from http://rosettacode.org/wiki/Pascal%27s_triangle#AutoHotkey
pascalstriangle(n=8) ; n rows of Pascal's triangle
{
p := Object(), z:=Object()
Loop, % n
Loop, % row := A_Index
col := A_Index
, p[row, col] := row = 1 and col = 1
? 1
: (p[row-1, col-1] = "" ; math operations on blanks return blanks; I want to assume zero
? 0
: p[row-1, col-1])
- (p[row-1, col] = ""
? 0
: p[row-1, col])
Return p
}

; 2. Use the function to show here the polynomial expansions of p for p in the range 0 to at least 7, inclusive.
For k, v in pascalstriangle()
{
s .= "n(x-1)^" k-1 . "="
For k, w in v
s .= "+" w "x^" k-1
}
s := RegExReplace(s, "\+-", "-")
s := RegExReplace(s, "x\^0", "")
s := RegExReplace(s, "x\^1", "x")
Msgbox % clipboard := s

; 3. Use the previous function in creating another function that when given p returns whether p is prime using the AKS test.
aks(n)
{
isnotprime := False
For k, v in pascalstriangle(n+1)[n+1]
(k != 1 and k != n+1) ? isnotprime |= !(v // n = v / n) ; if any is not divisible, returns true
Return !isnotprime
}

; 4. Use your AKS test to generate a list of all primes under 35.
i := 49
p := pascalstriangle(i+1)
Loop, % i
{
n := A_Index
isnotprime := False
For k, v in p[n+1]
(k != 1 and k != n+1) ? isnotprime |= !(v // n = v / n) ; if any is not divisible, returns true
t .= isnotprime ? "" : A_Index " "
}
Msgbox % t
Return

Output:
(x-1)^0=+1
(x-1)^1=-1+1x
(x-1)^2=+1-2x+1x^2
(x-1)^3=-1+3x-3x^2+1x^3
(x-1)^4=+1-4x+6x^2-4x^3+1x^4
(x-1)^5=-1+5x-10x^2+10x^3-5x^4+1x^5
(x-1)^6=+1-6x+15x^2-20x^3+15x^4-6x^5+1x^6
(x-1)^7=-1+7x-21x^2+35x^3-35x^4+21x^5-7x^6+1x^7

1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47

Function maxes out at i = 61 as AutoHotkey supports up to 64-bit signed integers.

Bracmat

Bracmat automatically normalizes symbolic expressions with the algebraic binary operators +, *, ^ and \L (logartithm). It can differentiate such expressions using the \D binary operator. (These operators were implemented in Bracmat before all other operators!). Some algebraic values can exist in two evaluated forms. The equivalent x*(a+b) and x*a+x*b are both considered "normal", but x*(a+b)+-1 is not, and therefore expanded to -1+a*x+b*x. This is used in the forceExpansion function to convert e.g. x*(a+b) to x*a+x*b.

The primality test uses a pattern that looks for a fractional factor. If such a factor is found, the test fails. Otherwise it succeeds.

( (forceExpansion=.1+!arg+-1)
& (expandx-1P=.forceExpansion$((x+-1)^!arg)) & ( isPrime = . forceExpansion$ (!arg^-1*(expandx-1P$!arg+-1*(x^!arg+-1))) : ?+/*?+? & ~ | ) & out$"Polynomial representations of (x-1)^p for p <= 7 :"
& -1:?n
&   whl
' ( 1+!n:~>7:?n
& out$(str$("n=" !n ":") expandx-1P$!n) ) & 1:?n & :?primes & whl ' ( 1+!n:~>50:?n & ( isPrime$!n&!primes !n:?primes
|
)
)
& out$"2 <= Primes <= 50:" & out$!primes
);

Output:

Polynomial representations of (x-1)^p for p <= 7 :
n=0: 1
n=1: -1+x
n=2: 1+-2*x+x^2
n=3: -1+3*x+-3*x^2+x^3
n=4: 1+-4*x+6*x^2+-4*x^3+x^4
n=5: -1+5*x+-10*x^2+10*x^3+-5*x^4+x^5
n=6: 1+-6*x+15*x^2+-20*x^3+15*x^4+-6*x^5+x^6
n=7:
-1
+ 7*x
+ -21*x^2
+ 35*x^3
+ -35*x^4
+ 21*x^5
+ -7*x^6
+ x^7
2 <= Primes <= 50:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47

The AKS test kan be written more concisely than the task describes. This prints the primes between 980 and 1000:

( out$"Primes between 980 and 1000, short version:" & 980:?n & whl ' ( !n+1:<1000:?n & ( 1+!n^-1*((x+-1)^!n+-1*(x^!n+-1))+-1:?+/*?+? | out$!n
)
)
);

Output:

Primes between 980 and 1000, short version:
983
991
997

C

#include <stdio.h>
#include <stdlib.h>

long long c[100];

void coef(int n)
{
int i, j;

if (n < 0 || n > 63) abort(); // gracefully deal with range issue

for (c[i=0] = 1; i < n; c[0] = -c[0], i++)
for (c[1 + (j=i)] = 1; j > 0; j--)
c[j] = c[j-1] - c[j];
}

int is_prime(int n)
{
int i;

coef(n);
c[0] += 1, c[i=n] -= 1;
while (i-- && !(c[i] % n));

return i < 0;
}

void show(int n)
{
do printf("%+lldx^%d", c[n], n); while (n--);
}

int main(void)
{
int n;

for (n = 0; n < 10; n++) {
coef(n);
printf("(x-1)^%d = ", n);
show(n);
putchar('\n');
}

printf("\nprimes (never mind the 1):");
for (n = 1; n <= 63; n++)
if (is_prime(n))
printf(" %d", n);

putchar('\n');
return 0;
}


The ugly output:

(x-1)^0 = +1x^0
(x-1)^1 = +1x^1-1x^0
(x-1)^2 = +1x^2-2x^1+1x^0
(x-1)^3 = +1x^3-3x^2+3x^1-1x^0
(x-1)^4 = +1x^4-4x^3+6x^2-4x^1+1x^0
(x-1)^5 = +1x^5-5x^4+10x^3-10x^2+5x^1-1x^0
(x-1)^6 = +1x^6-6x^5+15x^4-20x^3+15x^2-6x^1+1x^0
(x-1)^7 = +1x^7-7x^6+21x^5-35x^4+35x^3-21x^2+7x^1-1x^0
(x-1)^8 = +1x^8-8x^7+28x^6-56x^5+70x^4-56x^3+28x^2-8x^1+1x^0
(x-1)^9 = +1x^9-9x^8+36x^7-84x^6+126x^5-126x^4+84x^3-36x^2+9x^1-1x^0

primes (never mind the 1): 1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61


C#

Translation of: C
using System;
public class AksTest
{
static long[] c = new long[100];

static void Main(string[] args)
{
for (int n = 0; n < 10; n++) {
coef(n);
Console.Write("(x-1)^" + n + " = ");
show(n);
Console.WriteLine("");
}
Console.Write("Primes:");
for (int n = 1; n <= 63; n++)
if (is_prime(n))
Console.Write(n + " ");

Console.WriteLine('\n');
}

static void coef(int n)
{
int i, j;

if (n < 0 || n > 63) System.Environment.Exit(0);// gracefully deal with range issue

for (c[i = 0] = 1L; i < n; c[0] = -c[0], i++)
for (c[1 + (j = i)] = 1L; j > 0; j--)
c[j] = c[j - 1] - c[j];
}

static bool is_prime(int n)
{
int i;

coef(n);
c[0] += 1;
c[i = n] -= 1;

while (i-- != 0 && (c[i] % n) == 0) ;

return i < 0;
}

static void show(int n)
{
do {
Console.Write("+" + c[n] + "x^" + n);
}while (n-- != 0);
}
}


C++

Translation of: Pascal
#include <iomanip>
#include <iostream>
using namespace std;

const int pasTriMax = 61;

uint64_t pasTri[pasTriMax + 1];

void pascalTriangle(unsigned long n)
// Calculate the n'th line 0.. middle
{
unsigned long j, k;

pasTri[0] = 1;
j = 1;
while (j <= n)
{
j++;
k = j / 2;
pasTri[k] = pasTri[k - 1];
for ( ;k >= 1; k--)
pasTri[k] += pasTri[k - 1];
}
}

bool isPrime(unsigned long n)
{
if (n > pasTriMax)
{
cout << n << " is out of range" << endl;
exit(1);
}

pascalTriangle(n);
bool res = true;
int i = n / 2;
while (res && (i > 1))
{
res = res && (pasTri[i] % n == 0);
i--;
}
return res;
}

void expandPoly(unsigned long n)
{
const char vz[] = {'+', '-'};

if (n > pasTriMax)
{
cout << n << " is out of range" << endl;
exit(1);
}

switch (n)
{
case 0:
cout << "(x-1)^0 = 1" << endl;
break;
case 1:
cout << "(x-1)^1 = x-1" << endl;
break;
default:
pascalTriangle(n);
cout << "(x-1)^" << n << " = ";
cout << "x^" << n;
bool bVz = true;
int nDiv2 = n / 2;
for (unsigned long j = n - 1; j > nDiv2; j--, bVz = !bVz)
cout << vz[bVz] << pasTri[n - j] << "*x^" << j;
for (unsigned long j = nDiv2; j > 1; j--, bVz = !bVz)
cout << vz[bVz] << pasTri[j] << "*x^" << j;
cout << vz[bVz] << pasTri[1] << "*x";
bVz = !bVz;
cout << vz[bVz] << pasTri[0] << endl;
break;
}
}

int main()
{
for (unsigned long n = 0; n <= 9; n++)
expandPoly(n);
for (unsigned long n = 2; n <= pasTriMax; n++)
if (isPrime(n))
cout << setw(3) << n;
cout << endl;
}

Output:
(x-1)^0 = 1
(x-1)^1 = x-1
(x-1)^2 = x^2-2*x+1
(x-1)^3 = x^3-3*x^2+3*x-1
(x-1)^4 = x^4-4*x^3+6*x^2-4*x+1
(x-1)^5 = x^5-5*x^4+10*x^3-10*x^2+5*x-1
(x-1)^6 = x^6-6*x^5+15*x^4-20*x^3+15*x^2-6*x+1
(x-1)^7 = x^7-7*x^6+21*x^5-35*x^4+35*x^3-21*x^2+7*x-1
(x-1)^8 = x^8-8*x^7+28*x^6-56*x^5+70*x^4-56*x^3+28*x^2-8*x+1
(x-1)^9 = x^9-9*x^8+36*x^7-84*x^6+126*x^5-126*x^4+84*x^3-36*x^2+9*x-1
2  3  5  7 11 13 17 19 23 29 31 37 41 43 47 53 59 61


Clojure

 This example is incorrect. Please fix the code and remove this message.Details: (x-1)(x-1) is x^2-2x+1 so should the result 2 (-1 2 1) rather be 2 (1 2 1) and similarly for several others

The *' function is an arbitrary precision multiplication.

(defn c
"kth coefficient of (x - 1)^n"
[n k]
(/ (apply *' (range n (- n k) -1))
(apply *' (range k 0 -1))
(if (and (even? k) (< k n)) -1 1)))

(defn cs
"coefficient series for (x - 1)^n, k=[0..n]"
[n]
(map #(c n %) (range (inc n))))

(defn aks? [p] (->> (cs p) rest butlast (every? #(-> % (mod p) zero?))))

(println "coefficient series n (k[0] .. k[n])")
(doseq [n (range 10)] (println n (cs n)))
(println)
(println "primes < 50 per AKS:" (filter aks? (range 2 50)))

Output:
coefficient series n (k[0] .. k[n])
0 (1)
1 (-1 1)
2 (-1 2 1)
3 (-1 3 -3 1)
4 (-1 4 -6 4 1)
5 (-1 5 -10 10 -5 1)
6 (-1 6 -15 20 -15 6 1)
7 (-1 7 -21 35 -35 21 -7 1)
8 (-1 8 -28 56 -70 56 -28 8 1)
9 (-1 9 -36 84 -126 126 -84 36 -9 1)

primes < 50 per AKS: (2 3 5 7 11 13 17 19 23 29 31 37 41 43 47)

CoffeeScript

pascal = () ->
a = []
return () ->
if a.length is 0 then a = [1]
else
b = (a[i] + a[i+1] for i in [0 ... a.length - 1])
a = [1].concat(b).concat [1]

show = (a) ->
show_x = (e) ->
switch e
when 0 then ""
when 1 then "x"
else "x^#{e}"

degree = a.length - 1
str = "(x - 1)^#{degree} ="
sgn = 1

for i in [0...a.length]
str += ' ' + (if sgn > 0 then "+" else "-") + ' ' + a[i] + show_x(degree - i)
sgn = -sgn

return str

primerow = (row) ->
degree = row.length - 1
row[1 ... degree].every (x) -> x % degree is 0

p = pascal()
console.log show p() for i in [0..7]

p = pascal()
p(); p()  # skip 0 and 1

primes = (i+1 for i in [1..49] when primerow p())

console.log ""
console.log "The primes upto 50 are: #{primes}"

Output:
(x - 1)^0 = + 1
(x - 1)^1 = + 1x - 1
(x - 1)^2 = + 1x^2 - 2x + 1
(x - 1)^3 = + 1x^3 - 3x^2 + 3x - 1
(x - 1)^4 = + 1x^4 - 4x^3 + 6x^2 - 4x + 1
(x - 1)^5 = + 1x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1
(x - 1)^6 = + 1x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1
(x - 1)^7 = + 1x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1

The primes upto 50 are: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47

Common Lisp

(defun coefficients (p)
(cond
((= p 0) #(1))

(t (loop for i from 1 upto p
for result = #(1 -1) then (map 'vector
#'-
(concatenate 'vector result #(0))
(concatenate 'vector #(0) result))
finally (return result)))))

(defun primep (p)
(cond
((< p 2) nil)

(t (let ((c (coefficients p)))
(decf (elt c 0))
(loop for i from 0 upto (/ (length c) 2)
for x across c
never (/= (mod x p) 0))))))

(defun main ()
(format t "# p: (x-1)^p for small p:~%")
(loop for p from 0 upto 7
do (format t "~D: " p)
(loop for i from 0
for x across (reverse (coefficients p))
do (when (>= x 0) (format t "+"))
(format t "~D" x)
(if (> i 0)
(format t "X^~D " i)
(format t " ")))
(format t "~%"))
(loop for i from 0 to 50
do (when (primep i) (format t "~D " i)))
(format t "~%"))

Output:
# p: (x-1)^p for small p:
0: +1
1: -1 +1X^1
2: +1 -2X^1 +1X^2
3: -1 +3X^1 -3X^2 +1X^3
4: +1 -4X^1 +6X^2 -4X^3 +1X^4
5: -1 +5X^1 -10X^2 +10X^3 -5X^4 +1X^5
6: +1 -6X^1 +15X^2 -20X^3 +15X^4 -6X^5 +1X^6
7: -1 +7X^1 -21X^2 +35X^3 -35X^4 +21X^5 -7X^6 +1X^7
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 

Crystal

Translation of: Ruby
def x_minus_1_to_the(p)
p.times.reduce([1]) do |ex, _|
([0_i64] + ex).zip(ex + [0]).map { |x, y| x - y }
end
end

def prime?(p)
return false if p < 2
coeff = x_minus_1_to_the(p)[1..p//2] # only need half of coeff terms
coeff.all?{ |n| n%p == 0 }
end

8.times do |n|
puts "(x-1)^#{n} = " +
x_minus_1_to_the(n).map_with_index{ |c, p|
p.zero? ? c.to_s : (c < 0 ? " - " : " + ") + (c.abs == 1 ? "x" : "#{c.abs}x") + (p == 1 ? "" : "^#{p}")
}.join
end

puts "\nPrimes below 50:", 50.times.select { |n| prime? n }.join(',')

Output:
(x-1)^0 = 1
(x-1)^1 = -1 + x
(x-1)^2 = 1 - 2x + x^2
(x-1)^3 = -1 + 3x - 3x^2 + x^3
(x-1)^4 = 1 - 4x + 6x^2 - 4x^3 + x^4
(x-1)^5 = -1 + 5x - 10x^2 + 10x^3 - 5x^4 + x^5
(x-1)^6 = 1 - 6x + 15x^2 - 20x^3 + 15x^4 - 6x^5 + x^6
(x-1)^7 = -1 + 7x - 21x^2 + 35x^3 - 35x^4 + 21x^5 - 7x^6 + x^7

Primes below 50:
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47


D

Translation of: Python
import std.stdio, std.range, std.algorithm, std.string, std.bigint;

BigInt[] expandX1(in uint p) pure /*nothrow*/ {
if (p == 0) return [1.BigInt];
typeof(return) r = [1.BigInt, BigInt(-1)];
foreach (immutable _; 1 .. p)
r = zip(r~0.BigInt, 0.BigInt~r).map!(xy => xy[0]-xy[1]).array;
r.reverse();
return r;
}

bool aksTest(in uint p) pure /*nothrow*/ {
if (p < 2) return false;
auto ex = p.expandX1;
ex[0]++;
return !ex[0 .. $- 1].any!(mult => mult % p); } void main() { "# p: (x-1)^p for small p:".writeln; foreach (immutable p; 0 .. 12) writefln("%3d: %s", p, p.expandX1.zip(iota(p + 1)).retro .map!q{"%+dx^%d ".format(a[])}.join.replace("x^0", "") .replace("^1 ", " ").replace("+", "+ ") .replace("-", "- ").replace(" 1x", " x")[2 ..$]);

"\nSmall primes using the AKS test:".writeln;
101.iota.filter!aksTest.writeln;
}

Output:
# p: (x-1)^p for small p:
0: 1
1: x - 1
2: x^2 - 2x + 1
3: x^3 - 3x^2 + 3x - 1
4: x^4 - 4x^3 + 6x^2 - 4x + 1
5: x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1
6: x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1
7: x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1
8: x^8 - 8x^7 + 28x^6 - 56x^5 + 70x^4 - 56x^3 + 28x^2 - 8x + 1
9: x^9 - 9x^8 + 36x^7 - 84x^6 + 126x^5 - 126x^4 + 84x^3 - 36x^2 + 9x - 1
10: x^10 - 10x^9 + 45x^8 - 120x^7 + 210x^6 - 252x^5 + 210x^4 - 120x^3 + 45x^2 - 10x + 1
11: x^11 - 11x^10 + 55x^9 - 165x^8 + 330x^7 - 462x^6 + 462x^5 - 330x^4 + 165x^3 - 55x^2 + 11x - 1

Small primes using the AKS test:
[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]

See #Pascal.

EchoLisp

We use the math.lib library and the poly functions to compute and display the required polynomials. A polynomial P(x) = a0 +a1*x + .. an*x^n is a list of coefficients (a0 a1 .... an).

(lib 'math.lib)
;; 1 - x^p  :  P = (1 0 0 0 ... 0 -1)
(define (mono p) (append (list 1) (make-list (1- p) 0) (list -1)))

;; compute (x-1)^p ,  p >= 1
(define (aks-poly p)
(poly-pow (list -1 1) p))

;;
(define (show-them n)
(for ((p (in-range 1 n)))
(writeln 'p p (poly->string 'x (aks-poly p)))))

;; aks-test
;; P = (x-1)^p + 1 - x^p
(define (aks-test p)
(let ((P (poly-add (mono p) (aks-poly p)))
(test (lambda(a) (zero? (modulo a p))))) ;; p divides a[i] ?
(apply and (map test P)))) ;; returns #t if true for all a[i]

Output:
(show-them 13) →
p     1     x -1
p     2     x^2 -2x +1
p     3     x^3 -3x^2 +3x -1
p     4     x^4 -4x^3 +6x^2 -4x +1
p     5     x^5 -5x^4 +10x^3 -10x^2 +5x -1
p     6     x^6 -6x^5 +15x^4 -20x^3 +15x^2 -6x +1
p     7     x^7 -7x^6 +21x^5 -35x^4 +35x^3 -21x^2 +7x -1
p     8     x^8 -8x^7 +28x^6 -56x^5 +70x^4 -56x^3 +28x^2 -8x +1
p     9     x^9 -9x^8 +36x^7 -84x^6 +126x^5 -126x^4 +84x^3 -36x^2 +9x -1
p     10     x^10 -10x^9 +45x^8 -120x^7 +210x^6 -252x^5 +210x^4 -120x^3 +45x^2 -10x +1
p     11     x^11 -11x^10 +55x^9 -165x^8 +330x^7 -462x^6 +462x^5 -330x^4 +165x^3 -55x^2 +11x -1
p     12     x^12 -12x^11 +66x^10 -220x^9 +495x^8 -792x^7 +924x^6 -792x^5 +495x^4 -220x^3 +66x^2 -12x +1

(lib 'bigint)

(for ((p (in-range 2 100)))
(when (aks-test p) (write p)))  →

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


Elena

Translation of: C#

ELENA 6.x :

import extensions;

singleton AksTest
{
static long[] c := new long[](100);

coef(int n)
{
int i := 0;
int j := 0;

if ((n < 0) || (n > 63)) { AbortException.raise() }; // gracefully deal with range issue

c[i] := 1l;
for (int i := 0; i < n; i += 1) {
c[1 + i] := 1l;
for (int j := i; j > 0; j -= 1) {
c[j] := c[j - 1] - c[j]
};
c[0] := c[0].Negative
}
}

bool is_prime(int n)
{
int i := n;

self.coef(n);
c[0] := c[0] + 1;
c[i] := c[i] - 1;

i -= 1;
while (i + 1 != 0 && c[i+1].mod(n) == 0)
{
i -= 1
};

^ i < 0
}

show(int n)
{
int i := n;
i += 1;
while(i != 0)
{
i -= 1;
console.print("+",c[i],"x^",i)
}
}
}

public program()
{
for (int n := 0; n < 10; n += 1) {
AksTest.coef(n);

console.print("(x-1)^",n," = ");
AksTest.show(n);
console.printLine()
};

console.print("Primes:");
for (int n := 1; n <= 63; n += 1) {
if (AksTest.is_prime(n))
{
console.print(n," ")
}
};

}
Output:
(x-1)^0 = +1x^0
(x-1)^1 = +1x^1+-1x^0
(x-1)^2 = +1x^2+-2x^1+1x^0
(x-1)^3 = +1x^3+-3x^2+3x^1+-1x^0
(x-1)^4 = +1x^4+-4x^3+6x^2+-4x^1+1x^0
(x-1)^5 = +1x^5+-5x^4+10x^3+-10x^2+5x^1+-1x^0
(x-1)^6 = +1x^6+-6x^5+15x^4+-20x^3+15x^2+-6x^1+1x^0
(x-1)^7 = +1x^7+-7x^6+21x^5+-35x^4+35x^3+-21x^2+7x^1+-1x^0
(x-1)^8 = +1x^8+-8x^7+28x^6+-56x^5+70x^4+-56x^3+28x^2+-8x^1+1x^0
(x-1)^9 = +1x^9+-9x^8+36x^7+-84x^6+126x^5+-126x^4+84x^3+-36x^2+9x^1+-1x^0
Primes:1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61


Elixir

Translation of: Erlang
defmodule AKS do
def iterate(f, x), do: fn -> [x | iterate(f, f.(x))] end

def take(0, _lazy), do: []
def take(n, lazy) do
[value | next] = lazy.()
[value | take(n-1, next)]
end

def pascal, do: iterate(fn row -> [1 | sum_adj(row)] end, [1])

defp sum_adj([_] = l), do: l

def show_binomial(row) do
degree = length(row) - 1
["(x - 1)^",  to_char_list(degree), " =", binomial_rhs(row, 1, degree)]
end

defp show_x(0), do: ""
defp show_x(1), do: "x"
defp show_x(n), do: [?x, ?^ | to_char_list(n)]

defp binomial_rhs([], _, _), do: []
defp binomial_rhs([coef | coefs], sgn, exp) do
signchar = if sgn > 0, do: ?+, else: ?-
[0x20, signchar, 0x20, to_char_list(coef), show_x(exp) | binomial_rhs(coefs, -sgn, exp-1)]
end

def primerow(row, n), do: Enum.all?(row, fn coef -> (coef == 1) or (rem(coef, n) == 0) end)

def main do
for row <- take(8, pascal), do: IO.puts show_binomial(row)
IO.write "\nThe primes upto 50: "
IO.inspect for {row, n} <- Enum.zip(tl(tl(take(51, pascal))), 2..50), primerow(row, n), do: n
end
end

AKS.main

Output:
(x - 1)^0 = + 1
(x - 1)^1 = + 1x - 1
(x - 1)^2 = + 1x^2 - 2x + 1
(x - 1)^3 = + 1x^3 - 3x^2 + 3x - 1
(x - 1)^4 = + 1x^4 - 4x^3 + 6x^2 - 4x + 1
(x - 1)^5 = + 1x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1
(x - 1)^6 = + 1x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1
(x - 1)^7 = + 1x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1

The primes upto 50: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]


Erlang

Translation of: CoffeeScript

The Erlang io module can print out lists of characters with any level of nesting as a flat string. (e.g. ["Er", ["la", ["n"]], "g"] prints as "Erlang") which is useful when constructing the strings to print out for the binomial expansions. The program also shows how lazy lists can be implemented in Erlang.

