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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} is prime if and only if all the coefficients of the polynomial expansion of
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x-1)^p - (x^p - 1)}
are divisible by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} .
- Example
Using Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.
- Task
- Create a function/subroutine/method that given Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} generates the coefficients of the expanded polynomial representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x-1)^p} .
- Use the function to show here the polynomial expansions of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x-1)^p} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} in the range 0 to at least 7, inclusive.
- Use the previous function in creating another function that when given Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} returns whether Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} is prime using the theorem.
- Use your test to generate a list of all primes under 35.
- As a stretch goal, generate all primes under 50 (needs integers larger than 31-bit).
- References
- Agrawal-Kayal-Saxena (AKS) primality test (Wikipedia)
- Fool-Proof Test for Primes - Numberphile (Video). The accuracy of this video is disputed -- at best it is an oversimplification.
11l
<lang 11l>F expand_x_1(p)
V ex = [BigInt(1)]
L(i) 0 .< p
ex.append(ex.last * -(p - i) I/ (i + 1))
R reversed(ex)
F aks_test(p)
I p < 2
R 0B
V ex = expand_x_1(p)
ex[0]++
R !any(ex[0 .< (len)-1].map(mult -> mult % @p != 0))
print(‘# p: (x-1)^p for small p’) L(p) 12
print(‘#3: #.’.format(p, enumerate(expand_x_1(p)).map((n, e) -> ‘#.#.#.’.format(‘+’ * (e >= 0), e, I n {(‘x^#.’.format(n))} E ‘’)).join(‘ ’)))
print("\n# small primes using the aks test") print((0..100).filter(p -> aks_test(p)))</lang>
- Output:
# p: (x-1)^p for small p 0: +1 1: -1 +1x^1 2: +1 -2x^1 +1x^2 3: -1 +3x^1 -3x^2 +1x^3 4: +1 -4x^1 +6x^2 -4x^3 +1x^4 5: -1 +5x^1 -10x^2 +10x^3 -5x^4 +1x^5 6: +1 -6x^1 +15x^2 -20x^3 +15x^4 -6x^5 +1x^6 7: -1 +7x^1 -21x^2 +35x^3 -35x^4 +21x^5 -7x^6 +1x^7 8: +1 -8x^1 +28x^2 -56x^3 +70x^4 -56x^5 +28x^6 -8x^7 +1x^8 9: -1 +9x^1 -36x^2 +84x^3 -126x^4 +126x^5 -84x^6 +36x^7 -9x^8 +1x^9 10: +1 -10x^1 +45x^2 -120x^3 +210x^4 -252x^5 +210x^6 -120x^7 +45x^8 -10x^9 +1x^10 11: -1 +11x^1 -55x^2 +165x^3 -330x^4 +462x^5 -462x^6 +330x^7 -165x^8 +55x^9 -11x^10 +1x^11 # small primes using the aks test [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
8th
<lang 8th> with: a
- nextrow \ a -- a
len
[ ( drop [1] ),
( drop [1,1] ),
( ' n:+ y 1 slide 1 push ) ]
swap 2 min caseof ;
- with
with: n
- .x \ n --
dup
[ ( drop ),
( drop "x" . ),
( "x^" . . ) ]
swap 2 min caseof space ;
- .term \ coef exp -- ; omit coef for 1x^n when n > 0
over 1 = over 0 > and if nip .x else swap . .x then ;
- .sgn \ +/-1 --
[ "-", null, "+" ] swap 1+ caseof . space ;
- .lhs \ n --
"(x-1)^" . . ;
- .rhs \ a -- a
a:len 1- >r 1 swap ( third .sgn r@ rot - .term -1 * ) a:each nip rdrop ;
- .eqn \ a -- a
a:len 1- .lhs " = " . .rhs ;
- .binomials \ --
[] ( nextrow .eqn cr ) 8 times drop ;
- primerow? \ a -- a ?
a:len 3 < if false ;then 1 a:@ >r \ 2nd position is the number to check for primality true swap ( nip dup 1 = swap r@ mod 0 = or and ) a:each swap rdrop ;
- .primes-via-aks \ --
[] ( nextrow primerow? if 1 a:@ . space then ) 50 times drop ;
- with
.binomials cr "The primes upto 50 are (via AKS): " . .primes-via-aks cr
bye</lang>
- Output:
(x-1)^0 = + 1 (x-1)^1 = + x - 1 (x-1)^2 = + x^2 - 2x + 1 (x-1)^3 = + x^3 - 3x^2 + 3x - 1 (x-1)^4 = + x^4 - 4x^3 + 6x^2 - 4x + 1 (x-1)^5 = + x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1 (x-1)^6 = + x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1 (x-1)^7 = + x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1 The primes upto 50 are (via AKS): 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
AArch64 Assembly
<lang AArch64 Assembly> /* ARM assembly AARCH64 Raspberry PI 3B or android 64 bits */ /* program AKS64.s */
/*******************************************/ /* Constantes file */ /*******************************************/ /* for this file see task include a file in language AArch64 assembly*/ .include "../includeConstantesARM64.inc" .equ MAXI, 64 .equ NUMBERLOOP, 10
/*********************************/ /* Initialized data */ /*********************************/ .data szMessResult: .asciz " (x-1)^@ = " szMessResult1: .asciz " @ x^@ " szMessResPrime: .asciz "Number @ is prime. \n" szCarriageReturn: .asciz "\n"
/*********************************/ /* UnInitialized data */ /*********************************/ .bss sZoneConv: .skip 24 qTabCoef: .skip 8 * MAXI /*********************************/ /* code section */ /*********************************/ .text .global main main: // entry of program
mov x4,#1
1: // loop
mov x0,x4 bl computeCoef // compute coefficient ldr x0,qAdrqTabCoef mov x0,x4 bl displayCoef // display coefficient add x4,x4,1 cmp x4,NUMBERLOOP blt 1b
mov x4,1
2:
mov x0,x4 bl isPrime // is prime ? cmp x0,1 bne 3f mov x0,x4 ldr x1,qAdrsZoneConv bl conversion10 // call decimal conversion add x1,x1,x0 strb wzr,[x1] ldr x0,qAdrszMessResPrime ldr x1,qAdrsZoneConv // insert value conversion in message bl strInsertAtCharInc bl affichageMess
3:
add x4,x4,1 cmp x4,MAXI blt 2b
100: // standard end of the program
mov x0,0 // return code mov x8,EXIT // request to exit program svc 0 // perform the system call
qAdrszCarriageReturn: .quad szCarriageReturn qAdrsZoneConv: .quad sZoneConv qAdrqTabCoef: .quad qTabCoef qAdrszMessResPrime: .quad szMessResPrime /***************************************************/ /* display coefficients */ /***************************************************/ // x0 contains a number displayCoef:
stp x1,lr,[sp,-16]! // save registres stp x2,x3,[sp,-16]! // save registres stp x4,x5,[sp,-16]! // save registres stp x6,x7,[sp,-16]! // save registres mov x2,x0 ldr x1,qAdrsZoneConv // bl conversion10 // call decimal conversion add x1,x1,x0 strb wzr,[x1] ldr x0,qAdrszMessResult ldr x1,qAdrsZoneConv // insert value conversion in message bl strInsertAtCharInc bl affichageMess ldr x3,qAdrqTabCoef
1:
ldr x0,[x3,x2,lsl #3] ldr x1,qAdrsZoneConv // bl conversion10S // call decimal conversion
