# 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 p is prime if and only if all the coefficients of the polynomial expansion of
(x − 1)p − (xp − 1)

are divisible by p.

For example, trying p = 3:

(x − 1)3 − (x3 − 1) = (x3 − 3x2 + 3x − 1) − (x3 − 1) = − 3x2 + 3x
And all the coefficients are divisible by 3 so 3 is prime.
1. Create a function/subroutine/method that given p generates the coefficients of the expanded polynomial representation of (x − 1)p.
2. Use the function to show here the polynomial expansions of (x − 1)p for p in the range 0 to at least 7, inclusive.
3. Use the previous function in creating another function that when given p returns whether p is prime using the 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 greater than 31 bit integers).
Note

The task given here is related to the elementary theorem, not the actual AKS algorithm. Using the elementary theorem directly as a way of testing for primes is interesting as an exercise but impractical.

References

## ALGOL 68

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

BEGIN
COMMENT
Mathematical preliminaries.

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

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

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

Output:
(x-1)^0 = 1
(x-1)^1 = x - 1
(x-1)^2 = x^2 - 2x + 1
(x-1)^3 = x^3 - 3x^2 + 3x - 1
(x-1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1
(x-1)^5 = x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1
(x-1)^6 = x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1
(x-1)^7 = x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1
Primes < 50 are 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
And just to show off, the primes between 900 and 1000 are 907 911 919 929 937 941 947 953 967 971 977 983 991 997

## AutoHotkey

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

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

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

; 4. Use your AKS test to generate a list of all primes under 35.
i := 49
p := pascalstriangle(i+1)
Loop, % i
{
n := A_Index
isnotprime := False
For k, v in p[n+1]
(k != 1 and k != n+1) ? isnotprime |= !(v // n = v / n) ; if any is not divisible, returns true
t .= isnotprime ? "" : A_Index " "
}
Msgbox % t
Return
Output:
(x-1)^0=+1
(x-1)^1=-1+1x
(x-1)^2=+1-2x+1x^2
(x-1)^3=-1+3x-3x^2+1x^3
(x-1)^4=+1-4x+6x^2-4x^3+1x^4
(x-1)^5=-1+5x-10x^2+10x^3-5x^4+1x^5
(x-1)^6=+1-6x+15x^2-20x^3+15x^4-6x^5+1x^6
(x-1)^7=-1+7x-21x^2+35x^3-35x^4+21x^5-7x^6+1x^7

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

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

## Bracmat

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

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

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

Output:

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

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

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

Output:

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

## C

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

long long c[100];

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

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

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

int is_prime(int n)
{
int i;

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

return i < 0;
}

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

int main(void)
{
int n;

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

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

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

The ugly output:

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

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

## C#

Translation of: C

using System;
public class AksTest
{
static long[] c = new long[100];

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

Console.WriteLine('\n');
}

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

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

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

static bool is_prime(int n)
{
int i;

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

while (i-- != 0 && (c[i] % n) == 0) ;

return i < 0;
}

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

## 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

## 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]

## Erlang

Translation of: CoffeeScript

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

#! /usr/bin/escript

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

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

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

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

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

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

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

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

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

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

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

## Fortran

program aks
implicit none

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

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

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

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

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

if (allocated(coeffs)) deallocate(coeffs)

allocate(coeffs(n + 1))

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

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

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

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

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

if (.not. allocated(coeffs)) return

non_zero = .false.

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

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

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

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

write(*, *)
end subroutine

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

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

res = .true.