#! /usr/bin/escript

-import(lists, [all/2, seq/2, zip/2]).

iterate(F, X) -> fun() -> [X | iterate(F, F(X))] end.

take(0, _lazy) -> [];
take(N, Lazy) ->
[Value | Next] = Lazy(),
[Value | take(N-1, Next)].

pascal() -> iterate(fun (Row) -> [1 | sum_adj(Row)] end, [1]).

show_binomial(Row) ->
Degree = length(Row) - 1,
["(x - 1)^",  integer_to_list(Degree), " =", binomial_rhs(Row, 1, Degree)].

show_x(0) -> "";
show_x(1) -> "x";
show_x(N) -> [$x,$^ | integer_to_list(N)].

binomial_rhs([], _, _) -> [];
binomial_rhs([Coef | Coefs], Sgn, Exp) ->
SignChar = if Sgn > 0 -> $+; true ->$- end,
[$, SignChar,$ , integer_to_list(Coef), show_x(Exp) | binomial_rhs(Coefs, -Sgn, Exp-1)].

primerow(Row, N) -> all(fun (Coef) -> (Coef =:= 1) or (Coef rem N =:= 0) end, Row).

main(_) ->
[io:format("~s~n", [show_binomial(Row)]) || Row <- take(8, pascal())],
io:format("~nThe primes upto 50: ~p~n",
[[N || {Row, N} <- zip(tl(tl(take(51, pascal()))), seq(2, 50)),
primerow(Row, N)]]).

Output:
(x - 1)^0 = + 1
(x - 1)^1 = + 1x - 1
(x - 1)^2 = + 1x^2 - 2x + 1
(x - 1)^3 = + 1x^3 - 3x^2 + 3x - 1
(x - 1)^4 = + 1x^4 - 4x^3 + 6x^2 - 4x + 1
(x - 1)^5 = + 1x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1
(x - 1)^6 = + 1x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1
(x - 1)^7 = + 1x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1

The primes upto 50: [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47]


F#

// N-grams. Nigel Galloway: April 11th., 2024
let fN g=let n,g=g@[0I],0I::g in List.map2(fun n g->n-g) n g
Seq.unfold(fun g->Some(g,fN g))[1I]|>Seq.take 9|>Seq.iteri(fun n g->printfn "%d -> %A" n g); printfn ""
let fG (n::g) l=(n+1I)%l=0I && g|>List.forall(fun n->n%l=0I)
Seq.unfold(fun(n,g)->Some((n,g),(n+1I,fN g)))(0I,[1I])|>Seq.skip 2|>Seq.filter(fun(n,_::g)->fG (List.rev g) n)|>Seq.takeWhile(fun(n,_)->n<100I)|>Seq.iter(fun(n,_)->printf "%A " n); printfn ""

Output:
0 -> [1]
1 -> [1; -1]
2 -> [1; -2; 1]
3 -> [1; -3; 3; -1]
4 -> [1; -4; 6; -4; 1]
5 -> [1; -5; 10; -10; 5; -1]
6 -> [1; -6; 15; -20; 15; -6; 1]
7 -> [1; -7; 21; -35; 35; -21; 7; -1]
8 -> [1; -8; 28; -56; 70; -56; 28; -8; 1]

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


Factor

USING: combinators formatting io kernel make math math.parser
math.polynomials prettyprint sequences ;
IN: rosetta-code.aks-test

! Polynomials are represented by the math.polynomials vocabulary
! as sequences with the highest exponent on the right. Hence
! { -1 1 } represents x - 1.
: (x-1)^ ( n -- seq ) { -1 1 } swap p^ ;

: choose-exp ( n -- str )
{ { 0 [ "" ] } { 1 [ "x" ] } [ "x^%d" sprintf ] } case ;

: choose-coeff ( n -- str )
[ dup neg? [ neg "- " ] [ "+ " ] if % # ] "" make ;

: terms ( coeffs-seq -- terms-seq )
[ [ choose-coeff ] [ choose-exp append ] bi* ] map-index ;

: (.p) ( n -- str ) (x-1)^ terms <reversed> " " join 3 tail ;

: .p ( n -- ) dup zero? [ drop "1" ] [ (.p) ] if print ;

: show-poly ( n -- ) [ "(x-1)^%d = " printf ] [ .p ] bi ;

: part1 ( -- ) 8 <iota> [ show-poly ] each ;

: (prime?) ( n -- ? )
(x-1)^ rest but-last dup first [ mod 0 = not ] curry find
nip not ;

: prime? ( n -- ? ) dup 2 < [ drop f ] [ (prime?) ] if ;

: part2 ( -- )
"Primes up to 50 via AKS:" print
50 <iota> [ prime? ] filter . ;

: aks-test ( -- ) part1 nl part2 ;

MAIN: aks-test

Output:
(x-1)^0 = 1
(x-1)^1 = x - 1
(x-1)^2 = x^2 - 2x + 1
(x-1)^3 = x^3 - 3x^2 + 3x - 1
(x-1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1
(x-1)^5 = x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1
(x-1)^6 = x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1
(x-1)^7 = x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1

Primes up to 50 via AKS:
V{ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 }


Forth

: coeffs ( u -- nu ... n0 ) \ coefficients of (x-1)^u
1 swap 1+ dup 1 ?do over over i - i */ negate swap loop drop ;

: prime? ( u -- f )
dup 2 < if drop false exit then
dup >r coeffs 1+
\ if not prime, this loop consumes at most half the coefficients, otherwise all
begin dup 1 <> while
r@ mod 0= while
repeat then rdrop
dup 1 = >r
begin 1 = until
r> ;

: .monom ( u1 u2 -- )
dup 0> if [char] + emit then 0 .r ?dup if ." x^" . else space then ;
: .poly ( u -- )
dup >r coeffs 0 r> 1+ 0 ?do
tuck swap .monom 1+
loop ;

: main
11 0 ?do i . ." : " i .poly cr loop cr
50 1 ?do i prime? if i . then loop
cr ;

Output:
0 : +1
1 : -1 +1x^1
2 : +1 -2x^1 +1x^2
3 : -1 +3x^1 -3x^2 +1x^3
4 : +1 -4x^1 +6x^2 -4x^3 +1x^4
5 : -1 +5x^1 -10x^2 +10x^3 -5x^4 +1x^5
6 : +1 -6x^1 +15x^2 -20x^3 +15x^4 -6x^5 +1x^6
7 : -1 +7x^1 -21x^2 +35x^3 -35x^4 +21x^5 -7x^6 +1x^7
8 : +1 -8x^1 +28x^2 -56x^3 +70x^4 -56x^5 +28x^6 -8x^7 +1x^8
9 : -1 +9x^1 -36x^2 +84x^3 -126x^4 +126x^5 -84x^6 +36x^7 -9x^8 +1x^9
10 : +1 -10x^1 +45x^2 -120x^3 +210x^4 -252x^5 +210x^6 -120x^7 +45x^8 -10x^9 +1x^10

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47

Fortran

program aks
implicit none

! Coefficients of polynomial expansion
integer(kind=16), dimension(:), allocatable :: coeffs
integer(kind=16) :: n
! Character variable for I/O
character(len=40) :: tmp

! Point #2
do n = 0, 7
write(tmp, *) n
call polynomial_expansion(n, coeffs)
call print_polynom(coeffs)
end do

! Point #4
do n = 2, 35
if (is_prime(n)) write(*, '(I4)', advance='no') n
end do
write(*, *)

! Point #5
do n = 2, 124
if (is_prime(n)) write(*, '(I4)', advance='no') n
end do
write(*, *)

if (allocated(coeffs)) deallocate(coeffs)
contains
! Calculate coefficients of (x - 1)^n using binomial theorem
subroutine polynomial_expansion(n, coeffs)
integer(kind=16), intent(in) :: n
integer(kind=16), dimension(:), allocatable, intent(out) :: coeffs
integer(kind=16) :: i, j

if (allocated(coeffs)) deallocate(coeffs)

allocate(coeffs(n + 1))

do i = 1, n + 1
coeffs(i) = binomial(n, i - 1)*(-1)**(n - i - 1)
end do
end subroutine

! Calculate binomial coefficient using recurrent relation, as calculation
! using factorial overflows too quickly.
function binomial(n, k) result (res)
integer(kind=16), intent(in) :: n, k
integer(kind=16) :: res
integer(kind=16) :: i

if (k == 0) then
res = 1
return
end if

res = 1
do i = 0, k - 1
res = res*(n - i)/(i + 1)
end do
end function

! Outputs polynomial with given coefficients
subroutine print_polynom(coeffs)
integer(kind=16), dimension(:), allocatable, intent(in) :: coeffs
integer(kind=4) :: i, p
character(len=40) :: cbuf, pbuf
logical(kind=1) :: non_zero

if (.not. allocated(coeffs)) return

non_zero = .false.

do i = 1, size(coeffs)
if (coeffs(i) .eq. 0) cycle

p = i - 1
write(cbuf, '(I40)') abs(coeffs(i))
write(pbuf, '(I40)') p

if (non_zero) then
if (coeffs(i) .gt. 0) then
write(*, fmt='(A)', advance='no') ' + '
else
write(*, fmt='(A)', advance='no') ' - '
endif
else
if (coeffs(i) .gt. 0) then
else
write(*, fmt='(A)', advance='no') ' - '
endif
endif

if (p .eq. 0) then
elseif (p .eq. 1) then
if (coeffs(i) .eq. 1) then
else
end if
else
if (coeffs(i) .eq. 1) then
else
end if
end if
non_zero = .true.
end do

write(*, *)
end subroutine

! Test if n is prime using AKS test. Point #3.
function is_prime(n) result (res)
integer(kind=16), intent (in) :: n
logical(kind=1) :: res
integer(kind=16), dimension(:), allocatable :: coeffs
integer(kind=16) :: i

call polynomial_expansion(n, coeffs)
coeffs(1) = coeffs(1) + 1
coeffs(n + 1) = coeffs(n + 1) - 1

res = .true.

do i = 1, n + 1
res = res .and. (mod(coeffs(i), n) == 0)
end do

if (allocated(coeffs)) deallocate(coeffs)
end function
end program aks

Output:
(x - 1)^0 =   1
(x - 1)^1 = - 1 + x
(x - 1)^2 =   1 - 2x + x^2
(x - 1)^3 = - 1 + 3x - 3x^2 + x^3
(x - 1)^4 =   1 - 4x + 6x^2 - 4x^3 + x^4
(x - 1)^5 = - 1 + 5x - 10x^2 + 10x^3 - 5x^4 + x^5
(x - 1)^6 =   1 - 6x + 15x^2 - 20x^3 + 15x^4 - 6x^5 + x^6
(x - 1)^7 = - 1 + 7x - 21x^2 + 35x^3 - 35x^4 + 21x^5 - 7x^6 + x^7
2   3   5   7  11  13  17  19  23  29  31
2   3   5   7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79  83  89  97 101 103 107 109 113


FreeBASIC

'METHOD -- Use the Pascal triangle to retrieve the coefficients
'UPPER LIMIT OF FREEBASIC ULONGINT GETS PRIMES UP TO 70
Sub string_split(s_in As String,char As String,result() As String)
Dim As String s=s_in,var1,var2
Dim As Integer n,pst
#macro split(stri,char,var1,var2)
pst=Instr(stri,char)
var1="":var2=""
If pst<>0 Then
var1=Mid(stri,1,pst-1)
var2=Mid(stri,pst+1)
Else
var1=stri
End If
Redim Preserve result(1 To 1+n-((Len(var1)>0)+(Len(var2)>0)))
result(n+1)=var1
#endmacro
Do
split(s,char,var1,var2):n=n+1:s=var2
Loop Until var2=""
Redim Preserve result(1 To Ubound(result)-1)
End Sub

'Get Pascal triangle components
Function pasc(n As Integer,flag As Integer=0) As String
n+=1
Dim As Ulongint V(n):V(1)=1ul
Dim As String s,sign
For r  As Integer= 2 To n
s=""
For i As Integer = r To 1 Step -1
V(i) +=  V(i-1)
If i Mod 2=1 Then sign="" Else sign="-"
s+=sign+Str(V(i))+","
Next i
Next r
If flag Then 'formatted output
Dim As String i,i2,i3,g
Redim As String a(0)
string_split(s,",",a())
For n1 As Integer=1 To Ubound(a)
If Left(a(n1),1)="-" Then sign="" Else sign="+"
If n1=Ubound(a) Then i2="" Else i2=a(n1)
If n1=2 Then i3="x" Else i3="x^"+Str(n1-1)
If n1=1 Then i="":sign=" " Else i=i3
g+=sign+i2+i+" "
Next n1
g="(x-1)^"+Str(n-1)+" = "+g
Return g
End If
Return s
End Function

Function isprime(num As Integer) As Integer
Redim As String a(0)
string_split(pasc(num),",",a())
For n As Integer=Lbound(a)+1 To Ubound(a)-1
If (Valulng(Ltrim(a(n),"-"))) Mod num<>0 Then Return 0
Next n
Return -1
End Function
'====================================
'Formatted output
For n As Integer=1 To 9
Print pasc(n,1)
Next n

Print
'Limit of Freebasic Ulongint sets about 70 max
Print "Primes up to 70:"
For n As Integer=2 To 70
If isprime(n) Then Print n;
Next n

Sleep

Output:
(x-1)^1 =  -1 +x
(x-1)^2 =  1 -2x +x^2
(x-1)^3 =  -1 +3x -3x^2 +x^3
(x-1)^4 =  1 -4x +6x^2 -4x^3 +x^4
(x-1)^5 =  -1 +5x -10x^2 +10x^3 -5x^4 +x^5
(x-1)^6 =  1 -6x +15x^2 -20x^3 +15x^4 -6x^5 +x^6
(x-1)^7 =  -1 +7x -21x^2 +35x^3 -35x^4 +21x^5 -7x^6 +x^7
(x-1)^8 =  1 -8x +28x^2 -56x^3 +70x^4 -56x^5 +28x^6 -8x^7 +x^8
(x-1)^9 =  -1 +9x -36x^2 +84x^3 -126x^4 +126x^5 -84x^6 +36x^7 -9x^8 +x^9

Primes up to 70:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67

Go

package main

import "fmt"

func bc(p int) []int64 {
c := make([]int64, p+1)
r := int64(1)
for i, half := 0, p/2; i <= half; i++ {
c[i] = r
c[p-i] = r
r = r * int64(p-i) / int64(i+1)
}
for i := p - 1; i >= 0; i -= 2 {
c[i] = -c[i]
}
return c
}

func main() {
for p := 0; p <= 7; p++ {
fmt.Printf("%d:  %s\n", p, pp(bc(p)))
}
for p := 2; p < 50; p++ {
if aks(p) {
fmt.Print(p, " ")
}
}
fmt.Println()
}

var e = []rune("²³⁴⁵⁶⁷")

func pp(c []int64) (s string) {
if len(c) == 1 {
return fmt.Sprint(c[0])
}
p := len(c) - 1
if c[p] != 1 {
s = fmt.Sprint(c[p])
}
for i := p; i > 0; i-- {
s += "x"
if i != 1 {
s += string(e[i-2])
}
if d := c[i-1]; d < 0 {
s += fmt.Sprintf(" - %d", -d)
} else {
s += fmt.Sprintf(" + %d", d)
}
}
return
}

func aks(p int) bool {
c := bc(p)
c[p]--
c[0]++
for _, d := range c {
if d%int64(p) != 0 {
return false
}
}
return true
}

Output:
0:  1
1:  x - 1
2:  x² - 2x + 1
3:  x³ - 3x² + 3x - 1
4:  x⁴ - 4x³ + 6x² - 4x + 1
5:  x⁵ - 5x⁴ + 10x³ - 10x² + 5x - 1
6:  x⁶ - 6x⁵ + 15x⁴ - 20x³ + 15x² - 6x + 1
7:  x⁷ - 7x⁶ + 21x⁵ - 35x⁴ + 35x³ - 21x² + 7x - 1
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47


expand p = scanl (\z i -> z * (p-i+1) div i) 1 [1..p]

test p | p < 2     = False
| otherwise = and [mod n p == 0 | n <- init . tail $expand p] printPoly [1] = "1" printPoly p = concat [ unwords [pow i, sgn (l-i), show (p!!(i-1))] | i <- [l-1,l-2..1] ] where l = length p sgn i = if even i then "+" else "-" pow i = take i "x^" ++ if i > 1 then show i else "" main = do putStrLn "-- p: (x-1)^p for small p" putStrLn$ unlines [show i ++ ": " ++ printPoly (expand i) | i <- [0..10]]
putStrLn "-- Primes up to 100:"
print (filter test [1..100])

Output:
-- p: (x-1)^p for small p
0: 1
1: x - 1
2: x^2 - 2x + 1
3: x^3 - 3x^2 + 3x - 1
4: x^4 - 4x^3 + 6x^2 - 4x + 1
5: x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1
6: x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1
7: x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1
8: x^8 - 8x^7 + 28x^6 - 56x^5 + 70x^4 - 56x^3 + 28x^2 - 8x + 1
9: x^9 - 9x^8 + 36x^7 - 84x^6 + 126x^5 - 126x^4 + 84x^3 - 36x^2 + 9x - 1
10: x^10 - 10x^9 + 45x^8 - 120x^7 + 210x^6 - 252x^5 + 210x^4 - 120x^3 + 45x^2 - 10x + 1

-- Primes up to 100:
[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]

Idris

import Data.Vect

-- Computes Binomial Coefficients
binCoef : Nat -> Nat -> Nat
binCoef _ Z = (S Z)
binCoef (S n) (S k) =
if n == k then (S Z) else ((S n) * (binCoef n k)) div (S k)

-- Binomial Expansion Of (x - 1)^p
expansion : (n : Nat) -> Vect (S n) Integer
expansion n = expansion' n 1
where
expansion' : (n : Nat) -> Integer -> Vect (S n) Integer
expansion' (S m) s = s * (toIntegerNat $binCoef n (n minus (S m))) :: expansion' m (s * -1) expansion' Z s = [s] showExpansion : Vect n Integer -> String showExpansion [] = " " showExpansion (x::xs) {n = S k} = (if x < 0 then "-" else "") ++ term x k ++ showExpansion' xs where term : Integer -> Nat -> String term x n = if n == 0 then (show (abs x)) else (if (abs x) == 1 then "" else (show (abs x))) ++ "x" ++ (if n == 1 then "" else "^" ++ show n) sign : Integer -> String sign x = if x >= 0 then " + " else " - " showExpansion' : Vect m Integer -> String showExpansion' [] = "" showExpansion' (y::ys) {m = S k} = sign y ++ term y k ++ showExpansion' ys natToFin' : (m : Nat) -> Fin (S m) natToFin' n with (natToFin n (S n)) natToFin' n | Just y = y isPrime : Nat -> Bool isPrime Z = False isPrime (S Z ) = False isPrime n = foldl (\divs, term => divs && (term mod (toIntegerNat n)) == 0) True (fullExpansion$ expansion n)

-- (x - 1)^p - ((x^p) - 1)
where fullExpansion : Vect (S m) Integer -> Vect (S m) Integer
fullExpansion (x::xs) {m} = updateAt (natToFin' m) (+1) $(x-1)::xs printExpansions : Nat -> IO () printExpansions n = do putStrLn "-- p: (x-1)^p for small p" sequence_$ map printExpansion [0..n]
where printExpansion : Nat -> IO ()
printExpansion n = do
print n
putStr ": "
putStrLn $showExpansion$ expansion n

main : IO()
main = do
printExpansions 10
putStrLn "\n-- Primes Up To 100:"
putStrLn $show$ filter isPrime [0..100]

Output:
-- p: (x-1)^p for small p
0: 1
1: x - 1
2: x^2 - 2x + 1
3: x^3 - 3x^2 + 3x - 1
4: x^4 - 4x^3 + 6x^2 - 4x + 1
5: x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1
6: x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1
7: x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1
8: x^8 - 8x^7 + 28x^6 - 56x^5 + 70x^4 - 56x^3 + 28x^2 - 8x + 1
9: x^9 - 9x^8 + 36x^7 - 84x^6 + 126x^5 - 126x^4 + 84x^3 - 36x^2 + 9x - 1
10: x^10 - 10x^9 + 45x^8 - 120x^7 + 210x^6 - 252x^5 + 210x^4 - 120x^3 + 45x^2 - 10x + 1

-- Primes Up To 100:
[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]

J

Solution:

   binomialExpansion =:  (!~ * _1 ^ 2 | ]) i.&.:<:         NB. 1) Create a function that gives the coefficients of (x-1)^p.
testAKS           =:  0 *./ .= ] | binomialExpansion    NB. 3) Use that function to create another which determines whether p is prime using AKS.