2: // removing spaces
ldrb w6,[x1] cmp x6,' ' cinc x1,x1,eq beq 2b
ldr x0,qAdrszMessResult1 bl strInsertAtCharInc mov x4,x0 mov x0,x2 ldr x1,qAdrsZoneConv // else display odd message bl conversion10 // call decimal conversion add x1,x1,x0 strb wzr,[x1] mov x0,x4 ldr x1,qAdrsZoneConv // insert value conversion in message bl strInsertAtCharInc bl affichageMess subs x2,x2,#1 bge 1b ldr x0,qAdrszCarriageReturn bl affichageMess
100:
ldp x6,x7,[sp],16 // restaur des 2 registres ldp x4,x5,[sp],16 // restaur des 2 registres ldp x2,x3,[sp],16 // restaur des 2 registres ldp x1,lr,[sp],16 // restaur des 2 registres ret
qAdrszMessResult: .quad szMessResult qAdrszMessResult1: .quad szMessResult1 /***************************************************/ /* compute coefficient */ /***************************************************/ // x0 contains a number computeCoef:
stp x1,lr,[sp,-16]! // save registres stp x2,x3,[sp,-16]! // save registres stp x4,x5,[sp,-16]! // save registres stp x6,x7,[sp,-16]! // save registres ldr x1,qAdrqTabCoef // address coefficient array mov x2,1 str x2,[x1] // store 1 to coeff [0] mov x3,0 // indice 1
1:
add x4,x3,1 mov x5,1 str x5,[x1,x4,lsl #3] mov x6,x3 // indice 2 = indice 1
2:
cmp x6,0 // zero ? -> end loop ble 3f sub x4,x6,1 ldr x5,[x1,x4,lsl 3] ldr x4,[x1,x6,lsl 3] sub x5,x5,x4 str x5,[x1,x6,lsl 3] sub x6,x6,1 b 2b
3:
ldr x2,[x1] // inversion coeff [0] neg x2,x2 str x2,[x1] add x3,x3,1 cmp x3,x0 blt 1b
100:
ldp x6,x7,[sp],16 // restaur des 2 registres ldp x4,x5,[sp],16 // restaur des 2 registres ldp x2,x3,[sp],16 // restaur des 2 registres ldp x1,lr,[sp],16 // restaur des 2 registres ret
/***************************************************/ /* verify number is prime */ /***************************************************/ // x0 contains a number isPrime:
stp x1,lr,[sp,-16]! // save registres stp x2,x3,[sp,-16]! // save registres stp x4,x5,[sp,-16]! // save registres bl computeCoef ldr x4,qAdrqTabCoef // address coefficient array ldr x2,[x4] add x2,x2,1 str x2,[x4] ldr x2,[x4,x0,lsl 3] sub x2,x2,#1 str x2,[x4,x0,lsl 3] mov x5,x0 // number start
1:
ldr x1,[x4,x5,lsl 3] // load one coeff sdiv x2,x1,x0 msub x3,x2,x0,x1 // compute remainder cmp x3,#0 // remainder = zéro ? bne 99f // if <> no prime subs x5,x5,#1 // next coef bgt 1b // and loop mov x0,#1 // prime b 100f
99:
mov x0,0 // no prime
100:
ldp x4,x5,[sp],16 // restaur des 2 registres ldp x2,x3,[sp],16 // restaur des 2 registres ldp x1,lr,[sp],16 // restaur des 2 registres ret
/********************************************************/ /* File Include fonctions */ /********************************************************/ /* for this file see task include a file in language AArch64 assembly */ .include "../includeARM64.inc" </lang>
- Output:
(x-1)^1 = +1 x^1 -1 x^0 (x-1)^2 = +1 x^2 -2 x^1 +1 x^0 (x-1)^3 = +1 x^3 -3 x^2 +3 x^1 -1 x^0 (x-1)^4 = +1 x^4 -4 x^3 +6 x^2 -4 x^1 +1 x^0 (x-1)^5 = +1 x^5 -5 x^4 +10 x^3 -10 x^2 +5 x^1 -1 x^0 (x-1)^6 = +1 x^6 -6 x^5 +15 x^4 -20 x^3 +15 x^2 -6 x^1 +1 x^0 (x-1)^7 = +1 x^7 -7 x^6 +21 x^5 -35 x^4 +35 x^3 -21 x^2 +7 x^1 -1 x^0 (x-1)^8 = +1 x^8 -8 x^7 +28 x^6 -56 x^5 +70 x^4 -56 x^3 +28 x^2 -8 x^1 +1 x^0 (x-1)^9 = +1 x^9 -9 x^8 +36 x^7 -84 x^6 +126 x^5 -126 x^4 +84 x^3 -36 x^2 +9 x^1 -1 x^0 Number 1 is prime. Number 2 is prime. Number 3 is prime. Number 5 is prime. Number 7 is prime. Number 11 is prime. Number 13 is prime. Number 17 is prime. Number 19 is prime. Number 23 is prime. Number 29 is prime. Number 31 is prime. Number 37 is prime. Number 41 is prime. Number 43 is prime. Number 47 is prime. Number 53 is prime. Number 59 is prime. Number 61 is prime.
Ada
<lang Ada>with Ada.Text_IO;
procedure Test_For_Primes is
type Pascal_Triangle_Type is array (Natural range <>) of Long_Long_Integer;
function Calculate_Pascal_Triangle (N : in Natural) return Pascal_Triangle_Type is
Pascal_Triangle : Pascal_Triangle_Type (0 .. N);
begin
Pascal_Triangle (0) := 1;
for I in Pascal_Triangle'First .. Pascal_Triangle'Last - 1 loop
Pascal_Triangle (1 + I) := 1;
for J in reverse 1 .. I loop
Pascal_Triangle (J) := Pascal_Triangle (J - 1) - Pascal_Triangle (J);
end loop;
Pascal_Triangle (0) := -Pascal_Triangle (0);
end loop;
return Pascal_Triangle;
end Calculate_Pascal_Triangle;
function Is_Prime (N : Integer) return Boolean is
I : Integer;
Result : Boolean := True;
Pascal_Triangle : constant Pascal_Triangle_Type := Calculate_Pascal_Triangle (N);
begin
I := N / 2;
while Result and I > 1 loop
Result := Result and Pascal_Triangle (I) mod Long_Long_Integer (N) = 0;
I := I - 1;
end loop;
return Result;
end Is_Prime;
function Image (N : in Long_Long_Integer;
Sign : in Boolean := False) return String is
Image : constant String := N'Image;
begin
if N < 0 then
return Image;
else
if Sign then
return "+" & Image (Image'First + 1 .. Image'Last);
else
return Image (Image'First + 1 .. Image'Last);
end if;
end if;
end Image;
procedure Show (Triangle : in Pascal_Triangle_Type) is
use Ada.Text_IO;
Begin
for I in reverse Triangle'Range loop
Put (Image (Triangle (I), Sign => True));
Put ("x^");
Put (Image (Long_Long_Integer (I)));
Put (" ");
end loop;
end Show;
procedure Show_Pascal_Triangles is
use Ada.Text_IO;
begin
for N in 0 .. 9 loop
declare
Pascal_Triangle : constant Pascal_Triangle_Type := Calculate_Pascal_Triangle (N);
begin
Put ("(x-1)^" & Image (Long_Long_Integer (N)) & " = ");
Show (Pascal_Triangle);
New_Line;
end;
end loop;
end Show_Pascal_Triangles;
procedure Show_Primes is
use Ada.Text_IO;
begin
for N in 2 .. 63 loop
if Is_Prime (N) then
Put (N'Image);
end if;
end loop;
New_Line;
end Show_Primes;
begin
Show_Pascal_Triangles; Show_Primes;
end Test_For_Primes;</lang>
- Output:
(x-1)^0 = +1x^0 (x-1)^1 = +1x^1 -1x^0 (x-1)^2 = +1x^2 -2x^1 +1x^0 (x-1)^3 = +1x^3 -3x^2 +3x^1 -1x^0 (x-1)^4 = +1x^4 -4x^3 +6x^2 -4x^1 +1x^0 (x-1)^5 = +1x^5 -5x^4 +10x^3 -10x^2 +5x^1 -1x^0 (x-1)^6 = +1x^6 -6x^5 +15x^4 -20x^3 +15x^2 -6x^1 +1x^0 (x-1)^7 = +1x^7 -7x^6 +21x^5 -35x^4 +35x^3 -21x^2 +7x^1 -1x^0 (x-1)^8 = +1x^8 -8x^7 +28x^6 -56x^5 +70x^4 -56x^3 +28x^2 -8x^1 +1x^0 (x-1)^9 = +1x^9 -9x^8 +36x^7 -84x^6 +126x^5 -126x^4 +84x^3 -36x^2 +9x^1 -1x^0 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61
ALGOL 68
The code below uses Algol 68 Genie which provides arbitrary precision arithmetic for LONG LONG modes.