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

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

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

## 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 70
Sub string_split(s_in As String,char As String,result() As String)
Dim As String s=s_in,var1,var2
Dim As Integer n,pst
#macro split(stri,char,var1,var2)
pst=Instr(stri,char)
var1="":var2=""
If pst<>0 Then
var1=Mid(stri,1,pst-1)
var2=Mid(stri,pst+1)
Else
var1=stri
End If
Redim Preserve result(1 To 1+n-((Len(var1)>0)+(Len(var2)>0)))
result(n+1)=var1
#endmacro
Do
split(s,char,var1,var2):n=n+1:s=var2
Loop Until var2=""
Redim Preserve result(1 To Ubound(result)-1)
End Sub

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

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

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

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

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

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]

## J

Solution:
binomialExpansion =:  (!~ * _1 ^ 2 | ]) i.&.:<:         NB. 1) Create a function that gives the coefficients of (x-1)^p.
testAKS =: 0 *./ .= ] | binomialExpansion NB. 3) Use that function to create another which determines whether p is prime using AKS.
Examples:
binomialExpansion&.> i. 8   NB.  2) show the polynomial expansions p in the range 0 to at 7 inclusive.
+-++--+----+-------+-----------+---------------+------------------+
|0||_2|_3 3|_4 6 _4|_5 10 _10 5|_6 15 _20 15 _6|_7 21 _35 35 _21 7|
+-++--+----+-------+-----------+---------------+------------------+
(#~ testAKS&> ) 2+i. 35 NB. 4) Generate a list of all primes under 35.
2 3 5 7 11 13 17 19 23 29 31
(#~ testAKS&> ) 2+i. 50 NB. 5) [stretch] Generate all primes under 50
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
i.&.:(_1&p:) 50 NB. Double-check our results using built-in prime filter.
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47

## Java

Translation of: C
Solution:

public class AksTest
{
static Long[] c = new Long[100];

public static void main(String[] args)
{
for (int n = 0; n < 10; n++) {
coef(n);
System.out.print("(x-1)^" + n + " = ");
show(n);
System.out.println("");
}

System.out.print("Primes:");
for (int n = 1; n <= 63; n++)
if (is_prime(n))
System.out.printf(" %d", n);

System.out.println('\n');
}

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

if (n < 0 || n > 63) System.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 boolean 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 {
System.out.print("+" + c[n] + "x^"+ n);
}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: 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

## jq

Works with: jq version 1.5rc1

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

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

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

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

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

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

optpascals | pascalize;