Examples:

   binomialExpansion&.> i. 8   NB.  2) show the polynomial expansions p in the range 0 to at 7 inclusive.
+-++--+----+-------+-----------+---------------+------------------+
|0||_2|_3 3|_4 6 _4|_5 10 _10 5|_6 15 _20 15 _6|_7 21 _35 35 _21 7|
+-++--+----+-------+-----------+---------------+------------------+
(#~ testAKS&> ) 2+i. 35     NB. 4) Generate a list of all primes under 35.
2 3 5 7 11 13 17 19 23 29 31
(#~ testAKS&> ) 2+i. 50     NB. 5) [stretch] Generate all primes under 50
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
i.&.:(_1&p:) 50             NB. Double-check our results using built-in prime filter.
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47


Java

Translation of: C

Solution:

public class AksTest {
private static final long[] c = new long[64];

public static void main(String[] args) {
for (int n = 0; n < 10; n++) {
coeff(n);
show(n);
}

System.out.print("Primes:");
for (int n = 1; n < c.length; n++)
if (isPrime(n))
System.out.printf(" %d", n);

System.out.println();
}

static void coeff(int n) {
c[0] = 1;
for (int i = 0; i < n; c[0] = -c[0], i++) {
c[1 + i] = 1;
for (int j = i; j > 0; j--)
c[j] = c[j - 1] - c[j];
}
}

static boolean isPrime(int n) {
coeff(n);
c[0]++;
c[n]--;

int i = n;
while (i-- != 0 && c[i] % n == 0)
continue;
return i < 0;
}

static void show(int n) {
System.out.print("(x-1)^" + n + " =");
for (int i = n; i >= 0; i--) {
System.out.print(" + " + c[i] + "x^" + i);
}
System.out.println();
}
}


Output:

(x-1)^0 = +1x^0
(x-1)^1 = +1x^1+-1x^0
(x-1)^2 = +1x^2+-2x^1+1x^0
(x-1)^3 = +1x^3+-3x^2+3x^1+-1x^0
(x-1)^4 = +1x^4+-4x^3+6x^2+-4x^1+1x^0
(x-1)^5 = +1x^5+-5x^4+10x^3+-10x^2+5x^1+-1x^0
(x-1)^6 = +1x^6+-6x^5+15x^4+-20x^3+15x^2+-6x^1+1x^0
(x-1)^7 = +1x^7+-7x^6+21x^5+-35x^4+35x^3+-21x^2+7x^1+-1x^0
(x-1)^8 = +1x^8+-8x^7+28x^6+-56x^5+70x^4+-56x^3+28x^2+-8x^1+1x^0
(x-1)^9 = +1x^9+-9x^8+36x^7+-84x^6+126x^5+-126x^4+84x^3+-36x^2+9x^1+-1x^0
Primes: 1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61


JavaScript

Translation of: CoffeeScript
var i, p, pascal, primerow, primes, show, _i;

pascal = function() {
var a;
a = [];
return function() {
var b, i;
if (a.length === 0) {
return a = [1];
} else {
b = (function() {
var _i, _ref, _results;
_results = [];
for (i = _i = 0, _ref = a.length - 1; 0 <= _ref ? _i < _ref : _i > _ref; i = 0 <= _ref ? ++_i : --_i) {
_results.push(a[i] + a[i + 1]);
}
return _results;
})();
return a = [1].concat(b).concat([1]);
}
};
};

show = function(a) {
var degree, i, sgn, show_x, str, _i, _ref;
show_x = function(e) {
switch (e) {
case 0:
return "";
case 1:
return "x";
default:
return "x^" + e;
}
};
degree = a.length - 1;
str = "(x - 1)^" + degree + " =";
sgn = 1;
for (i = _i = 0, _ref = a.length; 0 <= _ref ? _i < _ref : _i > _ref; i = 0 <= _ref ? ++_i : --_i) {
str += ' ' + (sgn > 0 ? "+" : "-") + ' ' + a[i] + show_x(degree - i);
sgn = -sgn;
}
return str;
};

primerow = function(row) {
var degree;
degree = row.length - 1;
return row.slice(1, degree).every(function(x) {
return x % degree === 0;
});
};

p = pascal();

for (i = _i = 0; _i <= 7; i = ++_i) {
console.log(show(p()));
}

p = pascal();

p();

p();

primes = (function() {
var _j, _results;
_results = [];
for (i = _j = 1; _j <= 49; i = ++_j) {
if (primerow(p())) {
_results.push(i + 1);
}
}
return _results;
})();

console.log("");

console.log("The primes upto 50 are: " + primes);

Output:
(x - 1)^0 = + 1
(x - 1)^1 = + 1x - 1
(x - 1)^2 = + 1x^2 - 2x + 1
(x - 1)^3 = + 1x^3 - 3x^2 + 3x - 1
(x - 1)^4 = + 1x^4 - 4x^3 + 6x^2 - 4x + 1
(x - 1)^5 = + 1x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1
(x - 1)^6 = + 1x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1
(x - 1)^7 = + 1x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1

The primes upto 50 are: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47

Reviewed (ES6):

function pascal(n) {
var cs = []; if (n) while (n--) coef(); return coef
function coef() {
if (cs.length === 0) return cs = [1];
for (var t=[1,1], i=cs.length-1; i; i-=1) t.splice( 1, 0, cs[i-1]+cs[i] ); return cs = t
}
}

function show(cs) {
for (var s='', sgn=true, i=0, deg=cs.length-1; i<=deg; sgn=!sgn, i+=1) {
s += ' ' + (sgn ? '+' : '-') + cs[i] + (e => e==0 ? '' : e==1 ? 'x' : 'x<sup>' + e + '</sup>')(deg-i)
}
return '(x-1)<sup>' + deg + '</sup> =' + s;
}

function isPrime(cs) {
var deg=cs.length-1; return cs.slice(1, deg).every( function(c) { return c % deg === 0 } )
}

var coef=pascal(); for (var i=0; i<=7; i+=1) document.write(show(coef()), '<br>')

document.write('<br>Primes: ');
for (var coef=pascal(2), n=2; n<=50; n+=1) if (isPrime(coef())) document.write(' ', n)

Output:
(x-1)0 = +1
(x-1)1 = +1x -1
(x-1)2 = +1x2 -2x +1
(x-1)3 = +1x3 -3x2 +3x -1
(x-1)4 = +1x4 -4x3 +6x2 -4x +1
(x-1)5 = +1x5 -5x4 +10x3 -10x2 +5x -1
(x-1)6 = +1x6 -6x5 +15x4 -20x3 +15x2 -6x +1
(x-1)7 = +1x7 -7x6 +21x5 -35x4 +35x3 -21x2 +7x -1

Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

Translation of: C
function coef(n) {
for (var c=[1], i=0; i<n; c[0]=-c[0], i+=1) {
c[i+1]=1; for (var j=i; j; j-=1) c[j] = c[j-1]-c[j]
}
return c
}

function show(cs)	{
var s='', n=cs.length-1
do s += (cs[n]>0 ? ' +' : ' ') + cs[n] + (n==0 ? '' : n==1 ? 'x' :'x<sup>'+n+'</sup>'); while (n--)
return s
}

function isPrime(n) {
var cs=coef(n), i=n-1; while (i-- && cs[i]%n == 0);
return i < 1
}

for (var n=0; n<=7; n++) document.write('(x-1)<sup>',n,'</sup> = ', show(coef(n)), '<br>')

document.write('<br>Primes: ');
for (var n=2; n<=50; n++) if (isPrime(n)) document.write(' ', n)

Output:
(x-1)0 = +1
(x-1)1 = +1x -1
(x-1)2 = +1x2 -2x +1
(x-1)3 = +1x3 -3x2 +3x -1
(x-1)4 = +1x4 -4x3 +6x2 -4x +1
(x-1)5 = +1x5 -5x4 +10x3 -10x2 +5x -1
(x-1)6 = +1x6 -6x5 +15x4 -20x3 +15x2 -6x +1
(x-1)7 = +1x7 -7x6 +21x5 -35x4 +35x3 -21x2 +7x -1

Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47


jq

Works with: jq version 1.5rc1

In the #Prolog section of this page, it is shown how the symmetry of rows in a Pascal triangle can be used to yield a more efficient test of primality than is apparently envisioned by the problem statement. The key concept is the "OptPascal row", which is just the longest non-decreasing sequence of the corresponding Pascal row. In this article, the focus will therefore be on OptPascal rows.

NOTE: jq uses IEEE 754 64-bit numbers and thus if builtin arithmetic is used, is_prime will only be accurate up to 96 by this method because of loss of precision. The program below, however, can easily be adapted to use a BigInt library such as the one at https://github.com/joelpurra/jq-bigint

# add_pairs is a helper function for optpascal/0
# Input: an OptPascal array
# but if the last two items are unequal, then their sum is repeated)
if length <= 1 then .
elif length == 2 then (.[0] + .[1]) as $S | if (.[0] == .[1]) then [$S]
else [$S,$S]
end
else [.[0] + .[1]] + (.[1:]|add_pairs)
end;

# Input: an OptPascal row
# Output: the next OptPascalRow

# generate a stream of OptPascal arrays, beginning with []
def optpascals: [] | recurse(next_optpascal);

# generate a stream of Pascal arrays
def pascals:
# pascalize takes as input an OptPascal array and produces
# the corresponding Pascal array;
# if the input ends in a pair, then peel it off before reversing it.
def pascalize:
. + ((if .[-2] == .[-1] then .[0:-2] else .[0:-1] end) | reverse);

optpascals | pascalize;

# Input: integer n
# Output: the n-th Pascal row
def pascal: nth(.; pascals);

def optpascal: nth(.; optpascals);
Task 1: "A method to generate the coefficients of (x-1)^p"

def coefficients:
def alternate_signs: . as $in | reduce range(0; length) as$i ([]; . + [$in[$i] * (if $i % 2 == 0 then 1 else -1 end )]); (.+1) | pascal | alternate_signs; Task 2: "Show here the polynomial expansions of (x − 1)^p for p in the range 0 to at least 7, inclusive." range(0;8) | "Coefficient for (x - 1)^\(.): \(coefficients)" Output: Coefficients for (x - 1)^0: [1] Coefficients for (x - 1)^1: [1,-1] Coefficients for (x - 1)^2: [1,-2,1] Coefficients for (x - 1)^3: [1,-3,3,-1] Coefficients for (x - 1)^4: [1,-4,6,-4,1] Coefficients for (x - 1)^5: [1,-5,10,-10,5,-1] Coefficients for (x - 1)^6: [1,-6,15,-20,15,-6,1] Coefficients for (x - 1)^7: [1,-7,21,-35,35,-21,7,-1]  Task 3: Prime Number Test For brevity, we show here only the relatively efficient solution based on optpascal/0: def is_prime: . as$N
| if . < 2 then false
else (1+.) | optpascal
| all(  .[2:][]; . % $N == 0 ) end; Task 4: "Use your AKS test to generate a list of all primes under 35." range(0;36) | select(is_prime) Output: 2 3 5 7 11 13 17 19 23 29 31  Task 5: "As a stretch goal, generate all primes under 50." [range(0;50) | select(is_prime)] Output: [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47]  Julia Task 1 function polycoefs(n::Int64) pc = typeof(n)[] if n < 0 return pc end sgn = one(n) for k in n:-1:0 push!(pc, sgn*binomial(n, k)) sgn = -sgn end return pc end  Perhaps this should be done with a comprehension, but properly accounting for the sign is tricky in that case. Task 2 using Printf function stringpoly(n::Int64) if n < 0 return "" end st = @sprintf "(x - 1)^{%d} & = & " n for (i, c) in enumerate(polycoefs(n)) if i == 1 op = "" ac = c elseif c < 0 op = "-" ac = abs(c) else op = "+" ac = abs(c) end p = n + 1 - i if p == 0 st *= @sprintf " %s %d\\\\" op ac elseif ac == 1 st *= @sprintf " %s x^{%d}" op p else st *= @sprintf " %s %dx^{%d}" op ac p end end return st end  Of course this could be simpler, but this produces a nice payoff in typeset equations that do on include extraneous characters (leading pluses and coefficients of 1). Task 3 function isaksprime(n::Int64) if n < 2 return false end for c in polycoefs(n)[2:(end-1)] if c%n != 0 return false end end return true end  Task 4 println("$") println("\\begin{array}{lcl}") for i in 0:10 println(stringpoly(i)) end println("\\end{array}") println("$\n") L = 50 print("AKS primes less than ", L, ": ") sep = "" for i in 1:L if isaksprime(i) print(sep, i) sep = ", " end end println()  Output: ${\displaystyle {\begin{array}{lcl}(x-1)^{0}&=&1\\(x-1)^{1}&=&x^{1}-1\\(x-1)^{2}&=&x^{2}-2x^{1}+1\\(x-1)^{3}&=&x^{3}-3x^{2}+3x^{1}-1\\(x-1)^{4}&=&x^{4}-4x^{3}+6x^{2}-4x^{1}+1\\(x-1)^{5}&=&x^{5}-5x^{4}+10x^{3}-10x^{2}+5x^{1}-1\\(x-1)^{6}&=&x^{6}-6x^{5}+15x^{4}-20x^{3}+15x^{2}-6x^{1}+1\\(x-1)^{7}&=&x^{7}-7x^{6}+21x^{5}-35x^{4}+35x^{3}-21x^{2}+7x^{1}-1\\(x-1)^{8}&=&x^{8}-8x^{7}+28x^{6}-56x^{5}+70x^{4}-56x^{3}+28x^{2}-8x^{1}+1\\(x-1)^{9}&=&x^{9}-9x^{8}+36x^{7}-84x^{6}+126x^{5}-126x^{4}+84x^{3}-36x^{2}+9x^{1}-1\\(x-1)^{10}&=&x^{10}-10x^{9}+45x^{8}-120x^{7}+210x^{6}-252x^{5}+210x^{4}-120x^{3}+45x^{2}-10x^{1}+1\\\end{array}}}$ AKS primes less than 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 Kotlin // version 1.1 fun binomial(n: Int, k: Int): Long = when { n < 0 || k < 0 -> throw IllegalArgumentException("negative numbers not allowed") k == 0 -> 1L k == n -> 1L else -> { var prod = 1L var div = 1L for (i in 1..k) { prod *= (n + 1 - i) div *= i if (prod % div == 0L) { prod /= div div = 1L } } prod } } fun isPrime(n: Int): Boolean { if (n < 2) return false return (1 until n).none { binomial(n, it) % n.toLong() != 0L } } fun main(args: Array<String>) { var coeff: Long var sign: Int var op: String for (n in 0..9) { print("(x - 1)^$n = ")
sign = 1
for (k in n downTo 0) {
coeff = binomial(n, k)
op = if (sign == 1) " + " else " - "
when (k) {
n    -> print("x^$n") 0 -> println("${op}1")
else -> print("$op${coeff}x^$k") } if (n == 0) println() sign *= -1 } } // generate primes under 62 var p = 2 val primes = mutableListOf<Int>() do { if (isPrime(p)) primes.add(p) if (p != 2) p += 2 else p = 3 } while (p < 62) println("\nThe prime numbers under 62 are:") println(primes) }  Output: (x - 1)^0 = x^0 (x - 1)^1 = x^1 - 1 (x - 1)^2 = x^2 - 2x^1 + 1 (x - 1)^3 = x^3 - 3x^2 + 3x^1 - 1 (x - 1)^4 = x^4 - 4x^3 + 6x^2 - 4x^1 + 1 (x - 1)^5 = x^5 - 5x^4 + 10x^3 - 10x^2 + 5x^1 - 1 (x - 1)^6 = x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x^1 + 1 (x - 1)^7 = x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x^1 - 1 (x - 1)^8 = x^8 - 8x^7 + 28x^6 - 56x^5 + 70x^4 - 56x^3 + 28x^2 - 8x^1 + 1 (x - 1)^9 = x^9 - 9x^8 + 36x^7 - 84x^6 + 126x^5 - 126x^4 + 84x^3 - 36x^2 + 9x^1 - 1 The prime numbers under 62 are: [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61]  LFE Loosely translated from Erlang. (defun next-row (row) (cons 1 (cl:maplist (match-lambda (((list a)) a) (((cons a (cons b _))) (+ a b))) row))) (defun pascal (n) (pascal n '(()))) (defun pascal ((0 rows) (cdr (lists:reverse rows))) ((n (= (cons row _) rows)) (pascal (- n 1) (cons (next-row row) rows)))) (defun show-x ((0) "") ((1) "x") ((n) (++ "x^" (integer_to_list n)))) (defun rhs (('() _ _) "") (((cons coef coefs) sgn exp) (++ (if (< sgn 0) " - " " + ") (integer_to_list coef) (show-x exp) (rhs coefs (- sgn) (- exp 1))))) (defun binomial-text (row) (let ((degree (- (length row) 1))) (++ "(x - 1)^" (integer_to_list degree) " =" (rhs row 1 degree)))) (defun primerow (('() _) 'true) (((1 . ,rest) n) (primerow rest n)) (((cons a (cons a _)) n) ; stop when we've checked half the list (=:= 0 (rem a n))) (((cons a rest) n) (andalso (=:= 0 (rem a n)) (primerow rest n)))) (defun main (_) (list-comp ((<- row (pascal 8))) (lfe_io:format "~s~n" (list (binomial-text row)))) (lfe_io:format "~nThe primes upto 50: ~p~n" (list (list-comp ((<- (tuple row n) (lists:zip (cddr (pascal 51)) (lists:seq 2 50))) (primerow row n)) n))))  Output: (x - 1)^0 = + 1 (x - 1)^1 = + 1x - 1 (x - 1)^2 = + 1x^2 - 2x + 1 (x - 1)^3 = + 1x^3 - 3x^2 + 3x - 1 (x - 1)^4 = + 1x^4 - 4x^3 + 6x^2 - 4x + 1 (x - 1)^5 = + 1x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1 (x - 1)^6 = + 1x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1 (x - 1)^7 = + 1x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1 The primes upto 50: (2 3 5 7 11 13 17 19 23 29 31 37 41 43 47)  Lua -- AKS test for primes, in Lua, 6/23/2020 db local function coefs(n) local list = {[0]=1} for k = 0, n do list[k+1] = math.floor(list[k] * (n-k) / (k+1)) end for k = 1, n, 2 do list[k] = -list[k] end return list end local function isprimeaks(n) local c = coefs(n) c[0], c[n] = c[0]-1, c[n]+1 for i = 0, n do if (c[i] % n ~= 0) then return false end end return true end local function pprintcoefs(n, list) local result = "" for i = 0, n do local s = i==0 and "" or list[i]>=0 and " + " or " - " local c, e = math.abs(list[i]), n-i if (c==1 and e > 0) then c = "" end local x = e==0 and "" or e==1 and "x" or "x^"..e result = result .. s .. c .. x end print("(x-1)^" .. n .." : " .. result) end for i = 0, 9 do pprintcoefs(i, coefs(i)) end local primes = {} for i = 2, 53 do if (isprimeaks(i)) then primes[#primes+1] = i end end print(table.concat(primes, ", "))  Output: (x-1)^0 : 1 (x-1)^1 : x - 1 (x-1)^2 : x^2 - 2x + 1 (x-1)^3 : x^3 - 3x^2 + 3x - 1 (x-1)^4 : x^4 - 4x^3 + 6x^2 - 4x + 1 (x-1)^5 : x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1 (x-1)^6 : x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1 (x-1)^7 : x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1 (x-1)^8 : x^8 - 8x^7 + 28x^6 - 56x^5 + 70x^4 - 56x^3 + 28x^2 - 8x + 1 (x-1)^9 : x^9 - 9x^8 + 36x^7 - 84x^6 + 126x^5 - 126x^4 + 84x^3 - 36x^2 + 9x - 1 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53 Lambdatalk {require lib_BN} // for big numbers 1) pascalian binomial coefficient C(n,p) = n!/(p!(n-p)!) = (n*(n-1)...(n-p+1))/(p*(p-1)...2*1) {def coeff {lambda {:n :p} {BN.intPart {BN./ {S.reduce BN.* {S.serie :n {- :n :p -1} -1}} {S.reduce BN.* {S.serie :p 1 -1}}}}}} -> coeff 2) polynomial expansions of (x − 1)^p {def sign {lambda {:n} {if {= {% :n 2} 0} then + else -}}} -> sign {def coeffs {lambda {:n} {br}(x - 1)^:n = {if {= :n 0} then + 1x^0 else {if {= :n 1} then + 1x^1 - 1x^0 else {sign 0} 1x^:n {S.map {{lambda {:p :n} {sign {- :p :n}} {coeff :p :n}x^{- :p :n}} :n} {S.serie 1 {- :n 1}}} {sign :n} 1x^0}}}} -> coeffs {S.map coeffs {S.serie 0 7}} -> (x - 1)^0 = + 1x^0 (x - 1)^1 = + 1x^1 - 1x^0 (x - 1)^2 = + 1x^2 - 2x^1 + 1x^0 (x - 1)^3 = + 1x^3 + 3x^2 - 3x^1 - 1x^0 (x - 1)^4 = + 1x^4 - 4x^3 + 6x^2 - 4x^1 + 1x^0 (x - 1)^5 = + 1x^5 + 5x^4 - 10x^3 + 10x^2 - 5x^1 - 1x^0 (x - 1)^6 = + 1x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x^1 + 1x^0 (x - 1)^7 = + 1x^7 + 7x^6 - 21x^5 + 35x^4 - 35x^3 + 21x^2 - 7x^1 - 1x^0 3) primality test Taking into account the symmetry of the list of coefficients and the uselessness of the sign in the calculation of the divisibility, one can limit the tests to half of the list, and define a simplified function, aks_coeffs: {def aks_coeffs {lambda {:n} {S.map {coeff :n} {S.serie 1 {+ {/ {- :n 1} 2} 1}}}}} -> aks_coeffs {def divide {lambda {:a :b} {= {BN.compare {BN.% :b :a} 0} 0}}} -> divide {def isprime {lambda {:n} {if {and {S.map {divide :n} {aks_coeffs :n}}} then :n else .}}} -> isprime {S.map isprime {S.serie 2 100}} -> 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 . . .  Liberty BASIC Translation of: Pascal Works with: Just BASIC version any global pasTriMax pasTriMax = 61 dim pasTri(pasTriMax + 1) for n = 0 to 9 call expandPoly n next n for n = 2 to pasTriMax if isPrime(n) <> 0 then print using("###", n); end if next n print end sub expandPoly n n = int(n) dim vz$(1)
vz$(0) = "+" vz$(1) = "-"
if n > pasTriMax then
print n; " is out of range"
end
end if
select case n
case 0
print "(x-1)^0 = 1"
case 1
print "(x-1)^1 = x-1"
case else
call pascalTriangle n
print "(x-1)^"; n; " = ";
print "x^"; n;
bVz = 1
nDiv2 = int(n / 2)
for j = n - 1 to nDiv2 + 1 step -1
print vz$(bVz); pasTri(n - j); "*x^"; j; bVz = abs(1 - bVz) next j for j = nDiv2 to 2 step -1 print vz$(bVz); pasTri(j); "*x^"; j;
bVz = abs(1 - bVz)
next j
print vz$(bVz); pasTri(1); "*x"; bVz = abs(1 - bVz) print vz$(bVz); pasTri(0)
end select
end sub