<lang algol68> 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 </lang>
- Output:
(x-1)^0 = 1 (x-1)^1 = x - 1 (x-1)^2 = x^2 - 2x + 1 (x-1)^3 = x^3 - 3x^2 + 3x - 1 (x-1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1 (x-1)^5 = x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1 (x-1)^6 = x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1 (x-1)^7 = x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1 Primes < 50 are 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 And just to show off, the primes between 900 and 1000 are 907 911 919 929 937 941 947 953 967 971 977 983 991 997
ARM Assembly
<lang ARM Assembly> /* ARM assembly Raspberry PI or android 32 bits */ /* program AKS.s */
/* REMARK 1 : this program use routines in a include file
see task Include a file language arm assembly for the routine affichageMess conversion10 see at end of this program the instruction include */
/* for constantes see task include a file in arm assembly */ /************************************/ /* Constantes */ /************************************/ .include "../constantes.inc" .equ MAXI, 32 .equ NUMBERLOOP, 10
/*********************************/ /* Initialized data */ /*********************************/ .data szMessResult: .asciz " (x-1)^@ = " szMessResult1: .asciz " @ x^@ " szMessResPrime: .asciz "Number @ is prime. \n" szCarriageReturn: .asciz "\n"
/*********************************/ /* UnInitialized data */ /*********************************/ .bss sZoneConv: .skip 24 iTabCoef: .skip 4 * MAXI /*********************************/ /* code section */ /*********************************/ .text .global main main: @ entry of program
mov r4,#1
1: @ loop
mov r0,r4 bl computeCoef @ compute coefficient ldr r0,iAdriTabCoef mov r0,r4 bl displayCoef @ display coefficient add r4,r4,#1 cmp r4,#NUMBERLOOP blt 1b
mov r4,#1
2:
mov r0,r4 bl isPrime @ is prime ? cmp r0,#1 bne 3f mov r0,r4 ldr r1,iAdrsZoneConv bl conversion10 @ call decimal conversion add r1,r0 mov r5,#0 strb r5,[r1] ldr r0,iAdrszMessResPrime ldr r1,iAdrsZoneConv @ insert value conversion in message bl strInsertAtCharInc bl affichageMess
3:
add r4,r4,#1 cmp r4,#MAXI blt 2b
100: @ standard end of the program
mov r0, #0 @ return code mov r7, #EXIT @ request to exit program svc #0 @ perform the system call
iAdrszCarriageReturn: .int szCarriageReturn iAdrsZoneConv: .int sZoneConv iAdriTabCoef: .int iTabCoef iAdrszMessResPrime: .int szMessResPrime /***************************************************/ /* display coefficients */ /***************************************************/ // r0 contains a number displayCoef:
push {r1-r6,lr} @ save registers
mov r2,r0
ldr r1,iAdrsZoneConv @
bl conversion10 @ call decimal conversion
add r1,r0
mov r5,#0
strb r5,[r1]
ldr r0,iAdrszMessResult
ldr r1,iAdrsZoneConv @ insert value conversion in message
bl strInsertAtCharInc
bl affichageMess
ldr r3,iAdriTabCoef
1:
ldr r0,[r3,r2,lsl #2] ldr r1,iAdrsZoneConv @ bl conversion10S @ call decimal conversion
2: @ removing spaces
ldrb r6,[r1] cmp r6,#' ' addeq r1,#1 beq 2b
ldr r0,iAdrszMessResult1 bl strInsertAtCharInc mov r4,r0 mov r0,r2 ldr r1,iAdrsZoneConv @ else display odd message bl conversion10 @ call decimal conversion add r1,r0 mov r5,#0 strb r5,[r1] mov r0,r4 ldr r1,iAdrsZoneConv @ insert value conversion in message bl strInsertAtCharInc bl affichageMess subs r2,r2,#1 bge 1b ldr r0,iAdrszCarriageReturn bl affichageMess
100:
pop {r1-r6,lr} @ restaur registers
bx lr @ return
iAdrszMessResult: .int szMessResult iAdrszMessResult1: .int szMessResult1 /***************************************************/ /* compute coefficient */ /***************************************************/ // r0 contains a number computeCoef:
push {r1-r6,lr} @ save registers
ldr r1,iAdriTabCoef @ address coefficient array
mov r2,#1
str r2,[r1] @ store 1 to coeff [0]
mov r3,#0 @ indice 1
1:
add r4,r3,#1 mov r5,#1 str r5,[r1,r4,lsl #2] mov r6,r3 @ indice 2 = indice 1
2:
cmp r6,#0 @ zero ? -> end loop ble 3f sub r4,r6,#1 ldr r5,[r1,r4,lsl #2] ldr r4,[r1,r6,lsl #2] sub r5,r5,r4 str r5,[r1,r6,lsl #2] sub r6,r6,#1 b 2b
3:
ldr r2,[r1] @ inversion coeff [0] neg r2,r2 str r2,[r1] add r3,r3,#1 cmp r3,r0 blt 1b
100:
pop {r1-r6,lr} @ restaur registers
bx lr @ return
/***************************************************/ /* verify number is prime */ /***************************************************/ // r0 contains a number isPrime:
push {r1-r5,lr} @ save registers
bl computeCoef
ldr r4,iAdriTabCoef @ address coefficient array
ldr r2,[r4]
add r2,r2,#1
str r2,[r4]
ldr r2,[r4,r0,lsl #2]
sub r2,r2,#1
str r2,[r4,r0,lsl #2]
mov r5,r0 @ number start
mov r1,r0 @ divisor
1:
ldr r0,[r4,r5,lsl #2] @ load one coeff cmp r0,#0 @ if negative inversion neglt r0,r0 bl division @ because this routine is number positive only cmp r3,#0 @ remainder = zéro ? movne r0,#0 @ if <> no prime bne 100f subs r5,r5,#1 @ next coef bgt 1b mov r0,#1 @ prime
100:
pop {r1-r5,lr} @ restaur registers
bx lr @ return
/***************************************************/ /* ROUTINES INCLUDE */ /***************************************************/ .include "../affichage.inc" </lang>
- Output:
(x-1)^1 = +1 x^1 -1 x^0 (x-1)^2 = +1 x^2 -2 x^1 +1 x^0 (x-1)^3 = +1 x^3 -3 x^2 +3 x^1 -1 x^0 (x-1)^4 = +1 x^4 -4 x^3 +6 x^2 -4 x^1 +1 x^0 (x-1)^5 = +1 x^5 -5 x^4 +10 x^3 -10 x^2 +5 x^1 -1 x^0 (x-1)^6 = +1 x^6 -6 x^5 +15 x^4 -20 x^3 +15 x^2 -6 x^1 +1 x^0 (x-1)^7 = +1 x^7 -7 x^6 +21 x^5 -35 x^4 +35 x^3 -21 x^2 +7 x^1 -1 x^0 (x-1)^8 = +1 x^8 -8 x^7 +28 x^6 -56 x^5 +70 x^4 -56 x^3 +28 x^2 -8 x^1 +1 x^0 (x-1)^9 = +1 x^9 -9 x^8 +36 x^7 -84 x^6 +126 x^5 -126 x^4 +84 x^3 -36 x^2 +9 x^1 -1 x^0 Number 1 is prime. Number 2 is prime. Number 3 is prime. Number 5 is prime. Number 7 is prime. Number 11 is prime. Number 13 is prime. Number 17 is prime. Number 19 is prime. Number 23 is prime. Number 29 is prime. Number 31 is prime.
