AKS test for primes

AKS test for primes is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

The AKS test for primes states that a number $p$ is prime if all the coefficients of the polynomial expansion of

$(x-1)^{p}-(x^{p}-1)$

are divisible by $p$.

For example, trying $p=3$:

$(x-1)^{3}-(x^{3}-1)=(X^{3}-3X^{2}+3X-1)-(x^{3}-1)=-3X^{2}+3X$

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

The task

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

Reference

• Fool-Proof Test for Primes - Numberphile (Video).

C

<lang c>#include <stdio.h>

1. 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; }</lang>

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

Python

<lang python>def expand_x_1(p):

   if p == 0: return [1]
   ex = [1, -1]
   for i in range(1, p):
       ex = [x - y for x,y in zip(ex+[0], [0]+ex)]
   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)])</lang>

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]