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

def optpascal: nth(.; optpascals);
Task 1: "A method to generate the coefficients of (x-1)^p"
def coefficients:
def alternate_signs: . as $in | reduce range(0; length) as$i ([]; . + [$in[$i] * (if $i % 2 == 0 then 1 else -1 end )]); (.+1) | pascal | alternate_signs; Task 2: "Show here the polynomial expansions of (x − 1)^p for p in the range 0 to at least 7, inclusive." range(0;8) | "Coefficient for (x - 1)^\(.): \(coefficients)" Output: Coefficients for (x - 1)^0: [1] Coefficients for (x - 1)^1: [1,-1] Coefficients for (x - 1)^2: [1,-2,1] Coefficients for (x - 1)^3: [1,-3,3,-1] Coefficients for (x - 1)^4: [1,-4,6,-4,1] Coefficients for (x - 1)^5: [1,-5,10,-10,5,-1] Coefficients for (x - 1)^6: [1,-6,15,-20,15,-6,1] Coefficients for (x - 1)^7: [1,-7,21,-35,35,-21,7,-1] Task 3: Prime Number Test For brevity, we show here only the relatively efficient solution based on optpascal/0: def is_prime: . as$N
| if . < 2 then false
else (1+.) | optpascal
| all( .[2:][]; . % $N == 0 ) end; Task 4: "Use your AKS test to generate a list of all primes under 35." range(0;36) | select(is_prime) Output: 2 3 5 7 11 13 17 19 23 29 31 Task 5: "As a stretch goal, generate all primes under 50." [range(0;50) | select(is_prime)] Output: [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47] ## Julia Task 1 function polycoefs(n::Int64) pc = typeof(n)[] if n < 0 return pc end sgn = one(n) for k in n:-1:0 push!(pc, sgn*binomial(n, k)) sgn = -sgn end return pc end Perhaps this should be done with a comprehension, but properly accounting for the sign is tricky in that case. Task 2 function stringpoly(n::Int64) if n < 0 return "" end st = @sprintf "(x - 1)^{%d} & = & " n for (i, c) in enumerate(polycoefs(n)) if i == 1 op = "" ac = c elseif c < 0 op = "-" ac = abs(c) else op = "+" ac = abs(c) end p = n + 1 - i if p == 0 st *= @sprintf " %s %d\\\\" op ac elseif ac == 1 st *= @sprintf " %s x^{%d}" op p else st *= @sprintf " %s %dx^{%d}" op ac p end end return st end Of course this could be simpler, but this produces a nice payoff in typeset equations that do on include extraneous characters (leading pluses and coefficients of 1). Task 3 function isaksprime(n::Int64) if n < 2 return false end for c in polycoefs(n)[2:(end-1)] if c%n != 0 return false end end return true end Task 4 println("$") println("\\begin{array}{lcl}") for i in 0:10 println(stringpoly(i)) end println("\\end{array}") println("$\n") L = 50 print("AKS primes less than ", L, ": ") sep = "" for i in 1:L if isaksprime(i) print(sep, i) sep = ", " end end println() Output: $\begin{array}{lcl} (x - 1)^{0} & = & 1\\ (x - 1)^{1} & = & x^{1} - 1\\ (x - 1)^{2} & = & x^{2} - 2x^{1} + 1\\ (x - 1)^{3} & = & x^{3} - 3x^{2} + 3x^{1} - 1\\ (x - 1)^{4} & = & x^{4} - 4x^{3} + 6x^{2} - 4x^{1} + 1\\ (x - 1)^{5} & = & x^{5} - 5x^{4} + 10x^{3} - 10x^{2} + 5x^{1} - 1\\ (x - 1)^{6} & = & x^{6} - 6x^{5} + 15x^{4} - 20x^{3} + 15x^{2} - 6x^{1} + 1\\ (x - 1)^{7} & = & x^{7} - 7x^{6} + 21x^{5} - 35x^{4} + 35x^{3} - 21x^{2} + 7x^{1} - 1\\ (x - 1)^{8} & = & x^{8} - 8x^{7} + 28x^{6} - 56x^{5} + 70x^{4} - 56x^{3} + 28x^{2} - 8x^{1} + 1\\ (x - 1)^{9} & = & x^{9} - 9x^{8} + 36x^{7} - 84x^{6} + 126x^{5} - 126x^{4} + 84x^{3} - 36x^{2} + 9x^{1} - 1\\ (x - 1)^{10} & = & x^{10} - 10x^{9} + 45x^{8} - 120x^{7} + 210x^{6} - 252x^{5} + 210x^{4} - 120x^{3} + 45x^{2} - 10x^{1} + 1\\ \end{array}$ AKS primes less than 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 ## 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 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} ## Oforth func: nextCoef(prev) { | i | ListBuffer new dup add(0) prev size 1 - loop: i [ dup add(prev at(i) prev at(i 1 +) - ) ] dup add(0) } func: coefs(n) { [ 0, 1, 0 ] #nextCoef times(n) extract(2, n 2 + ) } func: isPrime(n) { coefs(n) extract(2, n) conform(#[n mod 0 == ]) } func: aks { | i | 0 10 for: i [ System.Out "(x-1)^" << i << " = " << coefs(i) << cr ] 50 seq filter(#isPrime) apply(#[ print " " print ]) 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 tested wth freepascal const pasTriMax = 61; type tpasTri =array[0..pasTriMax] of UInt64; var pasTri : tpasTri; procedure pastriangle(n:longInt); //calculate the n'th line 0.. middle var j,k: longWord; begin pasTri[0] := 1; j := 1; while (j<=n) do begin inc(j); k := j SHR 1; pasTri[k] :=pasTri[k-1]; For k := k downto 1 do inc(pasTri[k],pasTri[k-1]); end; end; function CheckPrime(n:longWord):boolean; var i : integer; res: boolean; Begin IF n > pasTriMax then begin writeln(n,' is out of range '); EXIT; end; pastriangle(n); res := true; i := n shr 1; while res AND (i >1) do Begin res := res AND(pasTri[i] mod n = 0); dec(i); end; CheckPrime := res; end; procedure ExpandPoly(n:longWord); const Vz :array[boolean] of char = ('+','-'); var j,k: longWord; bVz: Boolean; Begin IF n < 2 then Begin IF n = 0 then writeln('(x-1)^0 = 1') else writeln('(x-1)^1 = x-1'); EXIT; end; IF n > pasTriMax then begin writeln(n,' is out of range '); EXIT; end; pastriangle(n); write('(x-1)^',n,' = '); k := 0; j := n; bVz := false; repeat IF j=n then write('x^',j) else write(Vz[bVz],pasTri[k],'*x^',j); bVz := Not(bVz); inc(k); dec(j); until k>= j; k := j; while k > 0 do Begin IF j <> 1 then write(Vz[bVz],pasTri[k],'*x^',j) else write(Vz[bVz],pasTri[k],'*x'); bVz := Not(bVz); dec(k); dec(j); end; write(Vz[bVz],pasTri[0]); writeln; end; var n: LongWord; Begin For n := 0 to 9 do ExpandPoly(n); For n := 2 to pasTriMax do IF CheckPrime(n) then write(n:3); 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($_) } 2..80), "]\n"; Output: expansions of (x-1)^p: 1 x - 1 x^2 - 2x + 1 x^3 - 3x^2 + 3x - 1 x^4 - 4x^3 + 6x^2 - 4x + 1 x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1 x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1 x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1 x^8 - 8x^7 + 28x^6 - 56x^5 + 70x^4 - 56x^3 + 28x^2 - 8x + 1 x^9 - 9x^8 + 36x^7 - 84x^6 + 126x^5 - 126x^4 + 84x^3 - 36x^2 + 9x - 1 Primes to 80: [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79] ## Perl 6 constant expansions = [1], [1,-1], -> @prior { [@prior,0 Z- 0,@prior] } ... *; sub polyprime($p where 2..*) { so expansions[$p].[1 ..^ */2].all %%$p }