AutoHotkey
<lang autohotkey>; 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</lang>
- 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. <lang bracmat>( (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 );</lang> 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: <lang bracmat>( 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
)
)
);</lang> Output:
Primes between 980 and 1000, short version: 983 991 997
C
<lang 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; }</lang>
The ugly output:
(x-1)^0 = +1x^0 (x-1)^1 = +1x^1-1x^0 (x-1)^2 = +1x^2-2x^1+1x^0 (x-1)^3 = +1x^3-3x^2+3x^1-1x^0 (x-1)^4 = +1x^4-4x^3+6x^2-4x^1+1x^0 (x-1)^5 = +1x^5-5x^4+10x^3-10x^2+5x^1-1x^0 (x-1)^6 = +1x^6-6x^5+15x^4-20x^3+15x^2-6x^1+1x^0 (x-1)^7 = +1x^7-7x^6+21x^5-35x^4+35x^3-21x^2+7x^1-1x^0 (x-1)^8 = +1x^8-8x^7+28x^6-56x^5+70x^4-56x^3+28x^2-8x^1+1x^0 (x-1)^9 = +1x^9-9x^8+36x^7-84x^6+126x^5-126x^4+84x^3-36x^2+9x^1-1x^0 primes (never mind the 1): 1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61
C#
<lang csharp> 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); }
}
</lang>
C++
<lang cpp>
- 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;
} </lang>
- Output:
(x-1)^0 = 1 (x-1)^1 = x-1 (x-1)^2 = x^2-2*x+1 (x-1)^3 = x^3-3*x^2+3*x-1 (x-1)^4 = x^4-4*x^3+6*x^2-4*x+1 (x-1)^5 = x^5-5*x^4+10*x^3-10*x^2+5*x-1 (x-1)^6 = x^6-6*x^5+15*x^4-20*x^3+15*x^2-6*x+1 (x-1)^7 = x^7-7*x^6+21*x^5-35*x^4+35*x^3-21*x^2+7*x-1 (x-1)^8 = x^8-8*x^7+28*x^6-56*x^5+70*x^4-56*x^3+28*x^2-8*x+1 (x-1)^9 = x^9-9*x^8+36*x^7-84*x^6+126*x^5-126*x^4+84*x^3-36*x^2+9*x-1 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61
Clojure
The *' function is an arbitrary precision multiplication. <lang clojure>(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)))</lang>
- 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
<lang 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}"</lang>
- 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
<lang 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 "~%"))</lang>
- Output:
# p: (x-1)^p for small p: 0: +1 1: -1 +1X^1 2: +1 -2X^1 +1X^2 3: -1 +3X^1 -3X^2 +1X^3 4: +1 -4X^1 +6X^2 -4X^3 +1X^4 5: -1 +5X^1 -10X^2 +10X^3 -5X^4 +1X^5 6: +1 -6X^1 +15X^2 -20X^3 +15X^4 -6X^5 +1X^6 7: -1 +7X^1 -21X^2 +35X^3 -35X^4 +21X^5 -7X^6 +1X^7 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Crystal
<lang ruby>def x_minus_1_to_the(p)
p.times.reduce([1]) do |ex, _|
([0_i64] + ex).zip(ex + [0]).map { |x, y| x - y }
end
end
def prime?(p)
return false if p < 2
coeff = x_minus_1_to_the(p)[1..p//2] # only need half of coeff terms
coeff.all?{ |n| n%p == 0 }
end
8.times do |n|
puts "(x-1)^#{n} = " +
x_minus_1_to_the(n).map_with_index{ |c, p|
p.zero? ? c.to_s : (c < 0 ? " - " : " + ") + (c.abs == 1 ? "x" : "#{c.abs}x") + (p == 1 ? "" : "^#{p}")
}.join
end
puts "\nPrimes below 50:", 50.times.select { |n| prime? n }.join(',') </lang>
- 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
<lang d>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;
}</lang>
- 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]
Delphi
See #Pascal.
EchoLisp
We use the math.lib library and the poly functions to compute and display the required polynomials. A polynomial P(x) = a0 +a1*x + .. an*x^n is a list of coefficients (a0 a1 .... an). <lang lisp> (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] </lang>
- Output:
<lang lisp> (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 </lang>
Elena
ELENA 4.x : <lang elena>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()
}</lang>
- 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
<lang elixir>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
end
end
AKS.main</lang>
- 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
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.
<lang 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",
| {Row, N} <- zip(tl(tl(take(51, pascal()))), seq(2, 50)),
primerow(Row, N)).
</lang>
- 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
<lang factor>USING: combinators formatting io kernel make math math.parser math.polynomials prettyprint sequences ; IN: rosetta-code.aks-test
! Polynomials are represented by the math.polynomials vocabulary ! as sequences with the highest exponent on the right. Hence ! { -1 1 } represents x - 1.
- (x-1)^ ( n -- seq ) { -1 1 } swap p^ ;
- choose-exp ( n -- str )
{ { 0 [ "" ] } { 1 [ "x" ] } [ "x^%d" sprintf ] } case ;
- choose-coeff ( n -- str )
[ dup neg? [ neg "- " ] [ "+ " ] if % # ] "" make ;
- terms ( coeffs-seq -- terms-seq )
[ [ choose-coeff ] [ choose-exp append ] bi* ] map-index ;
- (.p) ( n -- str ) (x-1)^ terms <reversed> " " join 3 tail ;
- .p ( n -- ) dup zero? [ drop "1" ] [ (.p) ] if print ;
- show-poly ( n -- ) [ "(x-1)^%d = " printf ] [ .p ] bi ;
- part1 ( -- ) 8 <iota> [ show-poly ] each ;
- (prime?) ( n -- ? )
(x-1)^ rest but-last dup first [ mod 0 = not ] curry find nip not ;
- prime? ( n -- ? ) dup 2 < [ drop f ] [ (prime?) ] if ;
- part2 ( -- )
"Primes up to 50 via AKS:" print 50 <iota> [ prime? ] filter . ;
- aks-test ( -- ) part1 nl part2 ;
MAIN: aks-test</lang>
- 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
<lang forth>: coeffs ( u -- nu ... n0 ) \ coefficients of (x-1)^u
1 swap 1+ dup 1 ?do over over i - i */ negate swap loop drop ;
- prime? ( u -- f )
dup 2 < if drop false exit then
dup >r coeffs 1+
\ if not prime, this loop consumes at most half the coefficients, otherwise all
begin dup 1 <> while
r@ mod 0= while
repeat then rdrop
dup 1 = >r
begin 1 = until
r> ;
- .monom ( u1 u2 -- )
dup 0> if [char] + emit then 0 .r ?dup if ." x^" . else space then ;
- .poly ( u -- )
dup >r coeffs 0 r> 1+ 0 ?do
tuck swap .monom 1+
loop ;
- main
11 0 ?do i . ." : " i .poly cr loop cr 50 1 ?do i prime? if i . then loop cr ;</lang>
- 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
<lang 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 function
end program aks </lang>
- Output:
(x - 1)^0 = 1 (x - 1)^1 = - 1 + x (x - 1)^2 = 1 - 2x + x^2 (x - 1)^3 = - 1 + 3x - 3x^2 + x^3 (x - 1)^4 = 1 - 4x + 6x^2 - 4x^3 + x^4 (x - 1)^5 = - 1 + 5x - 10x^2 + 10x^3 - 5x^4 + x^5 (x - 1)^6 = 1 - 6x + 15x^2 - 20x^3 + 15x^4 - 6x^5 + x^6 (x - 1)^7 = - 1 + 7x - 21x^2 + 35x^3 - 35x^4 + 21x^5 - 7x^6 + x^7 2 3 5 7 11 13 17 19 23 29 31 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113
FreeBASIC
<lang 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</lang>
- Output:
(x-1)^1 = -1 +x (x-1)^2 = 1 -2x +x^2 (x-1)^3 = -1 +3x -3x^2 +x^3 (x-1)^4 = 1 -4x +6x^2 -4x^3 +x^4 (x-1)^5 = -1 +5x -10x^2 +10x^3 -5x^4 +x^5 (x-1)^6 = 1 -6x +15x^2 -20x^3 +15x^4 -6x^5 +x^6 (x-1)^7 = -1 +7x -21x^2 +35x^3 -35x^4 +21x^5 -7x^6 +x^7 (x-1)^8 = 1 -8x +28x^2 -56x^3 +70x^4 -56x^5 +28x^6 -8x^7 +x^8 (x-1)^9 = -1 +9x -36x^2 +84x^3 -126x^4 +126x^5 -84x^6 +36x^7 -9x^8 +x^9 Primes up to 70: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67
Go
<lang 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
}</lang>
- 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
Haskell
<lang haskell>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])</lang>
- 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
<lang idris>import Data.Vect
-- Computes Binomial Coefficients binCoef : Nat -> Nat -> Nat binCoef _ Z = (S Z) binCoef (S n) (S k) =
if n == k then (S Z) else ((S n) * (binCoef n k)) `div` (S k)
-- Binomial Expansion Of (x - 1)^p expansion : (n : Nat) -> Vect (S n) Integer expansion n = expansion' n 1
where
expansion' : (n : Nat) -> Integer -> Vect (S n) Integer
expansion' (S m) s = s * (toIntegerNat $ binCoef n (n `minus` (S m))) ::
expansion' m (s * -1)
expansion' Z s = [s]
showExpansion : Vect n Integer -> String
showExpansion [] = " "
showExpansion (x::xs) {n = S k} = (if x < 0 then "-" else "") ++
term x k ++ showExpansion' xs
where
term : Integer -> Nat -> String
term x n = if n == 0 then (show (abs x)) else
(if (abs x) == 1 then "" else
(show (abs x))) ++ "x" ++
(if n == 1 then "" else "^" ++ show n)
sign : Integer -> String
sign x = if x >= 0 then " + " else " - "
showExpansion' : Vect m Integer -> String
showExpansion' [] = ""
showExpansion' (y::ys) {m = S k} = sign y ++ term y k ++
showExpansion' ys
natToFin' : (m : Nat) -> Fin (S m)
natToFin' n with (natToFin n (S n))
natToFin' n | Just y = y
isPrime : Nat -> Bool
isPrime Z = False
isPrime (S Z ) = False
isPrime n = foldl (\divs, term => divs && (term `mod` (toIntegerNat n)) == 0)
True (fullExpansion $ expansion n)
-- (x - 1)^p - ((x^p) - 1)
where fullExpansion : Vect (S m) Integer -> Vect (S m) Integer
fullExpansion (x::xs) {m} = updateAt (natToFin' m) (+1) $ (x-1)::xs
printExpansions : Nat -> IO ()