function isPrime(n)
n = int(n)
if n > pasTriMax then
print n; " is out of range"
end
end if
call pascalTriangle n
res = 1
i = int(n / 2)
while res and (i > 1)
res = res and (pasTri(i) mod n = 0)
i = i - 1
wend
isPrime = res
end function

sub pascalTriangle n
rem Calculate the n'th line 0.. middle
n = int(n)
pasTri(0) = 1
j = 1
while j <= n
j = j + 1
k = int(j / 2)
pasTri(k) = pasTri(k - 1)
for k = k to 1 step -1
pasTri(k) = pasTri(k) + pasTri(k - 1)
next k
wend
end sub
Output:
(x-1)^0 = 1
(x-1)^1 = x-1
(x-1)^2 = x^2-2*x+1
(x-1)^3 = x^3-3*x^2+3*x-1
(x-1)^4 = x^4-4*x^3+6*x^2-4*x+1
(x-1)^5 = x^5-5*x^4+10*x^3-10*x^2+5*x-1
(x-1)^6 = x^6-6*x^5+15*x^4-20*x^3+15*x^2-6*x+1
(x-1)^7 = x^7-7*x^6+21*x^5-35*x^4+35*x^3-21*x^2+7*x-1
(x-1)^8 = x^8-8*x^7+28*x^6-56*x^5+70*x^4-56*x^3+28*x^2-8*x+1
(x-1)^9 = x^9-9*x^8+36*x^7-84*x^6+126*x^5-126*x^4+84*x^3-36*x^2+9*x-1
2  3  5  7 11 13 17 19 23 29 31 37 41 43 47 53 59 61


Maple

Maple handles algebraic manipulation of polynomials natively.

> for xpr in seq( expand( (x-1)^p ), p = 0 .. 7 ) do print( xpr ) end:
1

x - 1

2
x  - 2 x + 1

3      2
x  - 3 x  + 3 x - 1

4      3      2
x  - 4 x  + 6 x  - 4 x + 1

5      4       3       2
x  - 5 x  + 10 x  - 10 x  + 5 x - 1

6      5       4       3       2
x  - 6 x  + 15 x  - 20 x  + 15 x  - 6 x + 1

7      6       5       4       3       2
x  - 7 x  + 21 x  - 35 x  + 35 x  - 21 x  + 7 x - 1

To implement the primality test, we write the following procedure that uses the (built-in) polynomial expansion to generate a list of coefficients of the expanded polynomial.

 polc := p -> [coeffs]( expand( (x-1)^p - (x^p-1) ) ):

Use polc to implement prime? which does the primality test.

prime? := n -> n > 1 and {op}( map( modp, polc( n ), n ) ) = {0}

Of course, rather than calling polc, we can inline it, just for the sake of making the whole thing a one-liner (while adding argument type-checking for good measure):

prime? := (n::posint) -> n > 1 and {op}( map( modp, [coeffs]( expand( (x-1)^n - (x^n-1) ) ), n ) ) = {0}

This agrees with the built-in primality test isprime:

> evalb( seq( prime?(i), i = 1 .. 1000 ) = seq( isprime( i ), i = 1 .. 1000 ) );
true

Use prime? with the built-in Maple select procedure to pick off the primes up to 50:

> select( prime?, [seq](1..50) );
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]

Mathematica / Wolfram Language

Algebraic manipulation is built into Mathematica, so there's no need to create a function to do (x-1)^p

Print["powers of (x-1)"]
(x - 1)^( Range[0, 7]) // Expand // TableForm
Print["primes under 50"]
poly[p_] := (x - 1)^p - (x^p - 1) // Expand;
coefflist[p_Integer] := Coefficient[poly[p], x, #] & /@ Range[0, p - 1];
AKSPrimeQ[p_Integer] := (Mod[coefflist[p] , p] // Union) == {0};
Select[Range[1, 50], AKSPrimeQ]

Output:
powers of (x-1)
1
-1+x
1-2 x+x^2
-1+3 x-3 x^2+x^3
1-4 x+6 x^2-4 x^3+x^4
-1+5 x-10 x^2+10 x^3-5 x^4+x^5
1-6 x+15 x^2-20 x^3+15 x^4-6 x^5+x^6
-1+7 x-21 x^2+35 x^3-35 x^4+21 x^5-7 x^6+x^7

primes under 50
{1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47}

Maxima

/* Function that lists coefficients given the exponent */
pol_to_list(n):=if n=0 then [1] else block(pol:expand((x-1)^n),
makelist(numfactor(part(pol,i)),i,1,length(pol)))$/* Function to expand the polynomial (x-1)^n */ expansion(n):=block( coeflist:pol_to_list(n), makelist(x^(length(%%)-i),i,1,length(%%)), %%*coeflist, apply("+",%%))$

/* AKS based predicate function */
aks_primep(n):=if n=1 then false else block(sample:expansion(n)-(x^n-1),
makelist(numfactor(part(sample,i)),i,1,length(sample)),
if not member(false,makelist(integerp(%%[i]/n),i,1,length(%%))) then true)$/* Test cases */ makelist(expansion(i),i,0,7); block( makelist(aks_primep(i),i,1,50), sublist_indices(%%,lambda([x],x=true)));  Output:  [1,x-1,x^2-2*x+1,x^3-3*x^2+3*x-1,x^4-4*x^3+6*x^2-4*x+1,x^5-5*x^4+10*x^3-10*x^2+5*x-1,x^6-6*x^5+15*x^4-20*x^3+15*x^2-6*x+1,x^7-7*x^6+21*x^5-35*x^4+35*x^3-21*x^2+7*x-1] [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47]  Nim from math import binom import strutils # Table of unicode superscript characters. const Exponents: array[0..9, string] = ["⁰", "¹", "²", "³", "⁴", "⁵", "⁶", "⁷", "⁸", "⁹"] iterator coeffs(n: int): int = ## Yield the coefficients of the expansion of (x - 1)ⁿ. var sign = 1 for k in 0..n: yield binom(n, k) * sign sign = -sign iterator polyExpansion(n: int): tuple[c, e: int] = ## Yield the coefficients and the exponents of the expansion of (x - 1)ⁿ. var e = n for c in coeffs(n): yield(c, e) dec e proc termString(c, e: int): string = ## Return the string for the term c * e^n. if e == 0: result.addInt(c) else: if c != 1: result.addInt(c) result.add('x') if e != 1: result.add(Exponents[e]) proc polyString(n: int): string = ## Return the string for the expansion of (x - 1)ⁿ. for (c, e) in polyExpansion(n): if c < 0: result.add(" - ") elif e != n: result.add(" + ") result.add(termString(abs(c), e)) proc isPrime(n: int): bool = ## Check if a number is prime using the polynome expansion. result = true for (c, e) in polyExpansion(n): if e in 1..(n-1): # xⁿ and 1 are eliminated by the subtraction. if c mod n != 0: return false #--------------------------------------------------------------------------------------------------- echo "Polynome expansions:" for p in 0..9: echo "(x - 1)$1 = $2".format(Exponents[p], polyString(p)) var primes: string for p in 2..34: if p.isPrime(): primes.addSep(", ", 0) primes.addInt(p) echo "\nPrimes under 35: ", primes for p in 35..50: if p.isPrime(): primes.add(", ") primes.addInt(p) echo "\nPrimes under 50: ", primes  Output: Polynome expansions: (x - 1)⁰ = 1 (x - 1)¹ = x - 1 (x - 1)² = x² - 2x + 1 (x - 1)³ = x³ - 3x² + 3x - 1 (x - 1)⁴ = x⁴ - 4x³ + 6x² - 4x + 1 (x - 1)⁵ = x⁵ - 5x⁴ + 10x³ - 10x² + 5x - 1 (x - 1)⁶ = x⁶ - 6x⁵ + 15x⁴ - 20x³ + 15x² - 6x + 1 (x - 1)⁷ = x⁷ - 7x⁶ + 21x⁵ - 35x⁴ + 35x³ - 21x² + 7x - 1 (x - 1)⁸ = x⁸ - 8x⁷ + 28x⁶ - 56x⁵ + 70x⁴ - 56x³ + 28x² - 8x + 1 (x - 1)⁹ = x⁹ - 9x⁸ + 36x⁷ - 84x⁶ + 126x⁵ - 126x⁴ + 84x³ - 36x² + 9x - 1 Primes under 35: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 Primes under 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47  Objeck Translation of: Java class AksTest { @c : static : Int[]; function : Main(args : String[]) ~ Nil { @c := Int->New[100]; for(n := 0; n < 10; n++;) { Coef(n); "(x-1)^ {$n} = "->Print();
Show(n);
'\n'->Print();
};

"\nPrimes:"->PrintLine();
for(n := 2; n <= 63; n++;) {
if(IsPrime(n)) {
" {$n}"->Print(); }; }; '\n'->Print(); } function : native : Coef(n : Int) ~ Nil { i := 0; j := 0; if (n < 0 | n > 63) { Runtime->Exit(0); }; for(@c[0] := 1; i < n; i++;) { j := i; for(@c[1 + j] := 1; j > 0; j--;) { @c[j] := @c[j-1] - @c[j]; }; @c[0] := @c[0] * -1; }; } function : native : IsPrime(n : Int) ~ Bool { Coef(n); @c[0] += 1; @c[n] -= 1; i:=n; while (i <> 0 & (@c[i] % n) = 0) { i--; }; return i = 0; } function : Show(n : Int) ~ Nil { do { value := @c[n]; "+{$value}x^{$n}"->Print(); } while (n-- <> 0); } } Output: (x-1)^ 0 = +1x^0 (x-1)^ 1 = +1x^1+-1x^0 (x-1)^ 2 = +1x^2+-2x^1+1x^0 (x-1)^ 3 = +1x^3+-3x^2+3x^1+-1x^0 (x-1)^ 4 = +1x^4+-4x^3+6x^2+-4x^1+1x^0 (x-1)^ 5 = +1x^5+-5x^4+10x^3+-10x^2+5x^1+-1x^0 (x-1)^ 6 = +1x^6+-6x^5+15x^4+-20x^3+15x^2+-6x^1+1x^0 (x-1)^ 7 = +1x^7+-7x^6+21x^5+-35x^4+35x^3+-21x^2+7x^1+-1x^0 (x-1)^ 8 = +1x^8+-8x^7+28x^6+-56x^5+70x^4+-56x^3+28x^2+-8x^1+1x^0 (x-1)^ 9 = +1x^9+-9x^8+36x^7+-84x^6+126x^5+-126x^4+84x^3+-36x^2+9x^1+-1x^0 Primes: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 OCaml Translation of: Clojure Uses gen library for lazy streams and zarith for arbitrarily sized integers. Runs as is through the utop REPL. #require "gen" #require "zarith" open Z let range ?(step=one) i j = if i = j then Gen.empty else Gen.unfold (fun k -> if compare i j = compare k j then Some (k, (add step k)) else None) i (* kth coefficient of (x - 1)^n *) let coeff n k = let numer = Gen.fold mul one (range n (sub n k) ~step:minus_one) in let denom = Gen.fold mul one (range k zero ~step:minus_one) in div numer denom |> mul @@ if compare k n < 0 && is_even k then minus_one else one (* coefficient series for (x - 1)^n, k=[0..n] *) let coeff_series n = Gen.map (coeff n) (range zero (succ n)) let middle g = Gen.drop 1 g |> Gen.peek |> Gen.filter_map (function (_, None) -> None | (e, _) -> Some e) let is_mod_p ~p n = rem n p = zero let aks p = coeff_series p |> middle |> Gen.for_all (is_mod_p ~p) let _ = print_endline "coefficient series n (k[0] .. k[n])"; Gen.iter (fun n -> Format.printf "%d (%s)\n" (to_int n) (Gen.map to_string (coeff_series n) |> Gen.to_list |> String.concat " ")) (range zero (of_int 10)); print_endline ""; print_endline ("primes < 50 per AKS: " ^ (Gen.filter aks (range (of_int 2) (of_int 50)) |> Gen.map to_string |> Gen.to_list |> String.concat " "))  Output: coefficient series n (k[0] .. k[n]) 0 (1) 1 (-1 1) 2 (-1 2 1) 3 (-1 3 -3 1) 4 (-1 4 -6 4 1) 5 (-1 5 -10 10 -5 1) 6 (-1 6 -15 20 -15 6 1) 7 (-1 7 -21 35 -35 21 -7 1) 8 (-1 8 -28 56 -70 56 -28 8 1) 9 (-1 9 -36 84 -126 126 -84 36 -9 1) primes < 50 per AKS: (2 3 5 7 11 13 17 19 23 29 31 37 41 43 47) Oforth import: mapping : nextCoef( prev -- [] ) | i | Array new 0 over dup prev size 1- loop: i [ prev at(i) prev at(i 1+) - over add ] 0 over add ; : coefs( n -- [] ) [ 0, 1, 0 ] #nextCoef times(n) extract(2, n 2 + ) ; : prime?( n -- b) coefs( n ) extract(2, n) conform?( #[n mod 0 == ] ) ; : aks | i | 0 10 for: i [ System.Out "(x-1)^" << i << " = " << coefs( i ) << cr ] 50 seq filter( #prime? ) apply(#.) printcr ; Output: (x-1)^0 = [1] (x-1)^1 = [-1, 1] (x-1)^2 = [1, -2, 1] (x-1)^3 = [-1, 3, -3, 1] (x-1)^4 = [1, -4, 6, -4, 1] (x-1)^5 = [-1, 5, -10, 10, -5, 1] (x-1)^6 = [1, -6, 15, -20, 15, -6, 1] (x-1)^7 = [-1, 7, -21, 35, -35, 21, -7, 1] (x-1)^8 = [1, -8, 28, -56, 70, -56, 28, -8, 1] (x-1)^9 = [-1, 9, -36, 84, -126, 126, -84, 36, -9, 1] (x-1)^10 = [1, -10, 45, -120, 210, -252, 210, -120, 45, -10, 1] 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47  PARI/GP getPoly(n)=('x-1)^n; vector(8,n,getPoly(n-1)) AKS_slow(n)=my(P=getPoly(n));for(i=1,n-1,if(polcoeff(P,i)%n,return(0))); 1; AKS(n)=my(X=('x-1)*Mod(1,n));X^n=='x^n-1; select(AKS, [1..50]) Output:  [1, x - 1, x^2 - 2*x + 1, x^3 - 3*x^2 + 3*x - 1, x^4 - 4*x^3 + 6*x^2 - 4*x + 1, x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x - 1, x^6 - 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x + 1, x^7 - 7*x^6 + 21*x^5 - 35*x^4 + 35*x^3 - 21*x^2 + 7*x - 1] [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47] Pascal Works with: Free Pascal const pasTriMax = 61; type TPasTri = array[0 .. pasTriMax] of UInt64; var pasTri: TPasTri; procedure PascalTriangle(n: LongWord); // Calculate the n'th line 0.. middle var j, k: LongWord; begin pasTri[0] := 1; j := 1; while j <= n do begin Inc(j); k := j div 2; pasTri[k] := pasTri[k - 1]; for k := k downto 1 do Inc(pasTri[k], pasTri[k - 1]); end; end; function IsPrime(n: LongWord): Boolean; var i: Integer; begin if n > pasTriMax then begin WriteLn(n, ' is out of range'); Halt; end; PascalTriangle(n); Result := true; i := n div 2; while Result and (i > 1) do begin Result := Result and (pasTri[i] mod n = 0); Dec(i); end; end; procedure ExpandPoly(n: LongWord); const Vz: array[Boolean] of Char = ('+', '-'); var j: LongWord; bVz: Boolean; begin if n > pasTriMax then begin WriteLn(n,' is out of range'); Halt; end; case n of 0: WriteLn('(x-1)^0 = 1'); 1: WriteLn('(x-1)^1 = x-1'); else PascalTriangle(n); Write('(x-1)^', n, ' = '); Write('x^', n); bVz := true; for j := n - 1 downto n div 2 + 1 do begin Write(vz[bVz], pasTri[n - j], '*x^', j); bVz := not bVz; end; for j := n div 2 downto 2 do begin Write(vz[bVz], pasTri[j], '*x^', j); bVz := not bVz; end; Write(vz[bVz], pasTri[1], '*x'); bVz := not bVz; WriteLn(vz[bVz], pasTri[0]); end; end; var n: LongWord; begin for n := 0 to 9 do ExpandPoly(n); for n := 2 to pasTriMax do if IsPrime(n) then Write(n:3); WriteLn; end.  output (x-1)^0 = 1 (x-1)^1 = x-1 (x-1)^2 = x^2-2*x+1 (x-1)^3 = x^3-3*x^2+3*x-1 (x-1)^4 = x^4-4*x^3+6*x^2-4*x+1 (x-1)^5 = x^5-5*x^4+10*x^3-10*x^2+5*x-1 (x-1)^6 = x^6-6*x^5+15*x^4-20*x^3+15*x^2-6*x+1 (x-1)^7 = x^7-7*x^6+21*x^5-35*x^4+35*x^3-21*x^2+7*x-1 (x-1)^8 = x^8-8*x^7+28*x^6-56*x^5+70*x^4-56*x^3+28*x^2-8*x+1 (x-1)^9 = x^9-9*x^8+36*x^7-84*x^6+126*x^5-126*x^4+84*x^3-36*x^2+9*x-1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 Perl use strict; use warnings; # Select one of these lines. Math::BigInt is in core, but quite slow. use Math::BigInt; sub binomial { Math::BigInt->new(shift)->bnok(shift) } # use Math::Pari "binomial"; # use ntheory "binomial"; sub binprime { my$p = shift;
return 0 unless $p >= 2; # binomial is symmetric, so only test half the terms for (1 .. ($p>>1)) { return 0 if binomial($p,$_) % $p } 1; } sub coef { # For prettier printing my($n,$e) = @_; return$n unless $e;$n = "" if $n==1;$e==1 ? "${n}x" : "${n}x^$e"; } sub binpoly { my$p = shift;
join(" ", coef(1,$p), map { join("",("+","-")[($p-$_)&1]," ",coef(binomial($p,$_),$_)) }
reverse 0..$p-1 ); } print "expansions of (x-1)^p:\n"; print binpoly($_),"\n" for 0..9;
print "Primes to 80: [", join(",", grep { binprime($_) } 2..80), "]\n";  Output: expansions of (x-1)^p: 1 x - 1 x^2 - 2x + 1 x^3 - 3x^2 + 3x - 1 x^4 - 4x^3 + 6x^2 - 4x + 1 x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1 x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1 x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1 x^8 - 8x^7 + 28x^6 - 56x^5 + 70x^4 - 56x^3 + 28x^2 - 8x + 1 x^9 - 9x^8 + 36x^7 - 84x^6 + 126x^5 - 126x^4 + 84x^3 - 36x^2 + 9x - 1 Primes to 80: [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79] Real AKS The ntheory module has implementations of the full AKS algorithm in Perl, C, and C+GMP. This is vastly faster than the method used in this task and is polynomial time, but like all current AKS implementations is still much slower than other methods such as BPSW, APR-CL, and ECPP. Library: ntheory use ntheory ":all"; # Uncomment next line to see the r and s values used. Set to 2 for more detail. # prime_set_config(verbose => 1); say join(" ", grep { is_aks_prime($_) } 1_000_000_000 .. 1_000_000_100);

Output:
1000000007 1000000009 1000000021 1000000033 1000000087 1000000093 1000000097

Phix

-- demo/rosetta/AKSprimes.exw
-- Does not work for primes above 53, which is actually beyond the original task anyway.
-- Translated from the C version, just about everything is (working) out-by-1, what fun.

sequence c = repeat(0,100)

procedure coef(integer n)
-- out-by-1, ie coef(1)==^0, coef(2)==^1, coef(3)==^2 etc.
c[n] = 1
for i=n-1 to 2 by -1 do
c[i] = c[i]+c[i-1]
end for
end procedure

function is_aks_prime(integer n)
coef(n+1); -- (I said it was out-by-1)
for i=2 to n-1 do   -- (technically "to n" is more correct)
if remainder(c[i],n)!=0 then
return 0
end if
end for
return 1
end function

procedure show(integer n)
-- (As per coef, this is (working) out-by-1)
object ci
for i=n to 1 by -1 do
ci = c[i]
if ci=1 then
if remainder(n-i,2)=0 then
if i=1 then
if n=1 then
ci = "1"
else
ci = "+1"
end if
else
ci = ""
end if
else
ci = "-1"
end if
else
if remainder(n-i,2)=0 then
ci = sprintf("+%d",ci)
else
ci = sprintf("-%d",ci)
end if
end if
if i=1 then -- ie ^0
printf(1,"%s",{ci})
elsif i=2 then -- ie ^1
printf(1,"%sx",{ci})
else
printf(1,"%sx^%d",{ci,i-1})
end if
end for
end procedure

procedure main()
for n=1 to 10 do -- (0 to 9 really)
coef(n);
printf(1,"(x-1)^%d = ", n-1);
show(n);
puts(1,'\n');
end for

puts(1,"\nprimes (<=53):");
--  coef(2); -- (needed to reset c, if we want to avoid saying 1 is prime...)
c[2] = 1 -- (this manages "", which is all that call did anyway...)
for n = 2 to 53 do
if is_aks_prime(n) then
printf(1," %d", n);
end if
end for
puts(1,'\n');
if getc(0) then end if
end procedure
main()

Output:
(x-1)^0 = 1
(x-1)^1 = x-1
(x-1)^2 = x^2-2x+1
(x-1)^3 = x^3-3x^2+3x-1
(x-1)^4 = x^4-4x^3+6x^2-4x+1
(x-1)^5 = x^5-5x^4+10x^3-10x^2+5x-1
(x-1)^6 = x^6-6x^5+15x^4-20x^3+15x^2-6x+1
(x-1)^7 = x^7-7x^6+21x^5-35x^4+35x^3-21x^2+7x-1
(x-1)^8 = x^8-8x^7+28x^6-56x^5+70x^4-56x^3+28x^2-8x+1
(x-1)^9 = x^9-9x^8+36x^7-84x^6+126x^5-126x^4+84x^3-36x^2+9x-1

primes (<=53): 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53


Picat

pascal([]) = [1].