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. :-)

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;
}
}
)
}
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

And testing the function:

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

## PicoLisp

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

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

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

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

## Prolog

### Prolog(ue)

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

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

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

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

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

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

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

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

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

### Pascal Triangle Generator

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

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

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

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

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

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

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

### Solutions

Solutions with output from SWI-Prolog:

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

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

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

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

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

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

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

The following would be more efficient because backtracking saves recomputation:

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

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

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

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

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

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

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

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

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

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

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

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

## 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
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
p (x − 1)p Prime(p)?
0 + 1 False
1 − 1 + x False
2 + 1 − 2x + x2 True
3 − 1 + 3x − 3x2 + x3 True
4 + 1 − 4x + 6x2 − 4x3 + x4 False
5 − 1 + 5x − 10x2 + 10x3 − 5x4 + x5 True
6 + 1 − 6x + 15x2 − 20x3 + 15x4 − 6x5 + x6 False
7 − 1 + 7x − 21x2 + 35x3 − 35x4 + 21x5 − 7x6 + x7 True
8 + 1 − 8x + 28x2 − 56x3 + 70x4 − 56x5 + 28x6 − 8x7 + x8 False
9 − 1 + 9x − 36x2 + 84x3 − 126x4 + 126x5 − 84x6 + 36x7 − 9x8 + x9 False
10 + 1 − 10x + 45x2 − 120x3 + 210x4 − 252x5 + 210x6 − 120x7 + 45x8 − 10x9 + x10 False
11 − 1 + 11x − 55x2 + 165x3 − 330x4 + 462x5 − 462x6 + 330x7 − 165x8 + 55x9 − 11x10 + x11 True

## R

Borrowing heavily from Python listing. Optimized for the fact that the vector of the coefficients is a palindrome.