printExpansions n = do
putStrLn "-- p: (x-1)^p for small p"
sequence_ $ map printExpansion [0..n]
where printExpansion : Nat -> IO ()
printExpansion n = do
print n
putStr ": "
putStrLn $ showExpansion $ expansion n
main : IO()
main = do
printExpansions 10 putStrLn "\n-- Primes Up To 100:" putStrLn $ show $ filter isPrime [0..100]</lang>
- 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:<lang j> 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.</lang>
Examples:<lang j> 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</lang>
Java
Solution:<lang java>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();
}
}</lang> 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
<lang javascript>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);</lang>
- 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): <lang javascript>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' + e + '')(deg-i) } return '(x-1)' + deg + ' =' + 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()), '
')
document.write('
Primes: ');
for (var coef=pascal(2), n=2; n<=50; n+=1) if (isPrime(coef())) document.write(' ', n)</lang>
- 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
<lang JavaScript>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'+n+''); 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)',n,' = ', show(coef(n)), '
')
document.write('
Primes: ');
for (var n=2; n<=50; n++) if (isPrime(n)) document.write(' ', n)</lang>
- 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
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 <lang jq># 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 OptPascalRow
def next_optpascal: [1] + add_pairs;
- 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);</lang>
Task 1: "A method to generate the coefficients of (x-1)^p"
<lang jq>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;</lang>
Task 2: "Show here the polynomial expansions of (x − 1)^p for p in the range 0 to at least 7, inclusive." <lang jq>range(0;8) | "Coefficient for (x - 1)^\(.): \(coefficients)"</lang>
- Output:
<lang sh>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]</lang>
Task 3: Prime Number Test
For brevity, we show here only the relatively efficient solution based on optpascal/0: <lang jq>def is_prime:
. as $N | if . < 2 then false else (1+.) | optpascal | all( .[2:][]; . % $N == 0 ) end;</lang>
Task 4: "Use your AKS test to generate a list of all primes under 35." <lang jq>range(0;36) | select(is_prime)</lang>
- Output:
<lang sh>2 3 5 7 11 13 17 19 23 29 31</lang>
Task 5: "As a stretch goal, generate all primes under 50." <lang ja>[range(0;50) | select(is_prime)]</lang>
- Output:
<lang sh>[2,3,5,7,11,13,17,19,23,29,31,37,41,43,47]</lang>
Julia
Task 1
<lang Julia> 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 </lang>
Perhaps this should be done with a comprehension, but properly accounting for the sign is tricky in that case.
Task 2
<lang Julia>using Printf
function stringpoly(n::Int64)
if n < 0
return ""
end
st = @sprintf "(x - 1)^{%d} & = & " n
for (i, c) in enumerate(polycoefs(n))
if i == 1
op = ""
ac = c
elseif c < 0
op = "-"
ac = abs(c)
else
op = "+"
ac = abs(c)
end
p = n + 1 - i
if p == 0
st *= @sprintf " %s %d\\\\" op ac
elseif ac == 1
st *= @sprintf " %s x^{%d}" op p
else
st *= @sprintf " %s %dx^{%d}" op ac p
end
end
return st
end </lang>
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
<lang Julia> 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 </lang>
Task 4
<lang Julia> println("Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ") 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() </lang>
- Output:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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
<lang scala>// 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)
}</lang>
- 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]
Lua
<lang lua>-- AKS test for primes, in Lua, 6/23/2020 db local function coefs(n)
local list = {[0]=1}
for k = 0, n do list[k+1] = math.floor(list[k] * (n-k) / (k+1)) end
for k = 1, n, 2 do list[k] = -list[k] end
return list
end
local function isprimeaks(n)
local c = coefs(n) c[0], c[n] = c[0]-1, c[n]+1 for i = 0, n do if (c[i] % n ~= 0) then return false end end return true
end
local function pprintcoefs(n, list)
local result = ""
for i = 0, n do
local s = i==0 and "" or list[i]>=0 and " + " or " - "
local c, e = math.abs(list[i]), n-i
if (c==1 and e > 0) then c = "" end
local x = e==0 and "" or e==1 and "x" or "x^"..e
result = result .. s .. c .. x
end
print("(x-1)^" .. n .." : " .. result)
end
for i = 0, 9 do
pprintcoefs(i, coefs(i))
end
local primes = {} for i = 2, 53 do
if (isprimeaks(i)) then primes[#primes+1] = i end
end print(table.concat(primes, ", "))</lang>
- Output:
(x-1)^0 : 1 (x-1)^1 : x - 1 (x-1)^2 : x^2 - 2x + 1 (x-1)^3 : x^3 - 3x^2 + 3x - 1 (x-1)^4 : x^4 - 4x^3 + 6x^2 - 4x + 1 (x-1)^5 : x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1 (x-1)^6 : x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1 (x-1)^7 : x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1 (x-1)^8 : x^8 - 8x^7 + 28x^6 - 56x^5 + 70x^4 - 56x^3 + 28x^2 - 8x + 1 (x-1)^9 : x^9 - 9x^8 + 36x^7 - 84x^6 + 126x^5 - 126x^4 + 84x^3 - 36x^2 + 9x - 1 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53
Lambdatalk
<lang Scheme> {require lib_BN} // for big numbers
1) pascalian binomial coefficient C(n,p) = n!/(p!(n-p)!) = (n*(n-1)...(n-p+1))/(p*(p-1)...2*1)
{def coeff
{lambda {:n :p}
{BN.intPart
{BN./ {S.reduce BN.* {S.serie :n {- :n :p -1} -1}}
{S.reduce BN.* {S.serie :p 1 -1}}}}}}
-> coeff
2) polynomial expansions of (x − 1)^p
{def sign
{lambda {:n}
{if {= {% :n 2} 0} then + else -}}}
-> sign
{def coeffs
{lambda {:n}
{br}(x - 1)^:n =
{if {= :n 0}
then + 1x^0
else {if {= :n 1}
then + 1x^1 - 1x^0
else {sign 0} 1x^:n
{S.map {{lambda {:p :n} {sign {- :p :n}} {coeff :p :n}x^{- :p :n}} :n}
{S.serie 1 {- :n 1}}}
{sign :n} 1x^0}}}}
-> coeffs
{S.map coeffs {S.serie 0 7}} -> (x - 1)^0 = + 1x^0 (x - 1)^1 = + 1x^1 - 1x^0 (x - 1)^2 = + 1x^2 - 2x^1 + 1x^0 (x - 1)^3 = + 1x^3 + 3x^2 - 3x^1 - 1x^0 (x - 1)^4 = + 1x^4 - 4x^3 + 6x^2 - 4x^1 + 1x^0 (x - 1)^5 = + 1x^5 + 5x^4 - 10x^3 + 10x^2 - 5x^1 - 1x^0 (x - 1)^6 = + 1x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x^1 + 1x^0 (x - 1)^7 = + 1x^7 + 7x^6 - 21x^5 + 35x^4 - 35x^3 + 21x^2 - 7x^1 - 1x^0
3) primality test
Taking into account the symmetry of the list of coefficients and the uselessness of the sign in the calculation of the divisibility, one can limit the tests to half of the list, and define a simplified function, aks_coeffs:
{def aks_coeffs
{lambda {:n}
{S.map {coeff :n} {S.serie 1 {+ {/ {- :n 1} 2} 1}}}}}
-> aks_coeffs
{def divide
{lambda {:a :b}
{= {BN.compare {BN.% :b :a} 0} 0}}}
-> divide
{def isprime
{lambda {:n}
{if {and {S.map {divide :n} {aks_coeffs :n}}} then :n else .}}}
-> isprime
{S.map isprime {S.serie 2 100}} -> 2 3 . 5 . 7 . . . 11 . 13 . . . 17 . 19 . . . 23 . . . . . 29 . 31 . . . . . 37 . . . 41 . 43 . . . 47 . . . . . 53 . . . . . 59 . 61 . . . . . 67 . . . 71 . 73 . . . . . 79 . . . 83 . . . . . 89 . . . . . . . 97 . . . </lang>
Liberty BASIC
<lang lb> global pasTriMax pasTriMax = 61 dim pasTri(pasTriMax + 1)
for n = 0 to 9
call expandPoly n
next n for n = 2 to pasTriMax
if isPrime(n) <> 0 then
print using("###", n);
end if
next n print end
sub expandPoly n
n = int(n)
dim vz$(1)
vz$(0) = "+"
vz$(1) = "-"
if n > pasTriMax then
print n; " is out of range"
end
end if
select case n
case 0
print "(x-1)^0 = 1"
case 1
print "(x-1)^1 = x-1"
case else
call pascalTriangle n
print "(x-1)^"; n; " = ";
print "x^"; n;
bVz = 1
nDiv2 = int(n / 2)
for j = n - 1 to nDiv2 + 1 step -1
print vz$(bVz); pasTri(n - j); "*x^"; j;
bVz = abs(1 - bVz)
next j
for j = nDiv2 to 2 step -1
print vz$(bVz); pasTri(j); "*x^"; j;
bVz = abs(1 - bVz)
next j
print vz$(bVz); pasTri(1); "*x";
bVz = abs(1 - bVz)
print vz$(bVz); pasTri(0)
end select
end sub
function isPrime(n)
n = int(n) if n > pasTriMax then print n; " is out of range" end end if call pascalTriangle n res = 1 i = int(n / 2) while res and (i > 1) res = res and (pasTri(i) mod n = 0) i = i - 1 wend isPrime = res
end function
sub pascalTriangle n
rem Calculate the n'th line 0.. middle
n = int(n)
pasTri(0) = 1
j = 1
while j <= n
j = j + 1
k = int(j / 2)
pasTri(k) = pasTri(k - 1)
for k = k to 1 step -1
pasTri(k) = pasTri(k) + pasTri(k - 1)
next k
wend
end sub </lang>
- 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. <lang Maple>> 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
</lang>
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.
