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

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 [email protected] 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 [email protected] 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


ALGOL 68

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

 BEGINCOMMENT   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 speciallyCOMMENT   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]))      ESACOD;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


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#AutoHotkeypascalstriangle(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 := 49p := 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 % tReturn
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: 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


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');            Console.ReadLine();        }         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);	    }    }

Clojure

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 }  endend 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}")  }.joinend 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]

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)Lib: bigint.lib loaded. (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 4.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," ")        }    };     console.printLine().readChar()}
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  defp sum_adj([a, b | _] = row), do: [a+b | sum_adj(tl(row))]   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  endend 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]). sum_adj([_] = L) -> L;sum_adj([A, B | _] = Row) -> [A+B | sum_adj(tl(Row))].  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]


Factor

USING: combinators formatting io kernel make math math.parsermath.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      [email protected] 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)    write(*, fmt='(A)', advance='no') '(x - 1)^'//trim(adjustl(tmp))//' ='    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          write(*, fmt='(A)', advance='no') '   '        else          write(*, fmt='(A)', advance='no') ' - '        endif      endif       if (p .eq. 0) then        write(*, fmt='(A)', advance='no') trim(adjustl(cbuf))      elseif (p .eq. 1) then        if (coeffs(i) .eq. 1) then          write(*, fmt='(A)', advance='no') 'x'        else          write(*, fmt='(A)', advance='no') trim(adjustl(cbuf))//'x'        end if      else        if (coeffs(i) .eq. 1) then          write(*, fmt='(A)', advance='no') 'x^'//trim(adjustl(pbuf))        else          write(*, fmt='(A)', advance='no') &            trim(adjustl(cbuf))//'x^'//trim(adjustl(pbuf))        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 functionend 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


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


FreeBASIC

'METHOD -- Use the Pascal triangle to retrieve the coefficients'UPPER LIMIT OF FREEBASIC ULONGINT GETS PRIMES UP TO 70Sub 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 componentsFunction 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 sEnd 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 -1End Function'==================================== 'Formatted outputFor n As Integer=1 To 9    Print pasc(n,1)Next n Print'Limit of Freebasic Ulongint sets about 70 maxPrint "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

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 CoefficientsbinCoef : Nat -> Nat -> NatbinCoef _ 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)^pexpansion : (n : Nat) -> Vect (S n) Integerexpansion n = expansion' n 1  where     expansion' : (n : Nat) -> Integer -> Vect (S n) Integer    expansion' (S m) s = s * (toIntegerNat $binCoef n (n minus (S m))) :: expansion' m (s * -1) expansion' Z s = [s] showExpansion : Vect n Integer -> StringshowExpansion [] = " "showExpansion (x::xs) {n = S k} = (if x < 0 then "-" else "") ++ term x k ++ showExpansion' xs where term : Integer -> Nat -> String term x n = if n == 0 then (show (abs x)) else (if (abs x) == 1 then "" else (show (abs x))) ++ "x" ++ (if n == 1 then "" else "^" ++ show n) sign : Integer -> String sign x = if x >= 0 then " + " else " - " showExpansion' : Vect m Integer -> String showExpansion' [] = "" showExpansion' (y::ys) {m = S k} = sign y ++ term y k ++ showExpansion' ys natToFin' : (m : Nat) -> Fin (S m) natToFin' n with (natToFin n (S n)) natToFin' n | Just y = y isPrime : Nat -> BoolisPrime Z = FalseisPrime (S Z ) = FalseisPrime n = foldl (\divs, term => divs && (term mod (toIntegerNat n)) == 0) True (fullExpansion$ 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 502 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# Output: the next OptPascal array (obtained by adding adjacent items, # but if the last two items are unequal, then their sum is repeated)def add_pairs:  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 OptPascalRowdef next_optpascal: [1] + add_pairs; # generate a stream of OptPascal arrays, beginning with []def optpascals: [] | recurse(next_optpascal); # generate a stream of Pascal arraysdef 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 rowdef 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:][]; . % $N == 0 ) end; Task 4: "Use your AKS test to generate a list of all primes under 35." range(0;36) | select(is_prime) Output: 235711131719232931 Task 5: "As a stretch goal, generate all primes under 50." [range(0;50) | select(is_prime)] Output: [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47] Julia Task 1  function polycoefs(n::Int64) pc = typeof(n)[] if n < 0 return pc end sgn = one(n) for k in n:-1:0 push!(pc, sgn*binomial(n, k)) sgn = -sgn end return pcend  Perhaps this should be done with a comprehension, but properly accounting for the sign is tricky in that case. Task 2  function stringpoly(n::Int64) if n < 0 return "" end st = @sprintf "(x - 1)^{%d} & = & " n for (i, c) in enumerate(polycoefs(n)) if i == 1 op = "" ac = c elseif c < 0 op = "-" ac = abs(c) else op = "+" ac = abs(c) end p = n + 1 - i if p == 0 st *= @sprintf " %s %d\\\\" op ac elseif ac == 1 st *= @sprintf " %s x^{%d}" op p else st *= @sprintf " %s %dx^{%d}" op ac p end end return stend  Of course this could be simpler, but this produces a nice payoff in typeset equations that do on include extraneous characters (leading pluses and coefficients of 1). Task 3  function isaksprime(n::Int64) if n < 2 return false end for c in polycoefs(n)[2:(end-1)] if c%n != 0 return false end end return trueend  Task 4  println("$")println("\\begin{array}{lcl}")for i in 0:10 println(stringpoly(i))endprintln("\\end{array}")println("$\n") L = 50print("AKS primes less than ", L, ": ")sep = ""for i in 1:L if isaksprime(i) print(sep, i) sep = ", " endendprintln()  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]  Liberty BASIC Translation of: Pascal Works with: Just BASIC version any  global pasTriMaxpasTriMax = 61dim pasTri(pasTriMax + 1) for n = 0 to 9 call expandPoly nnext nfor n = 2 to pasTriMax if isPrime(n) <> 0 then print using("###", n); end ifnext nprintend 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 selectend 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 = resend 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  wendend 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}