Is.Prime<-function(x){
expand<-function(p){
ex = 1
for (i in 0:(p/2-1)){
ex<-c(ex[1]*(p-i)/(i+1),ex)
}
return(rev(ex)[-1])
}
return(as.logical(min(expand(x)%%x==0)))
}

## 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.=1
Numeric Digits 100
limit=200
pl=''
mmm=0
Do p=3 To limit
pm1=p-1
c.p.1=1
c.p.p=1
Do j=2 To p-1
jm1=j-1
c.p.j=c.pm1.jm1+c.pm1.j
mmm=max(mmm,c.p.j)
End
End
Say '(x-1)**0 = 1'
do i=2 To limit
im1=i-1
sign='+'
ol='(x-1)^'im1 '='
Do j=i to 2 by -1
If j=2 Then
term='x '
Else
term='x^'||(j-1)
If j=i Then
ol=ol term
Else
ol=ol sign c.i.j'*'term
sign=translate(sign,'+-','-+')
End
If i<10 then
Say ol sign 1
Do j=2 To i-1
If c.i.j//(i-1)>0 Then
Leave
End
If j>i-1 Then
pl=pl (i-1)
End
Say ' '
Say 'Primes:' subword(pl,2,27)
Say ' ' subword(pl,29)
Say 'Largest coefficient:' mmm
Say 'This has' length(mmm) 'digits'
Output:
(x-1)**0 = 1
(x-1)^1 = x   - 1
(x-1)^2 = x^2 - 2*x   + 1
(x-1)^3 = x^3 - 3*x^2 + 3*x   - 1
(x-1)^4 = x^4 - 4*x^3 + 6*x^2 - 4*x   + 1
(x-1)^5 = x^5 - 5*x^4 + 10*x^3 - 10*x^2 + 5*x   - 1
(x-1)^6 = x^6 - 6*x^5 + 15*x^4 - 20*x^3 + 15*x^2 - 6*x   + 1
(x-1)^7 = x^7 - 7*x^6 + 21*x^5 - 35*x^4 + 35*x^3 - 21*x^2 + 7*x   - 1
(x-1)^8 = x^8 - 8*x^7 + 28*x^6 - 56*x^5 + 70*x^4 - 56*x^3 + 28*x^2 - 8*x   + 1

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

### version 2

This REXX version is an optimized version (of version 1) and modified to address each of the requirements.
The program determines programmatically the required number of digits (precision) for the large coefficients.