<lang Maple> polc := p -> [coeffs]( expand( (x-1)^p - (x^p-1) ) ):</lang>
Use polc to implement prime? which does the primality test.
<lang Maple>prime? := n -> n > 1 and {op}( map( modp, polc( n ), n ) ) = {0}</lang>
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):
<lang Maple>prime? := (n::posint) -> n > 1 and {op}( map( modp, [coeffs]( expand( (x-1)^n - (x^n-1) ) ), n ) ) = {0}</lang>
This agrees with the built-in primality test isprime:
<lang Maple>> evalb( seq( prime?(i), i = 1 .. 1000 ) = seq( isprime( i ), i = 1 .. 1000 ) );
true
</lang>
Use prime? with the built-in Maple select procedure to pick off the primes up to 50:
<lang Maple>> select( prime?, [seq](1..50) );
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]
</lang>
Mathematica / Wolfram Language
Algebraic manipulation is built into Mathematica, so there's no need to create a function to do (x-1)^p <lang Mathematica>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]</lang>
- 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}
Nim
<lang Nim> from math import binom import strutils
- Table of unicode superscript characters.
const Exponents: array[0..9, string] = ["⁰", "¹", "²", "³", "⁴", "⁵", "⁶", "⁷", "⁸", "⁹"]
iterator coeffs(n: int): int =
## Yield the coefficients of the expansion of (x - 1)ⁿ. var sign = 1 for k in 0..n: yield binom(n, k) * sign sign = -sign
iterator polyExpansion(n: int): tuple[c, e: int] =
## Yield the coefficients and the exponents of the expansion of (x - 1)ⁿ. var e = n for c in coeffs(n): yield(c, e) dec e
proc termString(c, e: int): string =
## Return the string for the term c * e^n.
if e == 0:
result.addInt(c)
else:
if c != 1:
result.addInt(c)
result.add('x')
if e != 1:
result.add(Exponents[e])
proc polyString(n: int): string =
## Return the string for the expansion of (x - 1)ⁿ.
for (c, e) in polyExpansion(n):
if c < 0:
result.add(" - ")
elif e != n:
result.add(" + ")
result.add(termString(abs(c), e))
proc isPrime(n: int): bool =
## Check if a number is prime using the polynome expansion.
result = true
for (c, e) in polyExpansion(n):
if e in 1..(n-1): # xⁿ and 1 are eliminated by the subtraction.
if c mod n != 0:
return false
- ---------------------------------------------------------------------------------------------------
echo "Polynome expansions:" for p in 0..9:
echo "(x - 1)$1 = $2".format(Exponents[p], polyString(p))
var primes: string for p in 2..34:
if p.isPrime():
primes.addSep(", ", 0)
primes.addInt(p)
echo "\nPrimes under 35: ", primes
for p in 35..50:
if p.isPrime():
primes.add(", ")
primes.addInt(p)
echo "\nPrimes under 50: ", primes </lang>
- Output:
Polynome expansions: (x - 1)⁰ = 1 (x - 1)¹ = x - 1 (x - 1)² = x² - 2x + 1 (x - 1)³ = x³ - 3x² + 3x - 1 (x - 1)⁴ = x⁴ - 4x³ + 6x² - 4x + 1 (x - 1)⁵ = x⁵ - 5x⁴ + 10x³ - 10x² + 5x - 1 (x - 1)⁶ = x⁶ - 6x⁵ + 15x⁴ - 20x³ + 15x² - 6x + 1 (x - 1)⁷ = x⁷ - 7x⁶ + 21x⁵ - 35x⁴ + 35x³ - 21x² + 7x - 1 (x - 1)⁸ = x⁸ - 8x⁷ + 28x⁶ - 56x⁵ + 70x⁴ - 56x³ + 28x² - 8x + 1 (x - 1)⁹ = x⁹ - 9x⁸ + 36x⁷ - 84x⁶ + 126x⁵ - 126x⁴ + 84x³ - 36x² + 9x - 1 Primes under 35: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31 Primes under 50: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
Objeck
<lang objeck>class AksTest {
@c : static : Int[];
function : Main(args : String[]) ~ Nil {
@c := Int->New[100];
for(n := 0; n < 10; n++;) {
Coef(n);
"(x-1)^ {$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);
}
}</lang>
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
Uses gen library for lazy streams and zarith for arbitrarily sized integers. Runs as is through the utop REPL. <lang OCaml>#require "gen"
- require "zarith"
open Z let range ?(step=one) i j = if i = j then Gen.empty else Gen.unfold (fun k ->
if compare i j = compare k j then Some (k, (add step k)) else None) i
(* kth coefficient of (x - 1)^n *) let coeff n k =
let numer = Gen.fold mul one
(range n (sub n k) ~step:minus_one) in
let denom = Gen.fold mul one
(range k zero ~step:minus_one) in
div numer denom |> mul @@
if
compare k n < 0 && is_even k
then
minus_one
else
one
(* coefficient series for (x - 1)^n, k=[0..n] *) let coeff_series n =
Gen.map (coeff n) (range zero (succ n))
let middle g = Gen.drop 1 g |> Gen.peek |> Gen.filter_map
(function (_, None) -> None | (e, _) -> Some e)
let is_mod_p ~p n = rem n p = zero
let aks p =
coeff_series p |> middle |> Gen.for_all (is_mod_p ~p)
let _ =
print_endline "coefficient series n (k[0] .. k[n])";
Gen.iter
(fun n -> Format.printf "%d (%s)\n" (to_int n)
(Gen.map to_string (coeff_series n) |> Gen.to_list |> String.concat " "))
(range zero (of_int 10));
print_endline "";
print_endline ("primes < 50 per AKS: " ^
(Gen.filter aks (range (of_int 2) (of_int 50)) |>
Gen.map to_string |> Gen.to_list |> String.concat " "))</lang>
- 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
<lang 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
- </lang>
- 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
<lang parigp>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])</lang>
- 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
<lang pascal> const
pasTriMax = 61;
type
TPasTri = array[0 .. pasTriMax] of UInt64;
var
pasTri: TPasTri;
procedure PascalTriangle(n: LongWord); // Calculate the n'th line 0.. middle var
j, k: LongWord;
begin
pasTri[0] := 1;
j := 1;
while j <= n do
begin
Inc(j);
k := j div 2;
pasTri[k] := pasTri[k - 1];
for k := k downto 1 do
Inc(pasTri[k], pasTri[k - 1]);
end;
end;
function IsPrime(n: LongWord): Boolean; var
i: Integer;
begin
if n > pasTriMax then begin WriteLn(n, ' is out of range'); Halt; end;
PascalTriangle(n); Result := true; i := n div 2; while Result and (i > 1) do begin Result := Result and (pasTri[i] mod n = 0); Dec(i); end;
end;
procedure ExpandPoly(n: LongWord); const
Vz: array[Boolean] of Char = ('+', '-');
var
j: LongWord; bVz: Boolean;
begin
if n > pasTriMax then begin WriteLn(n,' is out of range'); Halt; end;
case n of
0: WriteLn('(x-1)^0 = 1');
1: WriteLn('(x-1)^1 = x-1');
else
PascalTriangle(n);
Write('(x-1)^', n, ' = ');
Write('x^', n);
bVz := true;
for j := n - 1 downto n div 2 + 1 do
begin
Write(vz[bVz], pasTri[n - j], '*x^', j);
bVz := not bVz;
end;
for j := n div 2 downto 2 do
begin
Write(vz[bVz], pasTri[j], '*x^', j);
bVz := not bVz;
end;
Write(vz[bVz], pasTri[1], '*x');
bVz := not bVz;
WriteLn(vz[bVz], pasTri[0]);
end;
end;
var
n: LongWord;
begin
for n := 0 to 9 do
ExpandPoly(n);
for n := 2 to pasTriMax do
if IsPrime(n) then
Write(n:3);
WriteLn;
end.</lang>
- 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
<lang 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";</lang>
- 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.