Next = L,
while (Row = [A,B|_])
L = [A+B|Rest],
L := Rest,
Row := tail(Row)
end,
L = Row.

show_x(0) = "".
show_x(1) = "x".
show_x(N) = S, N > 1 => S = [x, '^' | to_string(N)].

show_term(Coef, Exp) = cond((Coef != 1; Exp == 0), Coef.to_string, "") ++ show_x(Exp).

expansions(N) =>
Row = [],
foreach (I in 0..N-1)
Row := pascal(Row),
writef("(x - 1)^%d = ", I),
Exp = I,
Sgn = '+',
foreach (Coef in Row)
if Exp != I then
writef(" %w ", Sgn)
end,
writef("%s", show_term(Coef, Exp)),
Exp := Exp - 1,
Sgn := cond(Sgn == '+', '-', '+')
end,
nl
end.

primerow([], _).
primerow([A,A|_], N) :- A mod N == 0.  % end when we've seen half the list.
primerow([A|As], N) :- (A mod N == 0; A == 1), primerow(As, N).

primes_upto(N) = Primes =>
Primes = L,
Row = [1, 1],
foreach (K in 2..N)
Row := pascal(Row),
if primerow(Row, K) then
L = [K|Rest],
L := Rest
end
end,
L = [].

main =>
expansions(8),
writef("%nThe primes upto 50 (via AKS) are: %w%n", primes_upto(50)).
Output:
(x - 1)^0 = 1
(x - 1)^1 = x - 1
(x - 1)^2 = x^2 - 2x + 1
(x - 1)^3 = x^3 - 3x^2 + 3x - 1
(x - 1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1
(x - 1)^5 = x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1
(x - 1)^6 = x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1
(x - 1)^7 = x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1

The primes upto 50 (via AKS) are: [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47]


PicoLisp

(de pascal (N)
(let D 1
(make
(for X (inc N)
(setq D
(*/ D (- (inc N) X) (- X)) ) ) ) ) )

(for (X 0 (> 10 X) (inc X))
(println X '-> (pascal X) ) )

(println
(filter
'((X)
(fully
'((Y) (=0 (% Y X)))
(cdr (head -1 (pascal X))) ) )
(range 2 50) ) )

(bye)

Output:
0 -> (1)
1 -> (1 -1)
2 -> (1 -2 1)
3 -> (1 -3 3 -1)
4 -> (1 -4 6 -4 1)
5 -> (1 -5 10 -10 5 -1)
6 -> (1 -6 15 -20 15 -6 1)
7 -> (1 -7 21 -35 35 -21 7 -1)
8 -> (1 -8 28 -56 70 -56 28 -8 1)
9 -> (1 -9 36 -84 126 -126 84 -36 9 -1)
(2 3 5 7 11 13 17 19 23 29 31 37 41 43 47)


PL/I

AKS: procedure options (main, reorder); /* 16 September 2015, derived from Fortran */

/* Coefficients of polynomial expansion */
declare coeffs(*) fixed (31) controlled;
declare n fixed(3);

/* Point #2 */
do n = 0 to 7;
call polynomial_expansion(n, coeffs);
put edit ( '(x - 1)^', trim(n), ' =' ) (a);
call print_polynomial (coeffs);
end;

/* Point #4 */
put skip;
do n = 2 to 35;
if is_prime(n) then put edit ( trim (n) ) (x(1), a);
end;

/* Point #5 */
put skip;
do n = 2 to 97;
if is_prime(n) then put edit ( trim (n) ) (x(1), a);
end;
put skip;

/* Calculate coefficients of (x - 1)^n using binomial theorem */
polynomial_expansion: procedure (n, coeffs);
declare n fixed binary;
declare coeffs (*) fixed (31) controlled;
declare i fixed binary;

if allocation(coeffs) > 0 then free coeffs;
allocate coeffs (n+1);

do i = 1 to n + 1;
coeffs(i) = binomial(n, i - 1);
if iand(n - i - 1, 1) = 1 then coeffs(i) = -coeffs(i);
end;
end polynomial_expansion;

/* Calculate binomial coefficient using recurrent relation, as calculation	*/
/* using factorial overflows too quickly.					*/
binomial: procedure (n, k) returns (fixed(31));
declare (n, k) fixed;
declare i fixed;
declare result fixed (31) initial (n);

if k = 0 then return (1);

do i = 1 to k - 1;
result = (result*(n - i))/(i + 1);
end;
return (result);
end binomial;

/* Outputs polynomial with given coefficients */
print_polynomial: procedure (coeffs);
declare coeffs (*) fixed (31) controlled;
declare ( i, p ) fixed binary;
declare non_zero bit (1) aligned;
declare (true initial ('1'b), false initial ('0'b)) bit (1);

if allocation(coeffs) = 0 then return;

non_zero = false;

do i = 1 to hbound(coeffs);
if coeffs(i) = 0 then iterate;

p = i - 1;

if non_zero then
do;
if coeffs(i) > 0 then
put edit ( ' + ' ) (a);
else
put edit ( ' - ' ) (a);
end;
else
do;
if coeffs(i) > 0 then
put edit ( '   ' ) (a);
else
put edit ( ' - ' ) (a);
end;

if p = 0 then
put edit ( trim(abs(coeffs(i))) ) (a);
else if p = 1 then
do;
if coeffs(i) = 1 then
put edit ( 'x' ) (a);
else
put edit ( trim(abs(coeffs(i))), 'x' ) (a);
end;
else
do;
if coeffs(i) = 1 then
put edit ( 'x^', trim(p) ) (a);
else
put edit (  trim(abs(coeffs(i)) ), 'x^', trim(p)) (a);
end;

non_zero = true;
end;

put skip;
end print_polynomial;

/* Test if n is prime using AKS test. Point #3. */
is_prime: procedure (n) returns (bit (1));
declare n fixed (15);
declare result bit (1) aligned;
declare coeffs (*) fixed (31) controlled;
declare i fixed binary;

call polynomial_expansion(n, coeffs);
coeffs(1) = coeffs(1) + 1;
coeffs(n + 1) = coeffs(n + 1) - 1;

result = '1'b;

do i = 1 to n + 1;
result = result & (mod(coeffs(i), n) = 0);
end;

if allocation(coeffs) > 0 then free coeffs;
return (result);
end is_prime;

end AKS;

Results obtained:

(x - 1)^0 =   1
(x - 1)^1 = - 1 + x
(x - 1)^2 =   1 - 2x + x^2
(x - 1)^3 = - 1 + 3x - 3x^2 + x^3
(x - 1)^4 =   1 - 4x + 6x^2 - 4x^3 + x^4
(x - 1)^5 = - 1 + 5x - 10x^2 + 10x^3 - 5x^4 + x^5
(x - 1)^6 =   1 - 6x + 15x^2 - 20x^3 + 15x^4 - 6x^5 + x^6
(x - 1)^7 = - 1 + 7x - 21x^2 + 35x^3 - 35x^4 + 21x^5 - 7x^6 + x^7

2 3 5 7 11 13 17 19 23 29 31
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


Prolog

Prolog(ue)

The theorem as stated ties together two elementary concepts in mathematics: prime numbers and the Pascal triangle. The simplicity of the connection can be expressed directly in Prolog by the following prime number generator:

  prime(P) :-
pascal([1,P|Xs]),
append(Xs, [1], Rest),
forall( member(X,Xs), 0 is X mod P).


where pascal/1 is a generator of rows of the Pascal triangle, for example as defined below; the other predicates used above are standard.

This solution to the Rosetta Code problems will accordingly focus on the Pascal triangle, but to illustrate a number of points, we shall exploit its symmetry by representing each of its rows by the longest initial non-decreasing segment of that row, as illustrated in the third column of the following table:

Row   Pascal Row   optpascal
1       1           [1]
2      1 1          [1, 1]
3     1 2 1         [1, 2]
4    1 3 3 1        [1, 3, 3]


We shall refer to this condensed representation of a row as an "optpascal list". Using it, we can simplify and improve the above prime number generator by defining it as follows:

 prime(N) :- optpascal([1,N|Xs]), forall( member(X,Xs), 0 is X mod N).


Using SWI-Prolog without modifying any of the memory management parameters, this prime number generator was used to generate all primes up to and including 75,659.

Since Pascal triangles are the foundation of our approach to addressing the specific Rosetta Code problems, we begin by defining the generator pascal/2 that is required by the first problem, but we do so by defining it in terms of an efficient generator, optpascal/1.

Pascal Triangle Generator

% To generate the n-th row of a Pascal triangle
% pascal(+N, Row)
pascal(0, [1]).
pascal(N, Row) :-
N > 0, optpascal( [1, N|Xs] ),
!,
pascalize( [1, N|Xs], Row ).

pascalize( Opt, Row ) :-
% if Opt ends in a pair, then peel off the pair:
( append(X, [R,R], Opt) -> true ; append(X, [R], Opt) ),
reverse(X, Rs),
append( Opt, Rs, Row ).

% optpascal(-X) generates optpascal lines:
optpascal(X) :-
optpascal_successor( [], X).

% optpascal_successor(+P, -Q) is true if Q is an optpascal list beneath the optpascal list P:
optpascal_successor(P, Q) :-
optpascal(P, NextP),
(Q = NextP ; optpascal_successor(NextP, Q)).

% optpascal(+Row, NextRow) is true if Row and NextRow are adjacent rows in the Pascal triangle.
% optpascal(+Row, NextRow) where the optpascal representation is used
optpascal(X, [1|Y]) :-

% add_pairs(+OptPascal, NextOptPascal) is a helper function for optpascal/2.
% Given one OptPascal list, it generates the next by adding adjacent
% items, but if the last two items are unequal, then their sum is
% repeated.  This is intended to be a deterministic predicate, and to
% avoid a probable compiler limitation, we therefore use one cut.
S is X + Y,
(X = Y -> Ans=[S] ; Ans=[S,S]),
!.  % To overcome potential limitation of compiler

add_pairs( [X1, X2, X3|Xs], [S|Ys]) :-
S is X1 + X2,


Solutions

Solutions with output from SWI-Prolog:

%%% Task 1: "A method to generate the coefficients of (1-X)^p"

coefficients(N, Coefficients) :-
pascal(N, X),
alternate_signs(X, Coefficients).

alternate_signs( [], [] ).
alternate_signs( [A], [A] ).
alternate_signs( [A,B | X], [A, MB | Y] ) :-
MB is -B,
alternate_signs(X,Y).

%%% Task 2. "Show here the polynomial expansions of (x − 1)p for p in the range 0 to at least 7, inclusive."

coefficients(Coefficients) :-
optpascal( Opt),
pascalize( Opt, Row ),
alternate_signs(Row, Coefficients).

%  As required by the problem statement, but necessarily very inefficient:
:- between(0, 7, N), coefficients(N, Coefficients), writeln(Coefficients), fail ; true.

[1]
[1,-1]
[1,-2,1]
[1,-3,3,-1]
[1,-4,6,-4,1]
[1,-5,10,-10,5,-1]
[1,-6,15,-20,15,-6,1]
[1,-7,21,-35,35,-21,7,-1]


The following would be more efficient because backtracking saves recomputation:

:- coefficients(Coefficients),
writeln(Coefficients),
Coefficients = [_,N|_], N = -7.

%%% Task 3. Use the previous function in creating [sic]
%%% another function that when given p returns whether p is prime
%%% using the AKS test.

% Even for testing whether a given number, N, is prime,
% this approach is inefficient, but here is a Prolog implementation:

prime_test_per_requirements(N) :-
coefficients(N, [1|Coefficients]),
append(Cs, [_], Coefficients),
forall( member(C, Cs), 0 is C mod N).


The following is more efficient (because it relies on optpascal lists rather than the full array of coefficients), and more flexible (because it can be used to generate primes without requiring recomputation):

   prime(N) :- optpascal([1,N|Xs]), forall( member(X,Xs), 0 is X mod N).

%%% Task 4. Use your AKS test to generate a list of all primes under 35.

:-   prime(N), (N < 35 -> write(N), write(' '), fail ; nl).

% Output: 1 2 3 5 7 11 13 17 19 23 29 31

%%% Task 5. As a stretch goal, generate all primes under 50.

:-  prime(N), (N < 50 -> write(N), write(' '), fail ; nl).

% Output: 1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47


Alternate version using symbolic computation

This version uses Prolog's symbolic pattern matching to generate the polynomial equations that are the subject of the first part of the task. The rows of Pascal's triangle are generated, then combined with the "a" and "b" terms of the expression (a + b)^n. The expression is then algebraically simplified to remove things like x^0, x^1, 1*x, a + 0, etc.

It would be possible to go further and expand (a + b)^n and collect terms, but that turns out to be rather difficult because terms have to be reordered into a canonical form first -- for example, an expansion can produce both j*a*b and k*b*a, and these terms would have to be reordered to get (j+k)*a*b.

main :- task1(8), nl, task2(50), halt.

pascal(Z),
length(Rows, N),
prefix(Rows, Z),
forall(member(Row, Rows),
(length(Row, K), succ(DecK, K),
binomial(x, -1, Row, Expr),
format("(x-1)**~w = ~w~n", [DecK, Expr]))).

primes_upto(Upto, Ps),
format("The primes upto ~w (via AKS) are: ~p~n", [Upto, Ps]).

pascal(Lz) :-
lazy_list(pascal_row, [], Lz).

pascal_row([], R1, R1) :- R1 = [1], !.
pascal_row(R0, R1, R1) :-

sum_adj(L, L) :- L = [_], !.
As = [B|_], C is A + B,

% First part of task -- create textual representation of (x-1)^n
%  here we generate expression trees
%
binomial(A, B, Coefs, Expr) :-
length(Coefs, N), succ(DecN, N),
binomial(B, DecN, A, 0, Coefs, Exp0),
reduce(Exp0, Exp1),

binomial(_, _, _, _, [], 0) :- !.
binomial(A, PowA, B, PowB, [N|Ns], Ts + T) :-
T = N * A**PowA * B**PowB,
IncPow is PowB + 1,
DecPow is PowA - 1,
binomial(A, DecPow, B, IncPow, Ns, Ts).

(make_positive(B, D) -> X = C - D; X = C + B), !.

make_positive(N, Term) :- integer(N), N < 0, !, Term is -N.
make_positive(A*B, Term) :-
make_positive(A, PosA),
(PosA = 1 -> Term = B, !; Term = PosA*B).

reduce(A, C) :-
simplify(A, B),
(B = A -> C = A; reduce(B, C)).

simplify(_**0, 1) :- !.
simplify(1**_, 1) :- !.
simplify(-1**N, Z) :- integer(N), (0 is N /\ 1 -> Z = 1; Z = -1), !.
simplify(X**1, X) :- !.

simplify(0 + A, A) :- !.
simplify(A + 0, A) :- !.
simplify(A + B, C) :-
integer(A),
integer(B), !,
C is A + B.
simplify(A + B, C + D) :- !,
simplify(A, C),
simplify(B, D).

simplify(0 * _, 0) :- !.
simplify(_ * 0, 0) :- !.
simplify(1 * A, A) :- !.
simplify(A * 1, A) :- !.
simplify(A * B, C) :-
integer(A),
integer(B), !,
C is A * B.
simplify(A * B, C * D) :- !,
simplify(A, C),
simplify(B, D).

simplify(X, X).

% Second part of task -- Use the coefficients of Pascal's Triangle to check primality.
%

primerow([1, N| Rest]) :- primerow(N, Rest).

primerow(_, End) :- (End = []; End = [1]), !.
primerow(_, [A,A|_]) :- !.  % end when we've seen half the list.
primerow(N, [A|As]) :- A mod N =:= 0, primerow(N, As).

second([_,N|_], N).

primes_upto(N, Ps) :-
pascal(Z),
Z = [_, _ | Rows], % we only care about 2nd row on up. ([1,2,1])
succ(DecN, N), length(CheckRows, DecN), prefix(CheckRows, Rows),
include(primerow, CheckRows, PrimeRows),
maplist(second, PrimeRows, Ps).

?- main.