/*REXX pgm calculates primes via the Agrawal-Kayal-Saxena (AKS) primality test*/
parse arg Z .; if Z=='' then Z=200 /*Z not specified? Then use 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.*/
$.0='-';$.1="+" /*$.x is the sign (glyph) to be used*/ @.=1; big=1 /*define all coefficients to unity (1).*/ #= /*define list of prime numbers (so far)*/ do p=3 for Z; pm=p-1; pp=p+1 /*PM & PP: used as a coding convenience*/ do m=2 to pp%2; mm=m-1 /*calculate coefficients for a power. */ @.p.m=@.pm.mm + @.pm.m; h=pp-m /*calculate left side of coefficients*/ @.p.h=@.p.m /* " right " " " */ if @.p.m>big then big=@.p.m /*is this coefficient the biggest? */ end /*m*/ /* [↑] The M DO loop creates both */ end /*p*/ /* sides in the same loop, saving */ /* a bunch of execution time. */ if tell then say '(x-1)^0: 1' /*possibly display the first expression*/ /* [↓] test for primality by division.*/ do n=2 for Z; nh=n%2; nm=n-1 /*create expressions; find the primes.*/ do k=3 to nh until @.n.k//nm\==0 /*are coefficients divisible by N-1 ? */ end /*k*/ /* [↑] skip the 1st & 2nd coefficients*/ /* [↓] multiple THEN─IF faster than &s*/ if nm\==1 then if n\==5 then if k>nh then #=# nm /*add number to prime list*/ if \tell then iterate /*Don't tell? Don't show expressions. */ s=1 /*S: is the sign indicator (-1│+1).*/ y='(x-1)^'nm": " /*define first part of the expression. */ /* [↓] create the higher powers first.*/ do j=n to 2 by -1 /*proceed in a downward direction for J*/ if j==2 then xp='x' /*if power=1, then don't show the power*/ else xp='x^' || (j-1) /* ··· else show power with ^ */ if j==n then y=y xp /*no sign (+│-) for the 1st expression.*/ else y=y$.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,*/
/* and is displayed either - │ + */
say y $.s 1 /*just show the first N expressions, */ end /*n*/ /* [↑] ··· but only for negative Z. */ say /* [↓] Has Z a leading + ? Then show.*/ is="isn't"; if Z==word(. #,words(#)+1) then is='is' /*is or isn't a prime.*/ if left(OZ,1)=='+' then say Z is 'prime.' /*tell if OZ has a +. */ else say 'primes:' # /*display prime # list. */ say /* [↓] size of big 'un.*/ say 'Found ' words(#) ' primes and the largest coefficient has' , length(big) "decimal digits." /*stick a fork in it, we're all done. */ output for requirement #2, showing twenty expressions using as input: -20 (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 primes: 2 3 5 7 11 13 17 19 Found 8 primes and the largest coefficient has 6 decimal digits. output for 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; the version of Regina REXX used herein has a limit of around 2 Gbytes.) 2221 is prime. Found 331 primes and the largest coefficient has 668 decimal digits. output for requirement #4, showing all primes under 35 using for 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 for 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 for input: 500 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)**p end def prime?(p) return false if p < 2 (x_minus_1_to_the(p) - Polynomial.from_string("x**#{p}-1")).coefs.all?{|n| n%p==0} end 8.times do |n| # the default Polynomial#to_s would be OK here; the substitutions just make the # output match the other version below. puts "(x-1)^#{n} = #{x_minus_1_to_the(n).to_s.gsub(/\*\*/,'^').gsub(/\*/,'')}" end puts "\nPrimes below 50:", 50.times.select {|n| prime? n}.join(',') Or without the dependency: def x_minus_1_to_the(p) p.times.inject([1]) do |ex, _| ([0] + ex).zip(ex + [0]).map { |x,y| x - y } end end def prime?(p) return false if p < 2 coeff = x_minus_1_to_the(p) coeff[0] += coeff.pop coeff.all?{|n| n%p==0} end 8.times do |n| puts "(x-1)^#{n} = " + x_minus_1_to_the(n). each_with_index. map { |c, p| if p.zero? then c.to_s else (c<0 ? " - " : " + ") + (c.abs==1 ? "x" : "#{c.abs}x") + (p==1 ? "" : "^#{p}") end }.join end puts "\nPrimes below 50:", 50.times.select {|n| prime? n}.join(',') Output: (x-1)^0 = 1 (x-1)^1 = -1 + x (x-1)^2 = 1 - 2x + x^2 (x-1)^3 = -1 + 3x - 3x^2 + x^3 (x-1)^4 = 1 - 4x + 6x^2 - 4x^3 + x^4 (x-1)^5 = -1 + 5x - 10x^2 + 10x^3 - 5x^4 + x^5 (x-1)^6 = 1 - 6x + 15x^2 - 20x^3 + 15x^4 - 6x^5 + x^6 (x-1)^7 = -1 + 7x - 21x^2 + 35x^3 - 35x^4 + 21x^5 - 7x^6 + x^7 Primes below 50: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47 ## Rust #![feature(core)] use std::iter::{range_inclusive, 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); range_inclusive(1, (c.len() - 1) / 2).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 = ${poly2String( pascal(i) )}" ) } val primes = (2 to 50).filter(isPrime) println println(primes mkString " ") Output: (x-1)^0 = 1 (x-1)^1 = -1+x (x-1)^2 = 1-2x+x^2 (x-1)^3 = -1+3x-3x^2+x^3 (x-1)^4 = 1-4x+6x^2-4x^3+x^4 (x-1)^5 = -1+5x-10x^2+10x^3-5x^4+x^5 (x-1)^6 = 1-6x+15x^2-20x^3+15x^4-6x^5+x^6 (x-1)^7 = -1+7x-21x^2+35x^3-35x^4+21x^5-7x^6+x^7 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 ## Scilab clear xdel(winsid()) stacksize('max') sz=stacksize(); n=7; //For the expansion up to power of n g=50; //For test of primes up to g function X = pascal(g) //Pascal´s triangle X(1,1)=1; //Zeroth power X(2,1)=1; //First power X(2,2)=1; for q=3:1:g+1 //From second power use this loop X(q,1)=1; X(q,q)=1; for p=2:1:q-1 X(q,p)=X(q-1,p-1)+X(q-1,p); end end endfunction Z=pascal(g); //Generate Pascal's triangle up to g Q(0+1)="(x-1)^0 = 1"; //For nicer display Q(1+1)="(x-1)^1 = x^1-1"; //For nicer display disp(Q(1)) disp(Q(2)) function cf=coef(Z,q,p) //Return coeffiecents for nicer display of expansion without "ones" if Z(q,p)==1 then cf=""; else cf=string(Z(q,p)); end endfunction for q=3:n+1 //Generate and display the expansions Q(q)=strcat(["(x-1)^",string(q-1)," = "]); sing=""; //Sign of coeff. for p=1:q-1 //Number of coefficients equals power minus 1 Q(q)=strcat([Q(q),sing,coef(Z,q,p),"x^",string(q-p)]); if sing=="-" then sing="+"; else sing="-"; end end Q(q)=strcat([Q(q),sing,string(1)]); disp(Q(q)) clear Q end function prime=prime(Z,g) prime="true"; for p=2:g if abs(floor(Z(g+1,p)/g)-Z(g+1,p)/g)>0 then prime="false"; break; end end endfunction R="2"; //For nicer display for r=3:g if prime(Z,r)=="true" then R=strcat([R, ", ",string(r)]); end end disp(R) Output: (x-1)^0 = 1 (x-1)^1 = x^1-1 (x-1)^2 = x^2-2x^1+1 (x-1)^3 = x^3-3x^2+3x^1-1 (x-1)^4 = x^4-4x^3+6x^2-4x^1+1 (x-1)^5 = x^5-5x^4+10x^3-10x^2+5x^1-1 (x-1)^6 = x^6-6x^5+15x^4-20x^3+15x^2-6x^1+1 (x-1)^7 = x^7-7x^6+21x^5-35x^4+35x^3-21x^2+7x^1-1 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 ## Seed7$ include "seed7_05.s7i";