<lang perl>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);</lang>
- Output:
1000000007 1000000009 1000000021 1000000033 1000000087 1000000093 1000000097
Phix
-- demo/rosetta/AKSprimes.exw -- Does not work for primes above 53, which is actually beyond the original task anyway. -- Translated from the C version, just about everything is (working) out-by-1, what fun. sequence c = repeat(0,100) procedure coef(integer n) -- out-by-1, ie coef(1)==^0, coef(2)==^1, coef(3)==^2 etc. c[n] = 1 for i=n-1 to 2 by -1 do c[i] = c[i]+c[i-1] end for end procedure function is_aks_prime(integer n) coef(n+1); -- (I said it was out-by-1) for i=2 to n-1 do -- (technically "to n" is more correct) if remainder(c[i],n)!=0 then return 0 end if end for return 1 end function procedure show(integer n) -- (As per coef, this is (working) out-by-1) object ci for i=n to 1 by -1 do ci = c[i] if ci=1 then if remainder(n-i,2)=0 then if i=1 then if n=1 then ci = "1" else ci = "+1" end if else ci = "" end if else ci = "-1" end if else if remainder(n-i,2)=0 then ci = sprintf("+%d",ci) else ci = sprintf("-%d",ci) end if end if if i=1 then -- ie ^0 printf(1,"%s",{ci}) elsif i=2 then -- ie ^1 printf(1,"%sx",{ci}) else printf(1,"%sx^%d",{ci,i-1}) end if end for end procedure procedure main() for n=1 to 10 do -- (0 to 9 really) coef(n); printf(1,"(x-1)^%d = ", n-1); show(n); puts(1,'\n'); end for puts(1,"\nprimes (<=53):"); -- coef(2); -- (needed to reset c, if we want to avoid saying 1 is prime...) c[2] = 1 -- (this manages "", which is all that call did anyway...) for n = 2 to 53 do if is_aks_prime(n) then printf(1," %d", n); end if end for puts(1,'\n'); if getc(0) then end if end procedure main()
- Output:
(x-1)^0 = 1 (x-1)^1 = x-1 (x-1)^2 = x^2-2x+1 (x-1)^3 = x^3-3x^2+3x-1 (x-1)^4 = x^4-4x^3+6x^2-4x+1 (x-1)^5 = x^5-5x^4+10x^3-10x^2+5x-1 (x-1)^6 = x^6-6x^5+15x^4-20x^3+15x^2-6x+1 (x-1)^7 = x^7-7x^6+21x^5-35x^4+35x^3-21x^2+7x-1 (x-1)^8 = x^8-8x^7+28x^6-56x^5+70x^4-56x^3+28x^2-8x+1 (x-1)^9 = x^9-9x^8+36x^7-84x^6+126x^5-126x^4+84x^3-36x^2+9x-1 primes (<=53): 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53
Picat
<lang Picat> pascal([]) = [1]. pascal(L) = [1|sum_adj(L)].
sum_adj(Row) = Next =>
Next = L,
while (Row = [A,B|_])
L = [A+B|Rest],
L := Rest,
Row := tail(Row)
end,
L = Row.
show_x(0) = "". show_x(1) = "x". show_x(N) = S, N > 1 => S = [x, '^' | to_string(N)].
show_term(Coef, Exp) = cond((Coef != 1; Exp == 0), Coef.to_string, "") ++ show_x(Exp).
expansions(N) =>
Row = [],
foreach (I in 0..N-1)
Row := pascal(Row),
writef("(x - 1)^%d = ", I),
Exp = I,
Sgn = '+',
foreach (Coef in Row)
if Exp != I then
writef(" %w ", Sgn)
end,
writef("%s", show_term(Coef, Exp)),
Exp := Exp - 1,
Sgn := cond(Sgn == '+', '-', '+')
end,
nl
end.
primerow([], _). primerow([A,A|_], _). % end when we've seen half the list. primerow([A|As], N) :- (A mod N == 0; A == 1), primerow(As, N).
primes_upto(N) = Primes =>
Primes = L,
Row = [1, 1],
foreach (K in 2..N)
Row := pascal(Row),
if primerow(Row, K) then
L = [K|Rest],
L := Rest
end
end,
L = [].
main =>
expansions(8),
writef("%nThe primes upto 50 (via AKS) are: %w%n", primes_upto(50)).
</lang>
- Output:
(x - 1)^0 = 1 (x - 1)^1 = x - 1 (x - 1)^2 = x^2 - 2x + 1 (x - 1)^3 = x^3 - 3x^2 + 3x - 1 (x - 1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1 (x - 1)^5 = x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1 (x - 1)^6 = x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1 (x - 1)^7 = x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1 The primes upto 50 (via AKS) are: [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47]
PicoLisp
<lang>(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)</lang>
- 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
<lang 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; </lang> 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: <lang prolog>
prime(P) :- pascal([1,P|Xs]), append(Xs, [1], Rest), forall( member(X,Xs), 0 is X mod P).
</lang> 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
<lang prolog> % 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(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).
</lang>
Solutions
Solutions with output from SWI-Prolog:
<lang 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.
</lang>
[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.
<lang prolog> %%% 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).
</lang>
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):
<lang prolog>
prime(N) :- optpascal([1,N|Xs]), forall( member(X,Xs), 0 is X mod N).
</lang>
<lang prolog> %%% 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 </lang>
Alternate version using symbolic computation
This version uses Prolog's symbolic pattern matching to generate the polynomial equations that are the subject of the first part of the task. The rows of Pascal's triangle are generated, then combined with the "a" and "b" terms of the expression (a + b)^n. The expression is then algebraically simplified to remove things like x^0, x^1, 1*x, a + 0, etc.
It would be possible to go further and expand (a + b)^n and collect terms, but that turns out to be rather difficult because terms have to be reordered into a canonical form first -- for example, an expansion can produce both j*a*b and k*b*a, and these terms would have to be reordered to get (j+k)*a*b. <lang Prolog> main :- task1(8), nl, task2(50), halt.
task1(N) :-
pascal(Z),
length(Rows, N),
prefix(Rows, Z),
forall(member(Row, Rows),
(length(Row, K), succ(DecK, K),
binomial(x, -1, Row, Expr),
format("(x-1)**~w = ~w~n", [DecK, Expr]))).
task2(Upto) :-
primes_upto(Upto, Ps),
format("The primes upto ~w (via AKS) are: ~p~n", [Upto, Ps]).
pascal(Lz) :-
lazy_list(pascal_row, [], Lz).
pascal_row([], R1, R1) :- R1 = [1], !. pascal_row(R0, R1, R1) :-
sum_adj(R0, Next), R1 = [1|Next].
sum_adj(L, L) :- L = [_], !. sum_adj([A|As], [C|Cs]) :-
As = [B|_], C is A + B, sum_adj(As, Cs).