Output:
swipl aks.pl (x-1)**0 = 1 (x-1)**1 = x-1 (x-1)**2 = x**2-2*x+1 (x-1)**3 = x**3-3*x**2+3*x-1 (x-1)**4 = x**4-4*x**3+6*x**2-4*x+1 (x-1)**5 = x**5-5*x**4+10*x**3-10*x**2+5*x-1 (x-1)**6 = x**6-6*x**5+15*x**4-20*x**3+15*x**2-6*x+1 (x-1)**7 = x**7-7*x**6+21*x**5-35*x**4+35*x**3-21*x**2+7*x-1 The primes upto 50 (via AKS) are: [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47]  PureBasic EnableExplicit Define vzr.b = -1, vzc.b = ~vzr, nMAX.i = 10, n.i , k.i Procedure coeff(nRow.i, Array pd.i(2)) Define n.i, k.i For n=1 To nRow For k=0 To n If k=0 Or k=n : pd(n,k)=1 : Continue : EndIf pd(n,k)=pd(n-1,k-1)+pd(n-1,k) Next Next EndProcedure Procedure.b isPrime(n.i, Array pd.i(2)) Define m.i For m=1 To n-1 If Not pd(n,m) % n = 0 : ProcedureReturn #False : EndIf Next ProcedureReturn #True EndProcedure Dim pd.i(nMAX,nMAX) pd(0,0)=1 : coeff(nMAX, pd()) OpenConsole() For n=0 To nMAX Print(RSet(Str(n),3,Chr(32))+": ") If vzr : Print("+") : Else : Print("-") : EndIf For k=0 To n If k>0 : If vzc : Print("+") : Else : Print("-") : EndIf : vzc = ~vzc : EndIf Print(RSet(Str(pd(n,k)),3,Chr(32))+Space(3)) Next PrintN("") vzr = ~vzr : vzc = ~vzr Next PrintN("") nMAX=50 : Dim pd.i(nMAX,nMAX) Print("Primes n<=50 : ") : coeff(nMAX, pd()) For n=2 To 50 If isPrime(n,pd()) : Print(Str(n)+Space(2)) : EndIf Next Input()  Output:  0: + 1 1: - 1 + 1 2: + 1 - 2 + 1 3: - 1 + 3 - 3 + 1 4: + 1 - 4 + 6 - 4 + 1 5: - 1 + 5 - 10 + 10 - 5 + 1 6: + 1 - 6 + 15 - 20 + 15 - 6 + 1 7: - 1 + 7 - 21 + 35 - 35 + 21 - 7 + 1 8: + 1 - 8 + 28 - 56 + 70 - 56 + 28 - 8 + 1 9: - 1 + 9 - 36 + 84 -126 +126 - 84 + 36 - 9 + 1 10: + 1 - 10 + 45 -120 +210 -252 +210 -120 + 45 - 10 + 1 Primes n<=50 : 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 Python def expand_x_1(n): # This version uses a generator and thus less computations c =1 for i in range(n//2+1): c = c*(n-i)//(i+1) yield c def aks(p): if p==2: return True for i in expand_x_1(p): if i % p: # we stop without computing all possible solutions return False return True  or equivalently: def aks(p): if p==2:return True c=1 for i in range(p//2+1): c=c*(p-i)//(i+1) if c%p:return False return True  alternatively: def expand_x_1(p): ex = [1] for i in range(p): ex.append(ex[-1] * -(p-i) / (i+1)) return ex[::-1] def aks_test(p): if p < 2: return False ex = expand_x_1(p) ex[0] += 1 return not any(mult % p for mult in ex[0:-1]) print('# p: (x-1)^p for small p') for p in range(12): print('%3i: %s' % (p, ' '.join('%+i%s' % (e, ('x^%i' % n) if n else '') for n,e in enumerate(expand_x_1(p))))) print('\n# small primes using the aks test') print([p for p in range(101) if aks_test(p)])  Output: # p: (x-1)^p for small p 0: +1 1: -1 +1x^1 2: +1 -2x^1 +1x^2 3: -1 +3x^1 -3x^2 +1x^3 4: +1 -4x^1 +6x^2 -4x^3 +1x^4 5: -1 +5x^1 -10x^2 +10x^3 -5x^4 +1x^5 6: +1 -6x^1 +15x^2 -20x^3 +15x^4 -6x^5 +1x^6 7: -1 +7x^1 -21x^2 +35x^3 -35x^4 +21x^5 -7x^6 +1x^7 8: +1 -8x^1 +28x^2 -56x^3 +70x^4 -56x^5 +28x^6 -8x^7 +1x^8 9: -1 +9x^1 -36x^2 +84x^3 -126x^4 +126x^5 -84x^6 +36x^7 -9x^8 +1x^9 10: +1 -10x^1 +45x^2 -120x^3 +210x^4 -252x^5 +210x^6 -120x^7 +45x^8 -10x^9 +1x^10 11: -1 +11x^1 -55x^2 +165x^3 -330x^4 +462x^5 -462x^6 +330x^7 -165x^8 +55x^9 -11x^10 +1x^11 # small primes using the aks test [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] Python: Output formatted for wiki Using a wikitable and math features with the following additional code produces better formatted polynomial output: print(''' {| class="wikitable" style="text-align:left;" |+ Polynomial Expansions and AKS prime test |- ! $p$ ! $(x-1)^p$ |-''') for p in range(12): print('! $%i$\n| $%s$\n| %r\n|-' % (p, ' '.join('%s%s' % (('%+i' % e) if (e != 1 or not p or (p and not n) ) else '+', (('x^{%i}' % n) if n > 1 else 'x') if n else '') for n,e in enumerate(expand_x_1(p))), aks_test(p))) print('|}')  Output: Polynomial Expansions and AKS prime test ${\displaystyle p}$ ${\displaystyle (x-1)^{p}}$ Prime(p)? ${\displaystyle 0}$ ${\displaystyle +1}$ False ${\displaystyle 1}$ ${\displaystyle -1+x}$ False ${\displaystyle 2}$ ${\displaystyle +1-2x+x^{2}}$ True ${\displaystyle 3}$ ${\displaystyle -1+3x-3x^{2}+x^{3}}$ True ${\displaystyle 4}$ ${\displaystyle +1-4x+6x^{2}-4x^{3}+x^{4}}$ False ${\displaystyle 5}$ ${\displaystyle -1+5x-10x^{2}+10x^{3}-5x^{4}+x^{5}}$ True ${\displaystyle 6}$ ${\displaystyle +1-6x+15x^{2}-20x^{3}+15x^{4}-6x^{5}+x^{6}}$ False ${\displaystyle 7}$ ${\displaystyle -1+7x-21x^{2}+35x^{3}-35x^{4}+21x^{5}-7x^{6}+x^{7}}$ True ${\displaystyle 8}$ ${\displaystyle +1-8x+28x^{2}-56x^{3}+70x^{4}-56x^{5}+28x^{6}-8x^{7}+x^{8}}$ False ${\displaystyle 9}$ ${\displaystyle -1+9x-36x^{2}+84x^{3}-126x^{4}+126x^{5}-84x^{6}+36x^{7}-9x^{8}+x^{9}}$ False ${\displaystyle 10}$ ${\displaystyle +1-10x+45x^{2}-120x^{3}+210x^{4}-252x^{5}+210x^{6}-120x^{7}+45x^{8}-10x^{9}+x^{10}}$ False ${\displaystyle 11}$ ${\displaystyle -1+11x-55x^{2}+165x^{3}-330x^{4}+462x^{5}-462x^{6}+330x^{7}-165x^{8}+55x^{9}-11x^{10}+x^{11}}$ True Quackery  [ dup 0 peek swap [] 0 rot 0 join witheach [ abs tuck + rot join swap ] drop [] swap witheach [ rot negate dup dip unrot * join ] nip ] is nextcoeffs ( [ --> [ ) [ ' [ 1 ] swap times nextcoeffs ] is coeffs ( n --> [ ) [ dup 2 < iff [ drop false ] done true swap dup coeffs behead drop -1 split drop witheach [ over mod 0 != if [ dip not conclude ] ] drop ] is prime ( n --> b ) [ behead 0 < iff [ say "-" ] else sp dup size dup 0 = iff [ 1 echo 2drop ] done say "x" dup 1 = iff [ 2drop say " + 1" ] done say "^" echo witheach [ dup 0 < iff [ say " - " ] else [ say " + " ] abs echo i 0 = if done say "x" i 1 = if done say "^" i echo ] ] is echocoeffs ( [ --> ) 8 times [ i^ echo say ": " i^ coeffs echocoeffs cr ] cr 50 times [ i^ prime if [ i^ echo sp ] ] Output: 0: 1 1: -x + 1 2: x^2 - 2x + 1 3: -x^3 + 3x^2 - 3x + 1 4: x^4 - 4x^3 + 6x^2 - 4x + 1 5: -x^5 + 5x^4 - 10x^3 + 10x^2 - 5x + 1 6: x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1 7: -x^7 + 7x^6 - 21x^5 + 35x^4 - 35x^3 + 21x^2 - 7x + 1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47  R Working on the coefficients of the following expression (x-1)^p - (x^p-1). Excluding the coefficient of X^(p-1). AKS<-function(p){ i<-2:p-1 l<-unique(factorial(p) / (factorial(p-i) * factorial(i))) if(all(l%%p==0)){ print(noquote("It is prime.")) }else{ print(noquote("It isn't prime.")) } }  Racket With copious use of the math/number-theory library... #lang racket (require math/number-theory) ;; 1. coefficients of expanded polynomial (x-1)^p ;; produces a vector because in-vector can provide a start ;; and stop (of 1 and p) which allow us to drop the (-1)^p ;; and the x^p terms, respectively. ;; ;; (vector-ref (coefficients p) e) is the coefficient for p^e (define (coefficients p) (for/vector ((e (in-range 0 (add1 p)))) (define sign (expt -1 (- p e))) (* sign (binomial p e)))) ;; 2. Show the polynomial expansions from p=0 .. 7 (inclusive) ;; (it's possible some of these can be merged...) (define (format-coefficient c e leftmost?) (define (format-c.x^e c e) (define +c (abs c)) (match* (+c e) [(_ 0) (format "~a" +c)] [(1 _) (format "x^~a" e)] [(_ _) (format "~ax^~a" +c e)])) (define +/- (if (negative? c) "-" "+")) (define +c.x^e (format-c.x^e c e)) (match* (c e leftmost?) [(0 _ _) ""] [((? negative?) _ #t) (format "-~a" +c.x^e)] [(_ _ #t) +c.x^e] [(_ _ _) (format " ~a ~a" +/- +c.x^e)])) (define (format-polynomial cs) (define cs-length (sequence-length cs)) (apply string-append (reverse ; convention is to display highest exponent first (for/list ((c cs) (e (in-naturals))) (format-coefficient c e (= e (sub1 cs-length))))))) (for ((p (in-range 0 (add1 11)))) (printf "p=~a: ~a~%" p (format-polynomial (coefficients p)))) ;; 3. AKS primeality test (define (prime?/AKS p) (define cs (coefficients p)) (and (or (= (vector-ref cs 0) -1) ; c_0 = -1 -> c_0 - (-1) = 0 (divides? p 2)) ; c_0 = 1 -> c_0 - (-1) = 2 -> divides? (for/and ((c (in-vector cs 1 p))) (divides? p c)))) ;; there is some discussion (see Discussion) about what to do with the perennial "1" ;; case. This is my way of saying that I'm ignoring it (define lowest-tested-number 2) ;; 4. list of numbers < 35 that are prime (note that 1 is prime ;; by the definition of the AKS test for primes): (displayln (for/list ((i (in-range lowest-tested-number 35)) #:when (prime?/AKS i)) i)) ;; 5. stretch goal: all prime numbers under 50 (displayln (for/list ((i (in-range lowest-tested-number 50)) #:when (prime?/AKS i)) i)) (displayln (for/list ((i (in-range lowest-tested-number 100)) #:when (prime?/AKS i)) i))  Output: p=0: 1 p=1: x^1 - 1 p=2: x^2 - 2x^1 + 1 p=3: x^3 - 3x^2 + 3x^1 - 1 p=4: x^4 - 4x^3 + 6x^2 - 4x^1 + 1 p=5: x^5 - 5x^4 + 10x^3 - 10x^2 + 5x^1 - 1 p=6: x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x^1 + 1 p=7: x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x^1 - 1 p=8: x^8 - 8x^7 + 28x^6 - 56x^5 + 70x^4 - 56x^3 + 28x^2 - 8x^1 + 1 p=9: x^9 - 9x^8 + 36x^7 - 84x^6 + 126x^5 - 126x^4 + 84x^3 - 36x^2 + 9x^1 - 1 p=10: x^10 - 10x^9 + 45x^8 - 120x^7 + 210x^6 - 252x^5 + 210x^4 - 120x^3 + 45x^2 - 10x^1 + 1 p=11: x^11 - 11x^10 + 55x^9 - 165x^8 + 330x^7 - 462x^6 + 462x^5 - 330x^4 + 165x^3 - 55x^2 + 11x^1 - 1 (2 3 5 7 11 13 17 19 23 29 31) (2 3 5 7 11 13 17 19 23 29 31 37 41 43 47) (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) Raku (formerly Perl 6) Works with: Rakudo version 2015-12 The expansions are generated similarly to how most FP languages generate sequences that resemble Pascal's triangle, using a zipwith meta-operator (Z) with subtraction, applied between two lists that add a 0 on either end to the prior list. Here we define a constant infinite sequence using the ... sequence operator with a "whatever" endpoint. In fact, the second term [1,-1] could have been generated from the first term, but we put it in there for documentation so the reader can see what direction things are going. The polyprime function pretty much reads like the original description. Is it "so" that the p'th expansion's coefficients are all divisible by p? The .[1 ..^ */2] slice is done simply to weed out divisions by 1 or by factors we've already tested (since the coefficients are symmetrical in terms of divisibility). If we wanted to write polyprime even more idiomatically, we could have made it another infinite constant list that is just a mapping of the first list, but we decided that would just be showing off. :-) constant expansions = [1], [1,-1], -> @prior { [|@prior,0 Z- 0,|@prior] } ... *; sub polyprime(p where 2..*) { so expansions[$p].[1 ..^ */2].all %%$p }

# Showing the expansions:

say ' p: (x-1)ᵖ';
say '-----------';

sub super ($n) {$n.trans: '0123456789'
=> '⁰¹²³⁴⁵⁶⁷⁸⁹';
}

for ^13 -> $d { say$d.fmt('%2i: '), (
expansions[$d].kv.map: ->$i, $n { my$p = $d -$i;
[~] gather {
take < + - >[$n < 0] ~ ' ' unless$p == $d; take$n.abs                unless $p ==$d > 0;
take 'x'                   if $p > 0; take super$p - $i if$p > 1;
}
}
)
}

#  And testing the function:

print "\nPrimes up to 100:\n  { grep &polyprime, 2..100 }\n";

Output:
 p: (x-1)ᵖ
-----------
0: 1
1: x - 1
2: x² - 2x + 1
3: x³ - 3x² + 3x - 1
4: x⁴ - 4x³ + 6x² - 4x + 1
5: x⁵ - 5x⁴ + 10x³ - 10x² + 5x - 1
6: x⁶ - 6x⁵ + 15x⁴ - 20x³ + 15x² - 6x + 1
7: x⁷ - 7x⁶ + 21x⁵ - 35x⁴ + 35x³ - 21x² + 7x - 1
8: x⁸ - 8x⁷ + 28x⁶ - 56x⁵ + 70x⁴ - 56x³ + 28x² - 8x + 1
9: x⁹ - 9x⁸ + 36x⁷ - 84x⁶ + 126x⁵ - 126x⁴ + 84x³ - 36x² + 9x - 1
10: x¹⁰ - 10x⁹ + 45x⁸ - 120x⁷ + 210x⁶ - 252x⁵ + 210x⁴ - 120x³ + 45x² - 10x + 1
11: x¹¹ - 11x¹⁰ + 55x⁹ - 165x⁸ + 330x⁷ - 462x⁶ + 462x⁵ - 330x⁴ + 165x³ - 55x² + 11x - 1
12: x¹² - 12x¹¹ + 66x¹⁰ - 220x⁹ + 495x⁸ - 792x⁷ + 924x⁶ - 792x⁵ + 495x⁴ - 220x³ + 66x² - 12x + 1

Primes up to 100:
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

REXX

version 1

/* REXX ---------------------------------------------------------------
* 09.02.2014 Walter Pachl
* 22.02.2014 WP fix 'accounting' problem (courtesy GS)
*--------------------------------------------------------------------*/
c.=1
Numeric Digits 100
limit=200
pl=''
mmm=0
Do p=3 To limit
pm1=p-1
c.p.1=1
c.p.p=1
Do j=2 To p-1
jm1=j-1
c.p.j=c.pm1.jm1+c.pm1.j
mmm=max(mmm,c.p.j)
End
End
Say '(x-1)**0 = 1'
do i=2 To limit
im1=i-1
sign='+'
ol='(x-1)^'im1 '='
Do j=i to 2 by -1
If j=2 Then
term='x  '
Else
term='x^'||(j-1)
If j=i Then
ol=ol term
Else
ol=ol sign c.i.j'*'term
sign=translate(sign,'+-','-+')
End
If i<10 then
Say ol sign 1
Do j=2 To i-1
If c.i.j//(i-1)>0 Then
Leave
End
If j>i-1 Then
pl=pl (i-1)
End
Say ' '
Say 'Primes:' subword(pl,2,27)
Say '       ' subword(pl,29)
Say 'Largest coefficient:' mmm
Say 'This has' length(mmm) 'digits'

Output:
(x-1)**0 = 1
(x-1)^1 = x   - 1
(x-1)^2 = x^2 - 2*x   + 1
(x-1)^3 = x^3 - 3*x^2 + 3*x   - 1
(x-1)^4 = x^4 - 4*x^3 + 6*x^2 - 4*x   + 1
(x-1)^5 = x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x   - 1
(x-1)^6 = x^6 - 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x   + 1
(x-1)^7 = x^7 - 7*x^6 + 21*x^5 - 35*x^4 + 35*x^3 - 21*x^2 + 7*x   - 1
(x-1)^8 = x^8 - 8*x^7 + 28*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28*x^2 - 8*x   + 1

Primes: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103
107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199
Largest coefficient: 45274257328051640582702088538742081937252294837706668420660
This has 59 digits    

version 2

This REXX version is an optimized version (of version 1)   and modified to address all of the task requirements.

The program determines programmatically the required number of decimal digits (precision) for the large coefficients.