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 Zlet 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.. middlevar  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($_) } 1_000_000_000 .. 1_000_000_100); Output: 1000000007 1000000009 1000000021 1000000033 1000000087 1000000093 1000000097 Perl 6 Works with: Rakudo version 2015-12 The expansions are generated similarly to how most FP languages generate sequences that resemble Pascal's triangle, using a zipwith meta-operator (Z) with subtraction, applied between two lists that add a 0 on either end to the prior list. Here we define a constant infinite sequence using the ... sequence operator with a "whatever" endpoint. In fact, the second term [1,-1] could have been generated from the first term, but we put it in there for documentation so the reader can see what direction things are going. The polyprime function pretty much reads like the original description. Is it "so" that the p'th expansion's coefficients are all divisible by p? The .[1 ..^ */2] slice is done simply to weed out divisions by 1 or by factors we've already tested (since the coefficients are symmetrical in terms of divisibility). If we wanted to write polyprime even more idiomatically, we could have made it another infinite constant list that is just a mapping of the first list, but we decided that would just be showing off. :-) constant expansions = [1], [1,-1], -> @prior { [|@prior,0 Z- 0,|@prior] } ... *; sub polyprime($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

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. 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 forend procedure function is_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 1end 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 forend procedure procedure AKS_test_for_primes()    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_prime(n) then            printf(1," %d", n);        end if     end for    puts(1,'\n');    if getc(0) then end ifend procedure     AKS_test_for_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

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


PicoLisp

(de pascal (N)   (let D 1      (make         (for X (inc N)            (link D)            (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 usedoptpascal(X, [1|Y]) :-  add_pairs(X, 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.add_pairs([], []).add_pairs([X], [X]).add_pairs([X,Y], Ans) :-   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,  add_pairs( [X2, X3|Xs], Ys).

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

PureBasic

EnableExplicitDefine 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  NextEndProcedure 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 #TrueEndProcedure 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  NextPrintN("") 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)) : EndIfNextInput()
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

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)

REXX

version 1

/* REXX ---------------------------------------------------------------* 09.02.2014 Walter Pachl* 22.02.2014 WP fix 'accounting' problem (courtesy GS)*--------------------------------------------------------------------*/c.=1Numeric Digits 100limit=200pl=''mmm=0Do 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  EndSay '(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)  EndSay ' 'Say 'Primes:' subword(pl,2,27)Say '       ' subword(pl,29)Say 'Largest coefficient:' mmmSay '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)  @ddexit                                             /*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.  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)**pend 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 } endend 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}") }.joinend 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 use std::iter::repeat; fn aks_coefficients(k: usize) -> Vec<i64> { let mut coefficients = repeat(0i64).take(k + 1).collect::<Vec<_>>(); 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 .. (c.len() - 1) / 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..51).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)printlnprintln(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  clearxdel(winsid()) stacksize('max')sz=stacksize(); n=7; //For the expansion up to power of ng=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 endendfunction Z=pascal(g); //Generate Pascal's triangle up to g Q(0+1)="(x-1)^0 = 1"; //For nicer displayQ(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)); endendfunction 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 Qend 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 endendfunction R="2"; //For nicer displayfor r=3:g if prime(Z,r)=="true" then R=strcat([R, ", ",string(r)]); endenddisp(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 idx range 1 to pred(length(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.

matafunction 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


uBasic/4tH

For n = 0 To 9  Push n : Gosub _coef : Gosub _drop  Print "(x-1)^";n;" = ";  Push n : Gosub _show  PrintNext PrintPrint "primes (never mind the 1):"; For n = 1 To 34  Push n : Gosub _isprime  If Pop() Then Print " ";n;Next PrintEnd                                        ' show polynomial expansions_show                                  ' ( n --)  Do    If @(Tos()) > -1 Then Print "+";    Print @(Tos());"x^";Tos();  While (Tos())    Push Pop() - 1  Loop   Gosub _dropReturn                                        ' 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) = [email protected](0)    i = i + 1  LoopReturn                                        ' drop a value from the stack_drop                                  ' ( n --)  If Pop() EndifReturn
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 = 99Dim c(MAX + 1) As VariantPrivate 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 iEnd SubPrivate 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 = TrueEnd FunctionPrivate 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 iEnd SubPublic 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 nEnd 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



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)    nextend 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 ! mod(c(i), n) = 0 then            return 0        end if    next    return 1end 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    nextend 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    printend sub AKS_test_for_primes()

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)