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

const func boolean: aks_test (in integer: p) is func
result
var boolean: aks_test is FALSE;
local
var array integer: ex is 0 times 0;
var integer: idx is 0;
begin
if p >= 2 then
ex := expand_x_1(p);
ex[1] +:= 1;
for 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

## 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
Print
Next

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

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

Print
End

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

Gosub _drop
Return

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

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

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

Gosub _drop
Push (i = 0)
Return

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

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

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

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

' drop a value from the stack
_drop ' ( n --)
If Pop() Endif
Return
Output:
(x-1)^0 = +1x^0
(x-1)^1 = +1x^1-1x^0
(x-1)^2 = +1x^2-2x^1+1x^0
(x-1)^3 = +1x^3-3x^2+3x^1-1x^0
(x-1)^4 = +1x^4-4x^3+6x^2-4x^1+1x^0
(x-1)^5 = +1x^5-5x^4+10x^3-10x^2+5x^1-1x^0
(x-1)^6 = +1x^6-6x^5+15x^4-20x^3+15x^2-6x^1+1x^0
(x-1)^7 = +1x^7-7x^6+21x^5-35x^4+35x^3-21x^2+7x^1-1x^0
(x-1)^8 = +1x^8-8x^7+28x^6-56x^5+70x^4-56x^3+28x^2-8x^1+1x^0
(x-1)^9 = +1x^9-9x^8+36x^7-84x^6+126x^5-126x^4+84x^3-36x^2+9x^1-1x^0

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

## 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)