/*REXX program calculates  primes  via the  Agrawal─Kayal─Saxena  (AKS)  primality test.*/
parse arg Z .                                    /*obtain optional argument from the CL.*/
if Z=='' | Z=="," then Z= 200                    /*Not specified?  Then use the default.*/
OZ=Z;               tell= Z<0;       Z= abs(Z)   /*Is Z negative?  Then show expression.*/
numeric digits  max(9,  Z % 3)                   /*define a dynamic # of decimal digits.*/
call AKS                                         /*invoke the AKS funtion for coef. bld.*/
if left(OZ,1)=='+' then do; say Z isAksp(); exit /*display if  Z  is  or isn't  a prime.*/
end                      /* [↑]  call isAKSp if Z has leading +.*/
say;    say "primes found:"   #                  /*display the  prime number  list.     */
say;    if \datatype(#, 'W')  then exit          /* [↓]  the digit length of a big coef.*/
say 'Found '   words(#)   " primes and the largest coefficient has "   length(@.pm.h)  @dd
exit                                             /*stick a fork in it,  we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
AKS: $.0= '-';$.1= "+";           @. = 1   /*$.x: sign char; default coefficients.*/ q.= 1; q.1= 0; q.4= 0 /*sparse array for faster comparisons. */ #=; L= length(Z) /*define list of prime numbers (so far)*/ do p=3 for Z; pm=p - 1; pp=p + 1 /*PM & PP: used as a coding convenience*/ do m=2 for pp % 2 - 1; mm=m - 1 /*calculate coefficients for a power. */ @.p.m= @.pm.mm + @.pm.m; h=pp - m /*calculate left side of coefficients*/ @.p.h= @.p.m /* " right " " " */ end /*m*/ /* [↑] The M DO loop creates both */ end /*p*/ /* sides in the same loop. */ if tell then say '(x-1)^'right(0, L)": 1" /*possibly display the first expression*/ @dd= 'decimal digits.' /* [↓] test for primality by division.*/ do n=2 for Z; nh=n % 2; d= n - 1 /*create expressions; find the primes.*/ do k=3 to nh while @.n.k//d == 0 /*are coefficients divisible by N-1 ? */ end /*k*/ /* [↑] skip the 1st & 2nd coefficients*/ if k>nh then if q.d then #= # d /*add a number to the prime list. */ if \tell then iterate /*Don't tell? Don't show expressions.*/ y= '(x-1)^'right(d, L)":" /*define the 1st part of the expression*/ s=1 /*S: is the sign indicator (-1│+1).*/ do j=n for n-1 by -1 /*create the higher powers first. */ if j==2 then xp= 'x' /*if power=1, then don't show the power*/ else xp= 'x^' || j-1 /* ··· else show power with ^ */ if j==n then y=y xp /*no sign (+│-) for the 1st expression.*/ else y=y$.s || @.n.j'∙'xp /*build the expression with sign (+|-).*/
s= \s                                /*flip the sign for the next expression*/
end   /*j*/                          /* [↑]  the sign (now) is either 0 │ 1,*/
say  y  $.s'1' /*just show the first N expressions, */ end /*n*/ /* [↑] ··· but only for negative Z. */ if #=='' then #= "none"; return # /*if null, return "none"; else return #*/ /*──────────────────────────────────────────────────────────────────────────────────────*/ isAKSp: if z==word(#,words(#)) then return ' is a prime.'; else return " isn't a prime."  output for task requirement #2, showing thirty-one expressions using as input: -31 (Shown at five-sixth size.) (x-1)^ 0: 1 (x-1)^ 1: x -1 (x-1)^ 2: x^2 -2∙x +1 (x-1)^ 3: x^3 -3∙x^2 +3∙x -1 (x-1)^ 4: x^4 -4∙x^3 +6∙x^2 -4∙x +1 (x-1)^ 5: x^5 -5∙x^4 +10∙x^3 -10∙x^2 +5∙x -1 (x-1)^ 6: x^6 -6∙x^5 +15∙x^4 -20∙x^3 +15∙x^2 -6∙x +1 (x-1)^ 7: x^7 -7∙x^6 +21∙x^5 -35∙x^4 +35∙x^3 -21∙x^2 +7∙x -1 (x-1)^ 8: x^8 -8∙x^7 +28∙x^6 -56∙x^5 +70∙x^4 -56∙x^3 +28∙x^2 -8∙x +1 (x-1)^ 9: x^9 -9∙x^8 +36∙x^7 -84∙x^6 +126∙x^5 -126∙x^4 +84∙x^3 -36∙x^2 +9∙x -1 (x-1)^10: x^10 -10∙x^9 +45∙x^8 -120∙x^7 +210∙x^6 -252∙x^5 +210∙x^4 -120∙x^3 +45∙x^2 -10∙x +1 (x-1)^11: x^11 -11∙x^10 +55∙x^9 -165∙x^8 +330∙x^7 -462∙x^6 +462∙x^5 -330∙x^4 +165∙x^3 -55∙x^2 +11∙x -1 (x-1)^12: x^12 -12∙x^11 +66∙x^10 -220∙x^9 +495∙x^8 -792∙x^7 +924∙x^6 -792∙x^5 +495∙x^4 -220∙x^3 +66∙x^2 -12∙x +1 (x-1)^13: x^13 -13∙x^12 +78∙x^11 -286∙x^10 +715∙x^9 -1287∙x^8 +1716∙x^7 -1716∙x^6 +1287∙x^5 -715∙x^4 +286∙x^3 -78∙x^2 +13∙x -1 (x-1)^14: x^14 -14∙x^13 +91∙x^12 -364∙x^11 +1001∙x^10 -2002∙x^9 +3003∙x^8 -3432∙x^7 +3003∙x^6 -2002∙x^5 +1001∙x^4 -364∙x^3 +91∙x^2 -14∙x +1 (x-1)^15: x^15 -15∙x^14 +105∙x^13 -455∙x^12 +1365∙x^11 -3003∙x^10 +5005∙x^9 -6435∙x^8 +6435∙x^7 -5005∙x^6 +3003∙x^5 -1365∙x^4 +455∙x^3 -105∙x^2 +15∙x -1 (x-1)^16: x^16 -16∙x^15 +120∙x^14 -560∙x^13 +1820∙x^12 -4368∙x^11 +8008∙x^10 -11440∙x^9 +12870∙x^8 -11440∙x^7 +8008∙x^6 -4368∙x^5 +1820∙x^4 -560∙x^3 +120∙x^2 -16∙x +1 (x-1)^17: x^17 -17∙x^16 +136∙x^15 -680∙x^14 +2380∙x^13 -6188∙x^12 +12376∙x^11 -19448∙x^10 +24310∙x^9 -24310∙x^8 +19448∙x^7 -12376∙x^6 +6188∙x^5 -2380∙x^4 +680∙x^3 -136∙x^2 +17∙x -1 (x-1)^18: x^18 -18∙x^17 +153∙x^16 -816∙x^15 +3060∙x^14 -8568∙x^13 +18564∙x^12 -31824∙x^11 +43758∙x^10 -48620∙x^9 +43758∙x^8 -31824∙x^7 +18564∙x^6 -8568∙x^5 +3060∙x^4 -816∙x^3 +153∙x^2 -18∙x +1 (x-1)^19: x^19 -19∙x^18 +171∙x^17 -969∙x^16 +3876∙x^15 -11628∙x^14 +27132∙x^13 -50388∙x^12 +75582∙x^11 -92378∙x^10 +92378∙x^9 -75582∙x^8 +50388∙x^7 -27132∙x^6 +11628∙x^5 -3876∙x^4 +969∙x^3 -171∙x^2 +19∙x -1 (x-1)^20: x^20 -20∙x^19 +190∙x^18 -1140∙x^17 +4845∙x^16 -15504∙x^15 +38760∙x^14 -77520∙x^13 +125970∙x^12 -167960∙x^11 +184756∙x^10 -167960∙x^9 +125970∙x^8 -77520∙x^7 +38760∙x^6 -15504∙x^5 +4845∙x^4 -1140∙x^3 +190∙x^2 -20∙x +1 (x-1)^21: x^21 -21∙x^20 +210∙x^19 -1330∙x^18 +5985∙x^17 -20349∙x^16 +54264∙x^15 -116280∙x^14 +203490∙x^13 -293930∙x^12 +352716∙x^11 -352716∙x^10 +293930∙x^9 -203490∙x^8 +116280∙x^7 -54264∙x^6 +20349∙x^5 -5985∙x^4 +1330∙x^3 -210∙x^2 +21∙x -1 (x-1)^22: x^22 -22∙x^21 +231∙x^20 -1540∙x^19 +7315∙x^18 -26334∙x^17 +74613∙x^16 -170544∙x^15 +319770∙x^14 -497420∙x^13 +646646∙x^12 -705432∙x^11 +646646∙x^10 -497420∙x^9 +319770∙x^8 -170544∙x^7 +74613∙x^6 -26334∙x^5 +7315∙x^4 -1540∙x^3 +231∙x^2 -22∙x +1 (x-1)^23: x^23 -23∙x^22 +253∙x^21 -1771∙x^20 +8855∙x^19 -33649∙x^18 +100947∙x^17 -245157∙x^16 +490314∙x^15 -817190∙x^14 +1144066∙x^13 -1352078∙x^12 +1352078∙x^11 -1144066∙x^10 +817190∙x^9 -490314∙x^8 +245157∙x^7 -100947∙x^6 +33649∙x^5 -8855∙x^4 +1771∙x^3 -253∙x^2 +23∙x -1 (x-1)^24: x^24 -24∙x^23 +276∙x^22 -2024∙x^21 +10626∙x^20 -42504∙x^19 +134596∙x^18 -346104∙x^17 +735471∙x^16 -1307504∙x^15 +1961256∙x^14 -2496144∙x^13 +2704156∙x^12 -2496144∙x^11 +1961256∙x^10 -1307504∙x^9 +735471∙x^8 -346104∙x^7 +134596∙x^6 -42504∙x^5 +10626∙x^4 -2024∙x^3 +276∙x^2 -24∙x +1 (x-1)^25: x^25 -25∙x^24 +300∙x^23 -2300∙x^22 +12650∙x^21 -53130∙x^20 +177100∙x^19 -480700∙x^18 +1081575∙x^17 -2042975∙x^16 +3268760∙x^15 -4457400∙x^14 +5200300∙x^13 -5200300∙x^12 +4457400∙x^11 -3268760∙x^10 +2042975∙x^9 -1081575∙x^8 +480700∙x^7 -177100∙x^6 +53130∙x^5 -12650∙x^4 +2300∙x^3 -300∙x^2 +25∙x -1 (x-1)^26: x^26 -26∙x^25 +325∙x^24 -2600∙x^23 +14950∙x^22 -65780∙x^21 +230230∙x^20 -657800∙x^19 +1562275∙x^18 -3124550∙x^17 +5311735∙x^16 -7726160∙x^15 +9657700∙x^14 -10400600∙x^13 +9657700∙x^12 -7726160∙x^11 +5311735∙x^10 -3124550∙x^9 +1562275∙x^8 -657800∙x^7 +230230∙x^6 -65780∙x^5 +14950∙x^4 -2600∙x^3 +325∙x^2 -26∙x +1 (x-1)^27: x^27 -27∙x^26 +351∙x^25 -2925∙x^24 +17550∙x^23 -80730∙x^22 +296010∙x^21 -888030∙x^20 +2220075∙x^19 -4686825∙x^18 +8436285∙x^17 -13037895∙x^16 +17383860∙x^15 -20058300∙x^14 +20058300∙x^13 -17383860∙x^12 +13037895∙x^11 -8436285∙x^10 +4686825∙x^9 -2220075∙x^8 +888030∙x^7 -296010∙x^6 +80730∙x^5 -17550∙x^4 +2925∙x^3 -351∙x^2 +27∙x -1 (x-1)^28: x^28 -28∙x^27 +378∙x^26 -3276∙x^25 +20475∙x^24 -98280∙x^23 +376740∙x^22 -1184040∙x^21 +3108105∙x^20 -6906900∙x^19 +13123110∙x^18 -21474180∙x^17 +30421755∙x^16 -37442160∙x^15 +40116600∙x^14 -37442160∙x^13 +30421755∙x^12 -21474180∙x^11 +13123110∙x^10 -6906900∙x^9 +3108105∙x^8 -1184040∙x^7 +376740∙x^6 -98280∙x^5 +20475∙x^4 -3276∙x^3 +378∙x^2 -28∙x +1 (x-1)^29: x^29 -29∙x^28 +406∙x^27 -3654∙x^26 +23751∙x^25 -118755∙x^24 +475020∙x^23 -1560780∙x^22 +4292145∙x^21 -10015005∙x^20 +20030010∙x^19 -34597290∙x^18 +51895935∙x^17 -67863915∙x^16 +77558760∙x^15 -77558760∙x^14 +67863915∙x^13 -51895935∙x^12 +34597290∙x^11 -20030010∙x^10 +10015005∙x^9 -4292145∙x^8 +1560780∙x^7 -475020∙x^6 +118755∙x^5 -23751∙x^4 +3654∙x^3 -406∙x^2 +29∙x -1 (x-1)^30: x^30 -30∙x^29 +435∙x^28 -4060∙x^27 +27405∙x^26 -142506∙x^25 +593775∙x^24 -2035800∙x^23 +5852925∙x^22 -14307150∙x^21 +30045015∙x^20 -54627300∙x^19 +86493225∙x^18 -119759850∙x^17 +145422675∙x^16 -155117520∙x^15 +145422675∙x^14 -119759850∙x^13 +86493225∙x^12 -54627300∙x^11 +30045015∙x^10 -14307150∙x^9 +5852925∙x^8 -2035800∙x^7 +593775∙x^6 -142506∙x^5 +27405∙x^4 -4060∙x^3 +435∙x^2 -30∙x +1 (x-1)^31: x^31 -31∙x^30 +465∙x^29 -4495∙x^28 +31465∙x^27 -169911∙x^26 +736281∙x^25 -2629575∙x^24 +7888725∙x^23 -20160075∙x^22 +44352165∙x^21 -84672315∙x^20 +141120525∙x^19 -206253075∙x^18 +265182525∙x^17 -300540195∙x^16 +300540195∙x^15 -265182525∙x^14 +206253075∙x^13 -141120525∙x^12 +84672315∙x^11 -44352165∙x^10 +20160075∙x^9 -7888725∙x^8 +2629575∙x^7 -736281∙x^6 +169911∙x^5 -31465∙x^4 +4495∙x^3 -465∙x^2 +31∙x -1 primes: 2 3 5 7 11 13 17 19 23 29 31 Found 11 primes and the largest coefficient has 9 decimal digits.  output for task requirement #3, showing if 2221 is prime (or not) using for input: +2221 (Output note: this number is really pushing at the limits of REXX's use of virtual memory to store the coefficients; the version of Regina REXX used herein has a limit of around 2 Gbytes.) 2221 is prime.  output for task requirement #4, showing all primes under 35 using the input: 35 primes: 2 3 5 7 11 13 17 19 23 29 31 Found 11 primes and the largest coefficient has 10 decimal digits.  output for requirement #5 (stretch goal), showing all primes under 50 using the input: 50 primes: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 Found 15 primes and the largest coefficient has 15 decimal digits.  output when using the input: 500 (Shown at five-sixth size.) primes: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 Found 95 primes and the largest coefficient has 150 decimal digits.  RPL RPL code Comment ≪ → p ≪ { } 1 0 p FOR j p j COMB OVER * ROT SWAP + SWAP NEG NEXT DROP ≫ ≫ ‘XPM1’ STO ≪ DUP SIZE → coeffs size ≪ 0 1 size FOR j coeffs j GET 'X' size j - ^ * + NEXT ≫ ≫ ‘SHOWA’ STO ≪ DUP XPM1 → p coeffs ≪ 1 2 p FOR j coeffs j GET ABS p MOD NOT AND NEXT ≫ ≫ ‘PRIM?’ STO  XPM1 ( p -- { (x-1)^p_coefficients } ) coeffs = { } ; sign = 1 loop for j=0 to p append C(p,j)*sign to coeffs sign = -sign end loop return coeffs SHOWA ( { coeffs } -- 'polynom' ) loop for each coeff append jth polynomial term return polynom PRIM? ( p -- boolean ) res = true loop for j=2 to p res = res AND (mod(coeffs[j],p)=0) return res  Input:  ≪ { } 1 7 FOR n n XPM1 SHOWA + NEXT ≫ ≪ { } 2 35 FOR n IF n PRIM? THEN n + END NEXT ≫  Output: 8: '-1+X' 7: '1-2*X+X^2' 6: '-1+3*X-3*X^2+X^3' 5: '1-4*X-4*X^3+6*X^2+X^4' 4: '-1+5*X-5*X^4-10*X^2+10*X^3+X^5' 3: '1-6*X-6*X^5+15*X^2-20*X^3+15*X^4+X^6' 2: '-1+7*X-7*X^6-21*X^2+35*X^3-35*X^4+21*X^5+X^7' 1: { 2 3 5 7 11 13 17 19 23 29 31 }  Ruby Using the polynomial Rubygem, this can be written directly from the definition in the description: require 'polynomial' def x_minus_1_to_the(p) return Polynomial.new(-1,1)**p end def prime?(p) return false if p < 2 (x_minus_1_to_the(p) - Polynomial.from_string("x**#{p}-1")).coefs.all?{|n| n%p==0} end 8.times do |n| # the default Polynomial#to_s would be OK here; the substitutions just make the # output match the other version below. puts "(x-1)^#{n} = #{x_minus_1_to_the(n).to_s.gsub(/\*\*/,'^').gsub(/\*/,'')}" end puts "\nPrimes below 50:", 50.times.select {|n| prime? n}.join(',')  Or without the dependency: def x_minus_1_to_the(p) p.times.inject([1]) do |ex, _| ([0] + ex).zip(ex + [0]).map { |x,y| x - y } end end def prime?(p) return false if p < 2 coeff = x_minus_1_to_the(p)[1..p/2] # only need half of coeff terms coeff.all?{ |n| n%p == 0 } end 8.times do |n| puts "(x-1)^#{n} = " + x_minus_1_to_the(n).map.with_index { |c, p| p.zero? ? c.to_s : (c < 0 ? " - " : " + ") + (c.abs == 1 ? "x" : "#{c.abs}x") + (p == 1 ? "" : "^#{p}") }.join end puts "\nPrimes below 50:", 50.times.select {|n| prime? n}.join(',')  Output: (x-1)^0 = 1 (x-1)^1 = -1 + x (x-1)^2 = 1 - 2x + x^2 (x-1)^3 = -1 + 3x - 3x^2 + x^3 (x-1)^4 = 1 - 4x + 6x^2 - 4x^3 + x^4 (x-1)^5 = -1 + 5x - 10x^2 + 10x^3 - 5x^4 + x^5 (x-1)^6 = 1 - 6x + 15x^2 - 20x^3 + 15x^4 - 6x^5 + x^6 (x-1)^7 = -1 + 7x - 21x^2 + 35x^3 - 35x^4 + 21x^5 - 7x^6 + x^7 Primes below 50: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47 Rust fn aks_coefficients(k: usize) -> Vec<i64> { let mut coefficients = vec![0i64; k + 1]; coefficients[0] = 1; for i in 1..(k + 1) { coefficients[i] = -(1..i).fold(coefficients[0], |prev, j|{ let old = coefficients[j]; coefficients[j] = old - prev; old }); } coefficients } fn is_prime(p: usize) -> bool { if p < 2 { false } else { let c = aks_coefficients(p); (1..p / 2 + 1).all(|i| c[i] % p as i64 == 0) } } fn main() { for i in 0..8 { println!("{}: {:?}", i, aks_coefficients(i)); } for i in (1..=50).filter(|&i| is_prime(i)) { print!("{} ", i); } }  Output: 0: [1] 1: [1, -1] 2: [1, -2, 1] 3: [1, -3, 3, -1] 4: [1, -4, 6, -4, 1] 5: [1, -5, 10, -10, 5, -1] 6: [1, -6, 15, -20, 15, -6, 1] 7: [1, -7, 21, -35, 35, -21, 7, -1] 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47  An alternative version which computes the coefficients in a more functional but less efficient way. fn aks_coefficients(k: usize) -> Vec<i64> { if k == 0 { vec![1i64] } else { let zero = Some(0i64); range(1, k).fold(vec![1i64, -1], |r, _| { let a = r.iter().chain(zero.iter()); let b = zero.iter().chain(r.iter()); a.zip(b).map(|(x, &y)| x-y).collect() }) } }  Scala def powerMin1(n: BigInt) = if (n % 2 == 0) BigInt(1) else BigInt(-1) val pascal = (( Vector(Vector(BigInt(1))) /: (1 to 50)) { (rows, i) => val v = rows.head val newVector = ((1 until v.length) map (j => powerMin1(j+i) * (v(j-1).abs + v(j).abs)) ).toVector (powerMin1(i) +: newVector :+ powerMin1(i+v.length)) +: rows }).reverse def poly2String(poly: Vector[BigInt]) = ((0 until poly.length) map { i => (i, poly(i)) match { case (0, c) => c.toString case (_, c) => (if (c >= 0) "+" else "-") + (if (c == 1) "x" else c.abs + "x") + (if (i == 1) "" else "^" + i) } }) mkString "" def isPrime(n: Int) = { val poly = pascal(n) poly.slice(1, poly.length - 1).forall(i => i % n == 0) } for(i <- 0 to 7) { println( f"(x-1)^$i = ${poly2String( pascal(i) )}" ) } val primes = (2 to 50).filter(isPrime) println println(primes mkString " ")  Output: (x-1)^0 = 1 (x-1)^1 = -1+x (x-1)^2 = 1-2x+x^2 (x-1)^3 = -1+3x-3x^2+x^3 (x-1)^4 = 1-4x+6x^2-4x^3+x^4 (x-1)^5 = -1+5x-10x^2+10x^3-5x^4+x^5 (x-1)^6 = 1-6x+15x^2-20x^3+15x^4-6x^5+x^6 (x-1)^7 = -1+7x-21x^2+35x^3-35x^4+21x^5-7x^6+x^7 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 Scheme ;; implement mod m arithmetic with polnomials in x ;; as lists of coefficients, x^0 first. ;; ;; so x^3 + 5 is represented as (5 0 0 1) (define (+/m m a b) ;; add two polynomials (cond ((null? a) b) ((null? b) a) (else (cons (modulo (+ (car a) (car b)) m) (+/m m (cdr a) (cdr b)))))) (define (*c/m m c a) ;; multiplication by a constant (map (lambda (v) (modulo (* c v) m)) a)) (define (*/m m a b) ;; multiply two polynomials (let loop ((a a)) (if (null? a) '() (+/m m (*c/m m (car a) b) (cons 0 (*/m m (cdr a) b)))))) (define (x^n/m m n) (if (= n 0) '(1) (cons 0 (x^n/m m (- n 1))))) (define (^n/m m a n) ;; calculate the n'th power of polynomial a (cond ((= n 0) '(1)) ((= n 1) a) (else (*/m m a (^n/m m a (- n 1)))))) ;; test case ;; ;; ? lift(Mod((x^3 + 5)*(4 + 3*x + x^2),6)) ;; %13 = x^5 + 3*x^4 + 4*x^3 + 5*x^2 + 3*x + 2 ;; ;; > (*/m 6 '(5 0 0 1) '(4 3 1)) ;; '(2 3 5 4 3 1) ;; ;; working correctly (define (rosetta-aks-test p) (if (or (= p 0) (= p 1)) #f ;; u = (x - 1)^p ;; v = (x^p - 1) (let ((u (^n/m p (list -1 1) p)) (v (+/m p (x^n/m p p) (list -1)))) (every zero? (+/m p u (*c/m p -1 v)))))) ;; > (filter rosetta-aks-test (iota 50)) ;; '(2 3 5 7 11 13 17 19 23 29 31 37 41 43 47)  Scilab clear xdel(winsid()) stacksize('max') sz=stacksize(); n=7; //For the expansion up to power of n g=50; //For test of primes up to g function X = pascal(g) //Pascal´s triangle X(1,1)=1; //Zeroth power X(2,1)=1; //First power X(2,2)=1; for q=3:1:g+1 //From second power use this loop X(q,1)=1; X(q,q)=1; for p=2:1:q-1 X(q,p)=X(q-1,p-1)+X(q-1,p); end end endfunction Z=pascal(g); //Generate Pascal's triangle up to g Q(0+1)="(x-1)^0 = 1"; //For nicer display Q(1+1)="(x-1)^1 = x^1-1"; //For nicer display disp(Q(1)) disp(Q(2)) function cf=coef(Z,q,p) //Return coeffiecents for nicer display of expansion without "ones" if Z(q,p)==1 then cf=""; else cf=string(Z(q,p)); end endfunction for q=3:n+1 //Generate and display the expansions Q(q)=strcat(["(x-1)^",string(q-1)," = "]); sing=""; //Sign of coeff. for p=1:q-1 //Number of coefficients equals power minus 1 Q(q)=strcat([Q(q),sing,coef(Z,q,p),"x^",string(q-p)]); if sing=="-" then sing="+"; else sing="-"; end end Q(q)=strcat([Q(q),sing,string(1)]); disp(Q(q)) clear Q end function prime=prime(Z,g) prime="true"; for p=2:g if abs(floor(Z(g+1,p)/g)-Z(g+1,p)/g)>0 then prime="false"; break; end end endfunction R="2"; //For nicer display for r=3:g if prime(Z,r)=="true" then R=strcat([R, ", ",string(r)]); end end disp(R)  Output: (x-1)^0 = 1 (x-1)^1 = x^1-1 (x-1)^2 = x^2-2x^1+1 (x-1)^3 = x^3-3x^2+3x^1-1 (x-1)^4 = x^4-4x^3+6x^2-4x^1+1 (x-1)^5 = x^5-5x^4+10x^3-10x^2+5x^1-1 (x-1)^6 = x^6-6x^5+15x^4-20x^3+15x^2-6x^1+1 (x-1)^7 = x^7-7x^6+21x^5-35x^4+35x^3-21x^2+7x^1-1 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47  Seed7 $ include "seed7_05.s7i";

const func array integer: expand_x_1 (in integer: p) is func
result
var array integer: ex is [] (1);
local
var integer: i is 0;
begin
for i range 0 to p - 1 do
ex := [] (ex[1] * -(p - i) div (i + 1)) & ex;
end for;
end func;

const func boolean: aks_test (in integer: p) is func
result
var boolean: aks_test is FALSE;
local
var array integer: ex is 0 times 0;
var integer: idx is 0;
begin
if p >= 2 then
ex := expand_x_1(p);
ex[1] +:= 1;
for key idx range ex until ex[idx] rem p <> 0 do
noop;
end for;
aks_test := idx = length(ex);
end if;
end func;

const proc: main is func
local
var integer: p is 0;
var integer: n is 0;
var integer: e is 0;
begin
writeln("# p: (x-1)^p for small p");
for p range 0 to 11 do
write(p lpad 3 <& ": ");
for n key e range expand_x_1(p) do
write(" ");
if n >= 0 then
write("+");
end if;
write(n);
if e > 1 then
write("x^" <& pred(e));
end if;
end for;
writeln;
end for;
writeln;
writeln("# small primes using the aks test");
for p range 0 to 61 do
if aks_test(p) then
write(p <& " ");
end if;
end for;
writeln;
end func;
Output:
# p: (x-1)^p for small p
0:  +1
1:  -1 +1x^1
2:  +1 -2x^1 +1x^2
3:  -1 +3x^1 -3x^2 +1x^3
4:  +1 -4x^1 +6x^2 -4x^3 +1x^4
5:  -1 +5x^1 -10x^2 +10x^3 -5x^4 +1x^5
6:  +1 -6x^1 +15x^2 -20x^3 +15x^4 -6x^5 +1x^6
7:  -1 +7x^1 -21x^2 +35x^3 -35x^4 +21x^5 -7x^6 +1x^7
8:  +1 -8x^1 +28x^2 -56x^3 +70x^4 -56x^5 +28x^6 -8x^7 +1x^8
9:  -1 +9x^1 -36x^2 +84x^3 -126x^4 +126x^5 -84x^6 +36x^7 -9x^8 +1x^9
10:  +1 -10x^1 +45x^2 -120x^3 +210x^4 -252x^5 +210x^6 -120x^7 +45x^8 -10x^9 +1x^10
11:  -1 +11x^1 -55x^2 +165x^3 -330x^4 +462x^5 -462x^6 +330x^7 -165x^8 +55x^9 -11x^10 +1x^11

# small primes using the aks test
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61


Sidef

Translation of: Perl
func binprime(p) {
p >= 2 || return false
for i in (1 .. p>>1) {
(binomial(p, i) % p) && return false
}
return true
}

func coef(n, e) {
(e == 0) && return "#{n}"
(n == 1) && (n = "")
(e == 1) ? "#{n}x" : "#{n}x^#{e}"
}

func binpoly(p) {
join(" ", coef(1, p), ^p -> map {|i|
join(" ", %w(+ -)[(p-i)&1], coef(binomial(p, i), i))
}.reverse...)
}

say "expansions of (x-1)^p:"
for i in ^10 { say binpoly(i) }
say "Primes to 80: [#{2..80 -> grep { binprime(_) }.join(' ')}]"

Output:
expansions of (x-1)^p:
1
x - 1
x^2 - 2x + 1
x^3 - 3x^2 + 3x - 1
x^4 - 4x^3 + 6x^2 - 4x + 1
x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1
x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1
x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1
x^8 - 8x^7 + 28x^6 - 56x^5 + 70x^4 - 56x^3 + 28x^2 - 8x + 1
x^9 - 9x^8 + 36x^7 - 84x^6 + 126x^5 - 126x^4 + 84x^3 - 36x^2 + 9x - 1
Primes to 80: [2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79]


Stata

Using the moremata library to print the polynomial coefficients. They are in decreasing degree order. To install moremata, type ssc install moremata in Stata. Since Stata is using double precision floating-point instead of 32 bit integers, the polynomials are exact up to p=54.

mata
function pol(n) {
a=J(1,n+1,1)
r=1
s=1
for (k=0; k<n; k++) {
s=-s
r=(r*(n-k))/(k+1)
a[k+2]=r*s
}
return(a)
}

for (n=0; n<=7; n++) mm_matlist(pol(n))

1
+-------------+
1 |          1  |
+-------------+
1           2
+-------------------------+
1 |          1          -1  |
+-------------------------+
1           2           3
+-------------------------------------+
1 |          1          -2           1  |
+-------------------------------------+
1           2           3           4
+-------------------------------------------------+
1 |          1          -3           3          -1  |
+-------------------------------------------------+
1           2           3           4           5
+-------------------------------------------------------------+
1 |          1          -4           6          -4           1  |
+-------------------------------------------------------------+
1           2           3           4           5           6
+-------------------------------------------------------------------------+
1 |          1          -5          10         -10           5          -1  |
+-------------------------------------------------------------------------+
1           2           3           4           5           6           7
+-------------------------------------------------------------------------------------+
1 |          1          -6          15         -20          15          -6           1  |
+-------------------------------------------------------------------------------------+
1           2           3           4           5           6           7           8
+-------------------------------------------------------------------------------------------------+
1 |          1          -7          21         -35          35         -21           7          -1  |
+-------------------------------------------------------------------------------------------------+

function isprime(n) {
a=pol(n)
for (k=2; k<=n; k++) {
if (mod(a[k],n)) return(0)
}
return(1)
}

for (n=2; n<=50; n++) {
if (isprime(n)) printf("%f ",n)
}

2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
end


Swift

func polynomialCoeffs(n: Int) -> [Int] {
var result = [Int](count : n+1, repeatedValue : 0)

result[0]=1
for i in 1 ..< n/2+1 { //Progress up, until reaching the middle value
result[i] = result[i-1] * (n-i+1)/i;
}
for i in n/2+1 ..< n+1 { //Copy the inverse of the first part
result[i] = result[n-i];
}
// Take into account the sign
for i in stride(from: 1, through: n, by: 2) {
result[i] = -result[i]
}

return result
}

func isPrime(n: Int) -> Bool {

var coeffs = polynomialCoeffs(n)

coeffs[0]--
coeffs[n]++

for i in 1 ... n {
if coeffs[i]%n != 0 {
return false
}
}

return true
}

for i in 0...10 {

let coeffs = polynomialCoeffs(i)

print("(x-1)^\(i) = ")
if i == 0 {
print("1")
} else {
if i == 1 {
print("x")
} else {
print("x^\(i)")
if i == 2 {
print("\(coeffs[i-1])x")
} else {
for j in 1...(i - 2) {
if j%2 == 0 {
print("+\(coeffs[j])x^\(i-j)")
} else {
print("\(coeffs[j])x^\(i-j)")
}
}
if (i-1)%2 == 0 {
print("+\(coeffs[i-1])x")
} else {
print("\(coeffs[i-1])x")
}
}
}
if i%2 == 0 {
print("+\(coeffs[i])")
} else {
print("\(coeffs[i])")
}
}
println()
}

println()
print("Primes under 50 : ")

for i in 1...50 {
if isPrime(i) {
print("\(i) ")
}
}

Output:
(x-1)^0 = 1
(x-1)^1 = x-1
(x-1)^2 = x^2-2x+1
(x-1)^3 = x^3-3x^2+3x-1
(x-1)^4 = x^4-4x^3+6x^2-4x+1
(x-1)^5 = x^5-5x^4+10x^3-10x^2+5x-1
(x-1)^6 = x^6-6x^5+15x^4-20x^3+15x^2-6x+1
(x-1)^7 = x^7-7x^6+21x^5-35x^4+35x^3-21x^2+7x-1
(x-1)^8 = x^8-8x^7+28x^6-56x^5+70x^4-56x^3+28x^2-8x+1
(x-1)^9 = x^9-9x^8+36x^7-84x^6+126x^5-126x^4+84x^3-36x^2+9x-1
(x-1)^10 = x^10-10x^9+45x^8-120x^7+210x^6-252x^5+210x^4-120x^3+45x^2-10x+1

Primes under 50 : 1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47


Tcl

A recursive method with memorization would be more efficient, but this is sufficient for small-scale work.

proc coeffs {p {signs 1}} {
set clist 1
for {set i 0} {$i <$p} {incr i} {
set clist [lmap x [list 0 {*}$clist] y [list {*}$clist 0] {
expr {$x +$y}
}]
}
if {$signs} { set s -1 set clist [lmap c$clist {expr {[set s [expr {-$s}]] *$c}}]
}
return $clist } proc aksprime {p} { if {$p < 2} {
return false
}
foreach c [coeffs $p 0] { if {$c == 1} continue
if {$c %$p} {
return false
}
}
return true
}

for {set i 0} {$i <= 7} {incr i} { puts -nonewline "(x-1)^$i ="
set j $i foreach c [coeffs$i] {
puts -nonewline [format " %+dx^%d" $c$j]
incr j -1
}
puts ""
}

set sub35primes {}
for {set i 1} {$i < 35} {incr i} { if {[aksprime$i]} {
lappend sub35primes $i } } puts "primes under 35: [join$sub35primes ,]"

set sub50primes {}
for {set i 1} {$i < 50} {incr i} { if {[aksprime$i]} {
lappend sub50primes $i } } puts "primes under 50: [join$sub50primes ,]"

Output:
(x-1)^0 = +1x^0
(x-1)^1 = +1x^1 -1x^0
(x-1)^2 = +1x^2 -2x^1 +1x^0
(x-1)^3 = +1x^3 -3x^2 +3x^1 -1x^0
(x-1)^4 = +1x^4 -4x^3 +6x^2 -4x^1 +1x^0
(x-1)^5 = +1x^5 -5x^4 +10x^3 -10x^2 +5x^1 -1x^0
(x-1)^6 = +1x^6 -6x^5 +15x^4 -20x^3 +15x^2 -6x^1 +1x^0
(x-1)^7 = +1x^7 -7x^6 +21x^5 -35x^4 +35x^3 -21x^2 +7x^1 -1x^0
primes under 35: 2,3,5,7,11,13,17,19,23,29,31
primes under 50: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47


Transd

Translation of: Python
#lang transd

MainModule: {
poly: (λ n Long()
(with v Vector<Long>([1])
(for i in Range(n) do
(append v (/ (* (get v -1) (- (- n i))) (to-Long (+ i 1))))
)
(reverse v)
(ret v)
)
),

aks_test: (λ n Long() -> Bool()
(if (< n 2) (ret false))
(with v (poly n)
(set-el v 0 (+ (get v 0) 1))
(ret (not (any Range(in: v 0 -1) (λ (mod @it n)))))
)
),

_start: (λ (with v Vector<Long>()
(for p in Seq( 12 ) do
(set v (poly p))
(textout "(x - 1)^" p " = ")
(for i in v do (textout :sign i "x^" @idx " "))
(textout "\n")
)
(textout "\nList of primes in 2-62 interval:\n")
(for p in Seq( 2 62 ) do
(if (aks_test p) (textout p " "))
)
))
}

Output:
(x - 1)^0 = +1x^0
(x - 1)^1 = -1x^0 +1x^1
(x - 1)^2 = +1x^0 -2x^1 +1x^2
(x - 1)^3 = -1x^0 +3x^1 -3x^2 +1x^3
(x - 1)^4 = +1x^0 -4x^1 +6x^2 -4x^3 +1x^4
(x - 1)^5 = -1x^0 +5x^1 -10x^2 +10x^3 -5x^4 +1x^5
(x - 1)^6 = +1x^0 -6x^1 +15x^2 -20x^3 +15x^4 -6x^5 +1x^6
(x - 1)^7 = -1x^0 +7x^1 -21x^2 +35x^3 -35x^4 +21x^5 -7x^6 +1x^7
(x - 1)^8 = +1x^0 -8x^1 +28x^2 -56x^3 +70x^4 -56x^5 +28x^6 -8x^7 +1x^8
(x - 1)^9 = -1x^0 +9x^1 -36x^2 +84x^3 -126x^4 +126x^5 -84x^6 +36x^7 -9x^8 +1x^9
(x - 1)^10 = +1x^0 -10x^1 +45x^2 -120x^3 +210x^4 -252x^5 +210x^6 -120x^7 +45x^8 -10x^9 +1x^10
(x - 1)^11 = -1x^0 +11x^1 -55x^2 +165x^3 -330x^4 +462x^5 -462x^6 +330x^7 -165x^8 +55x^9 -11x^10 +1x^11

List of primes in 2-62 interval:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61


uBasic/4tH

For n = 0 To 9
Push n : Gosub _coef : Gosub _drop
Print "(x-1)^";n;" = ";
Push n : Gosub _show
Print
Next

Print
Print "primes (never mind the 1):";

For n = 1 To 34
Push n : Gosub _isprime
If Pop() Then Print " ";n;
Next

Print
End

' show polynomial expansions
_show                                  ' ( n --)
Do
If @(Tos()) > -1 Then Print "+";
Print @(Tos());"x^";Tos();
While (Tos())
Push Pop() - 1
Loop

Gosub _drop
Return

' test whether number is a prime
_isprime                               ' ( n --)
Gosub _coef

i = Tos()
@(0) = @(0) + 1
@(i) = @(i) - 1

Do While (i) * ((@(i) % Tos()) = 0)
i = i - 1
Loop

Gosub _drop
Push (i = 0)
Return

' generate coefficients
_coef                                  ' ( n -- n)
If (Tos() < 0) + (Tos() > 34) Then End
' gracefully deal with range issue
i = 0
@(i) = 1

Do While i < Tos()
j = i
@(j+1) = 1

Do While j > 0
@(j) = @(j-1) - @(j)
j = j - 1
Loop

@(0) = -@(0)
i = i + 1
Loop
Return

' drop a value from the stack
_drop                                  ' ( n --)
If Pop() Endif
Return

Output:
(x-1)^0 = +1x^0
(x-1)^1 = +1x^1-1x^0
(x-1)^2 = +1x^2-2x^1+1x^0
(x-1)^3 = +1x^3-3x^2+3x^1-1x^0
(x-1)^4 = +1x^4-4x^3+6x^2-4x^1+1x^0
(x-1)^5 = +1x^5-5x^4+10x^3-10x^2+5x^1-1x^0
(x-1)^6 = +1x^6-6x^5+15x^4-20x^3+15x^2-6x^1+1x^0
(x-1)^7 = +1x^7-7x^6+21x^5-35x^4+35x^3-21x^2+7x^1-1x^0
(x-1)^8 = +1x^8-8x^7+28x^6-56x^5+70x^4-56x^3+28x^2-8x^1+1x^0
(x-1)^9 = +1x^9-9x^8+36x^7-84x^6+126x^5-126x^4+84x^3-36x^2+9x^1-1x^0

primes (never mind the 1): 1 2 3 5 7 11 13 17 19 23 29 31


VBA

Translation of: Phix
'-- Does not work for primes above 97, which is actually beyond the original task anyway.
'-- Translated from the C version, just about everything is (working) out-by-1, what fun.
'-- This updated VBA version utilizes the Decimal datatype to handle numbers requiring
'-- more than 32 bits.
Const MAX = 99
Dim c(MAX + 1) As Variant
Private Sub coef(n As Integer)
'-- out-by-1, ie coef(1)==^0, coef(2)==^1, coef(3)==^2 etc.
c(n) = CDec(1) 'converts c(n) from Variant to Decimal, a 12 byte data type
For i = n - 1 To 2 Step -1
c(i) = c(i) + c(i - 1)
Next i
End Sub
Private Function is_prime(fn As Variant) As Boolean
fn = CDec(fn)
Call coef(fn + 1)   '-- (I said it was out-by-1)
For i = 2 To fn - 1 '-- (technically "to n" is more correct)
If c(i) - fn * Int(c(i) / fn) <> 0 Then 'c(i) Mod fn <> 0 Then --Mod works upto 32 bit numbers
is_prime = False: Exit Function
End If
Next i
is_prime = True
End Function
Private Sub show(n As Integer)
'-- (As per coef, this is (working) out-by-1)
Dim ci As Variant
For i = n To 1 Step -1
ci = c(i)
If ci = 1 Then
If (n - i) Mod 2 = 0 Then
If i = 1 Then
If n = 1 Then
ci = "1"
Else
ci = "+1"
End If
Else
ci = ""
End If
Else
ci = "-1"
End If
Else
If (n - i) Mod 2 = 0 Then
ci = "+" & ci
Else
ci = "-" & ci
End If
End If
If i = 1 Then '-- ie ^0
Debug.Print ci
Else
If i = 2 Then '-- ie ^1
Debug.Print ci & "x";
Else
Debug.Print ci & "x^" & i - 1;
End If
End If
Next i
End Sub
Public Sub AKS_test_for_primes()
Dim n As Integer
For n = 1 To 10 '-- (0 to 9 really)
coef n
Debug.Print "(x-1)^" & n - 1 & " = ";
show n
Next n
Debug.Print "primes (<="; MAX; "):"
coef 2 '-- (needed to reset c, if we want to avoid saying 1 is prime...)
For n = 2 To MAX
If is_prime(n) Then
Debug.Print n;
End If
Next n
End Sub

Output:

(x-1)^0 = 1
(x-1)^1 = x-1
(x-1)^2 = x^2-2x+1
(x-1)^3 = x^3-3x^2+3x-1
(x-1)^4 = x^4-4x^3+6x^2-4x+1
(x-1)^5 = x^5-5x^4+10x^3-10x^2+5x-1
(x-1)^6 = x^6-6x^5+15x^4-20x^3+15x^2-6x+1
(x-1)^7 = x^7-7x^6+21x^5-35x^4+35x^3-21x^2+7x-1
(x-1)^8 = x^8-8x^7+28x^6-56x^5+70x^4-56x^3+28x^2-8x+1
(x-1)^9 = x^9-9x^8+36x^7-84x^6+126x^5-126x^4+84x^3-36x^2+9x-1
primes (<= 99 ):

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



V (Vlang)

Translation of: go
fn bc(p int) []i64 {
mut c := []i64{len: p+1}
mut r := i64(1)
for i, half := 0, p/2; i <= half; i++ {
c[i] = r
c[p-i] = r
r = r * i64(p-i) / i64(i+1)
}
for i := p - 1; i >= 0; i -= 2 {
c[i] = -c[i]
}
return c
}

fn main() {
for p := 0; p <= 7; p++ {
println("$p:${pp(bc(p))}")
}
for p := 2; p < 50; p++ {
if aks(p) {
print("$p ") } } println('') } const e = [²,³,⁴,⁵,⁶,⁷] fn pp(c []i64) string { mut s := '' if c.len == 1 { return c[0].str() } p := c.len - 1 if c[p] != 1 { s = c[p].str() } for i := p; i > 0; i-- { s += "x" if i != 1 { s += e[i-2].str() } d := c[i-1] if d < 0 { s += " -${-d}"
} else {
s += " + $d" } } return s } fn aks(p int) bool { mut c := bc(p) c[p]-- c[0]++ for d in c { if d%i64(p) != 0 { return false } } return true } Output: 0: 1 1: x - 1 2: x² - 2x + 1 3: x³ - 3x² + 3x - 1 4: x⁴ - 4x³ + 6x² - 4x + 1 5: x⁵ - 5x⁴ + 10x³ - 10x² + 5x - 1 6: x⁶ - 6x⁵ + 15x⁴ - 20x³ + 15x² - 6x + 1 7: x⁷ - 7x⁶ + 21x⁵ - 35x⁴ + 35x³ - 21x² + 7x - 1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47  Wren Translation of: Go var bc = Fn.new { |p| var c = List.filled(p+1, 0) var r = 1 var half = (p/2).floor for (i in 0..half) { c[i] = r c[p-i] = r r = (r * (p-i) / (i+1)).floor } var j = p - 1 while (j >= 0) { c[j] = -c[j] j = j - 2 } return c } var e = "²³⁴⁵⁶⁷".codePoints.toList var pp = Fn.new { |c| if (c.count == 1) return "%(c[0])" var p = c.count - 1 var s = "" if (c[p] != 1) s = "%(c[p])" if (p == 0) return s for (i in p..1) { s = s + "x" if (i != 1) s = s + String.fromCodePoint(e[i-2]) var d = c[i-1] s = s + ((d < 0) ? " - %(-d)" : " + %(d)") } return s } var aks = Fn.new { |p| var c = bc.call(p) c[p] = c[p] - 1 c[0] = c[0] + 1 for (d in c) { if (d%p != 0) return false } return true } for (p in 0..7) System.print("%(p): %(pp.call(bc.call(p)))") System.print("\nAll primes under 50:") for (p in 2..49) if (aks.call(p)) System.write("%(p) ") System.print()  Output: 0: 1 1: x - 1 2: x² - 2x + 1 3: x³ - 3x² + 3x - 1 4: x⁴ - 4x³ + 6x² - 4x + 1 5: x⁵ - 5x⁴ + 10x³ - 10x² + 5x - 1 6: x⁶ - 6x⁵ + 15x⁴ - 20x³ + 15x² - 6x + 1 7: x⁷ - 7x⁶ + 21x⁵ - 35x⁴ + 35x³ - 21x² + 7x - 1 All primes under 50: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47  Yabasic Translation of: Phix // Does not work for primes above 53, which is actually beyond the original task anyway. // Translated from the C version, just about everything is (working) out-by-1, what fun. dim c(100) sub coef(n) local i // out-by-1, ie coef(1)==^0, coef(2)==^1, coef(3)==^2 etc. c(n) = 1 for i = n-1 to 2 step -1 c(i) = c(i) + c(i-1) next end sub sub is_prime(n) local i coef(n+1) // (I said it was out-by-1) for i = 2 to n-1 // (technically "to n" is more correct) if int(c(i)/n) <> c(i)/n then return 0 end if next return 1 end sub sub show(n) // (As per coef, this is (working) out-by-1) local ci, ci$, i

for i = n to 1 step -1
ci = c(i)
if ci = 1 then
if mod(n-i, 2) = 0 then
if i = 1 then
if n = 1 then
ci$= "1" else ci$ = "+1"
end if
else
ci$= "" end if else ci$ = "-1"
end if
else
if mod(n-i, 2) = 0 then
ci$= "+" + str$(ci)
else
ci$= "-" + str$(ci)
end if
end if
if i = 1 then // ie ^0
print ci$; elsif i=2 then // ie ^1 print ci$, "x";
else
print ci$, "x^", i-1; end if next end sub sub AKS_test_for_primes() local n for n = 1 to 10 // (0 to 9 really) coef(n) print "(x-1)^", n-1, " = "; show(n) print next print "\nprimes (<=53): "; c(2) = 1 // (this manages "", which is all that call did anyway...) for n = 2 to 53 if is_prime(n) then print " ", n; end if next print end sub AKS_test_for_primes() Zig Uses Zig's compile-time interpreter to create Pascal's triangle at compile-time. const std = @import("std"); const assert = std.debug.assert; pub fn main() !void { const stdout = std.io.getStdOut().writer(); var i: u6 = 0; while (i < 8) : (i += 1) try showBinomial(stdout, i); try stdout.print("\nThe primes upto 50 (via AKS) are: ", .{}); i = 2; while (i <= 50) : (i += 1) if (aksPrime(i)) try stdout.print("{} ", .{i}); try stdout.print("\n", .{}); } fn showBinomial(writer: anytype, n: u6) !void { const row = binomial(n).?; var sign: u8 = '+'; var exp = row.len; try writer.print("(x - 1)^{} =", .{n}); for (row) |coef| { try writer.print(" ", .{}); if (exp != row.len) try writer.print("{c} ", .{sign}); exp -= 1; if (coef != 1 or exp == 0) try writer.print("{}", .{coef}); if (exp >= 1) { try writer.print("x", .{}); if (exp > 1) try writer.print("^{}", .{exp}); } sign = if (sign == '+') '-' else '+'; } try writer.print("\n", .{}); } fn aksPrime(n: u6) bool { return for (binomial(n).?) |coef| { if (coef > 1 and coef % n != 0) break false; } else true; } pub fn binomial(n: u32) ?[]const u64 { if (n >= rmax) return null else { const k = n * (n + 1) / 2; return pascal[k .. k + n + 1]; } } const rmax = 68; // evaluated and created at compile-time const pascal = build: { @setEvalBranchQuota(100_000); var coefficients: [(rmax * (rmax + 1)) / 2]u64 = undefined; coefficients[0] = 1; var j: u32 = 0; var k: u32 = 1; var n: u32 = 1; while (n < rmax) : (n += 1) { var prev = coefficients[j .. j + n]; var next = coefficients[k .. k + n + 1]; next[0] = 1; var i: u32 = 1; while (i < n) : (i += 1) next[i] = prev[i] + prev[i - 1]; next[i] = 1; j = k; k += n + 1; } break :build coefficients; };  Output: $ zig run aks.zig
(x - 1)^0 = 1
(x - 1)^1 = x - 1
(x - 1)^2 = x^2 - 2x + 1
(x - 1)^3 = x^3 - 3x^2 + 3x - 1
(x - 1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1
(x - 1)^5 = x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1
(x - 1)^6 = x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1
(x - 1)^7 = x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1

The primes upto 50 (via AKS) are: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47


zkl

Translation of: Python
var BN=Import("zklBigNum");
fcn expand_x_1(p){
ex := L(BN(1));
foreach i in (p){ ex.append(ex[-1] * -(p-i) / (i+1)) }
return(ex.reverse())
}
fcn aks_test(p){
if (p < 2) return(False);
ex := expand_x_1(p);
ex[0] = ex[0] + 1;
return(not ex[0,-1].filter('%.fp1(p)));
}
println("# p: (x-1)^p for small p");
foreach p in (12){
println("%3d: ".fmt(p),expand_x_1(p).enumerate()
.pump(String,fcn([(n,e)]){"%+d%s ".fmt(e,n and "x^%d".fmt(n) or "")}));
}

println("\n# small primes using the aks test");
println([0..110].filter(aks_test).toString(*));
Output:
# p: (x-1)^p for small p
0: +1
1: -1 +1x^1
2: +1 -2x^1 +1x^2
3: -1 +3x^1 -3x^2 +1x^3
4: +1 -4x^1 +6x^2 -4x^3 +1x^4
5: -1 +5x^1 -10x^2 +10x^3 -5x^4 +1x^5
6: +1 -6x^1 +15x^2 -20x^3 +15x^4 -6x^5 +1x^6
7: -1 +7x^1 -21x^2 +35x^3 -35x^4 +21x^5 -7x^6 +1x^7
8: +1 -8x^1 +28x^2 -56x^3 +70x^4 -56x^5 +28x^6 -8x^7 +1x^8
9: -1 +9x^1 -36x^2 +84x^3 -126x^4 +126x^5 -84x^6 +36x^7 -9x^8 +1x^9
10: +1 -10x^1 +45x^2 -120x^3 +210x^4 -252x^5 +210x^6 -120x^7 +45x^8 -10x^9 +1x^10
11: -1 +11x^1 -55x^2 +165x^3 -330x^4 +462x^5 -462x^6 +330x^7 -165x^8 +55x^9 -11x^10 +1x^11

# small primes using the aks test
L(2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,101,103,107,109)
`