% First part of task -- create textual representation of (x-1)^n % here we generate expression trees % binomial(A, B, Coefs, Expr) :-
length(Coefs, N), succ(DecN, N), binomial(B, DecN, A, 0, Coefs, Exp0), reduce(Exp0, Exp1), addition_to_subtraction(Exp1, Expr).
binomial(_, _, _, _, [], 0) :- !. binomial(A, PowA, B, PowB, [N|Ns], Ts + T) :-
T = N * A**PowA * B**PowB, IncPow is PowB + 1, DecPow is PowA - 1, binomial(A, DecPow, B, IncPow, Ns, Ts).
addition_to_subtraction(A + B, X) :-
addition_to_subtraction(A, C), (make_positive(B, D) -> X = C - D; X = C + B), !.
addition_to_subtraction(X, X).
make_positive(N, Term) :- integer(N), N < 0, !, Term is -N. make_positive(A*B, Term) :-
make_positive(A, PosA), (PosA = 1 -> Term = B, !; Term = PosA*B).
reduce(A, C) :-
simplify(A, B), (B = A -> C = A; reduce(B, C)).
simplify(_**0, 1) :- !. simplify(1**_, 1) :- !. simplify(-1**N, Z) :- integer(N), (0 is N /\ 1 -> Z = 1; Z = -1), !. simplify(X**1, X) :- !.
simplify(0 + A, A) :- !. simplify(A + 0, A) :- !. simplify(A + B, C) :-
integer(A), integer(B), !, C is A + B.
simplify(A + B, C + D) :- !,
simplify(A, C), simplify(B, D).
simplify(0 * _, 0) :- !. simplify(_ * 0, 0) :- !. simplify(1 * A, A) :- !. simplify(A * 1, A) :- !. simplify(A * B, C) :-
integer(A), integer(B), !, C is A * B.
simplify(A * B, C * D) :- !,
simplify(A, C), simplify(B, D).
simplify(X, X).
% Second part of task -- Use the coefficients of Pascal's Triangle to check primality. %
primerow([1, N| Rest]) :- primerow(N, Rest).
primerow(_, End) :- (End = []; End = [1]), !. primerow(_, [A,A|_]) :- !. % end when we've seen half the list. primerow(N, [A|As]) :- A mod N =:= 0, primerow(N, As).
second([_,N|_], N).
primes_upto(N, Ps) :-
pascal(Z), Z = [_, _ | Rows], % we only care about 2nd row on up. ([1,2,1]) succ(DecN, N), length(CheckRows, DecN), prefix(CheckRows, Rows), include(primerow, CheckRows, PrimeRows), maplist(second, PrimeRows, Ps).
?- main. </lang>
- Output:
$ swipl aks.pl
(x-1)**0 = 1
(x-1)**1 = x-1
(x-1)**2 = x**2-2*x+1
(x-1)**3 = x**3-3*x**2+3*x-1
(x-1)**4 = x**4-4*x**3+6*x**2-4*x+1
(x-1)**5 = x**5-5*x**4+10*x**3-10*x**2+5*x-1
(x-1)**6 = x**6-6*x**5+15*x**4-20*x**3+15*x**2-6*x+1
(x-1)**7 = x**7-7*x**6+21*x**5-35*x**4+35*x**3-21*x**2+7*x-1
The primes upto 50 (via AKS) are: [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47]
</lang>
=={{header|PureBasic}}==
<lang purebasic>EnableExplicit
Define vzr.b = -1, vzc.b = ~vzr, nMAX.i = 10, n.i , k.i
Procedure coeff(nRow.i, Array pd.i(2))
Define n.i, k.i
For n=1 To nRow
For k=0 To n
If k=0 Or k=n : pd(n,k)=1 : Continue : EndIf
pd(n,k)=pd(n-1,k-1)+pd(n-1,k)
Next
Next
EndProcedure
Procedure.b isPrime(n.i, Array pd.i(2))
Define m.i
For m=1 To n-1
If Not pd(n,m) % n = 0 : ProcedureReturn #False : EndIf
Next
ProcedureReturn #True
EndProcedure
Dim pd.i(nMAX,nMAX)
pd(0,0)=1 : coeff(nMAX, pd())
OpenConsole()
For n=0 To nMAX
Print(RSet(Str(n),3,Chr(32))+": ")
If vzr : Print("+") : Else : Print("-") : EndIf
For k=0 To n
If k>0 : If vzc : Print("+") : Else : Print("-") : EndIf : vzc = ~vzc : EndIf
Print(RSet(Str(pd(n,k)),3,Chr(32))+Space(3))
Next
PrintN("")
vzr = ~vzr : vzc = ~vzr
Next
PrintN("")
nMAX=50 : Dim pd.i(nMAX,nMAX)
Print("Primes n<=50 : ") : coeff(nMAX, pd())
For n=2 To 50
If isPrime(n,pd()) : Print(Str(n)+Space(2)) : EndIf
Next
Input()</lang>
{{out}}
<pre> 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
<lang 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</lang>
or equivalently: <lang python>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</lang>
alternatively: <lang python>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)])</lang>
- Output:
# p: (x-1)^p for small p 0: +1 1: -1 +1x^1 2: +1 -2x^1 +1x^2 3: -1 +3x^1 -3x^2 +1x^3 4: +1 -4x^1 +6x^2 -4x^3 +1x^4 5: -1 +5x^1 -10x^2 +10x^3 -5x^4 +1x^5 6: +1 -6x^1 +15x^2 -20x^3 +15x^4 -6x^5 +1x^6 7: -1 +7x^1 -21x^2 +35x^3 -35x^4 +21x^5 -7x^6 +1x^7 8: +1 -8x^1 +28x^2 -56x^3 +70x^4 -56x^5 +28x^6 -8x^7 +1x^8 9: -1 +9x^1 -36x^2 +84x^3 -126x^4 +126x^5 -84x^6 +36x^7 -9x^8 +1x^9 10: +1 -10x^1 +45x^2 -120x^3 +210x^4 -252x^5 +210x^6 -120x^7 +45x^8 -10x^9 +1x^10 11: -1 +11x^1 -55x^2 +165x^3 -330x^4 +462x^5 -462x^6 +330x^7 -165x^8 +55x^9 -11x^10 +1x^11 # small primes using the aks test [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
Python: Output formatted for wiki
Using a wikitable and math features with the following additional code produces better formatted polynomial output:
<lang python>print(
for p in range(12): print('! Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle %i} \n| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle %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('|}')</lang>- Output:
| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x-1)^p} |
|---|
| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x-1)^p} | Prime(p)? |
|---|---|---|
| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +1} | False |
| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1 +x} | False |
| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +1 -2x +x^{2}} | True |
| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3} | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1 +3x -3x^{2} +x^{3}} | True |
| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4} | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +1 -4x +6x^{2} -4x^{3} +x^{4}} | False |
| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5} | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1 +5x -10x^{2} +10x^{3} -5x^{4} +x^{5}} | True |
| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 6} | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +1 -6x +15x^{2} -20x^{3} +15x^{4} -6x^{5} +x^{6}} | False |
| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 7} | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1 +7x -21x^{2} +35x^{3} -35x^{4} +21x^{5} -7x^{6} +x^{7}} | True |
| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 8} | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +1 -8x +28x^{2} -56x^{3} +70x^{4} -56x^{5} +28x^{6} -8x^{7} +x^{8}} | False |
| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 9} | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1 +9x -36x^{2} +84x^{3} -126x^{4} +126x^{5} -84x^{6} +36x^{7} -9x^{8} +x^{9}} | False |
| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10} | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +1 -10x +45x^{2} -120x^{3} +210x^{4} -252x^{5} +210x^{6} -120x^{7} +45x^{8} -10x^{9} +x^{10}} | False |
| Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 11} | Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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).
<lang R>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."))
}
}</lang>
Racket
With copious use of the math/number-theory library...
<lang racket>#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)) </lang>
- Output:
p=0: 1 p=1: x^1 - 1 p=2: x^2 - 2x^1 + 1 p=3: x^3 - 3x^2 + 3x^1 - 1 p=4: x^4 - 4x^3 + 6x^2 - 4x^1 + 1 p=5: x^5 - 5x^4 + 10x^3 - 10x^2 + 5x^1 - 1 p=6: x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x^1 + 1 p=7: x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x^1 - 1 p=8: x^8 - 8x^7 + 28x^6 - 56x^5 + 70x^4 - 56x^3 + 28x^2 - 8x^1 + 1 p=9: x^9 - 9x^8 + 36x^7 - 84x^6 + 126x^5 - 126x^4 + 84x^3 - 36x^2 + 9x^1 - 1 p=10: x^10 - 10x^9 + 45x^8 - 120x^7 + 210x^6 - 252x^5 + 210x^4 - 120x^3 + 45x^2 - 10x^1 + 1 p=11: x^11 - 11x^10 + 55x^9 - 165x^8 + 330x^7 - 462x^6 + 462x^5 - 330x^4 + 165x^3 - 55x^2 + 11x^1 - 1 (2 3 5 7 11 13 17 19 23 29 31) (2 3 5 7 11 13 17 19 23 29 31 37 41 43 47) (2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97)
Raku
(formerly Perl 6)
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. :-)
<lang perl6>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";</lang>
- Output:
p: (x-1)ᵖ ----------- 0: 1 1: x - 1 2: x² - 2x + 1 3: x³ - 3x² + 3x - 1 4: x⁴ - 4x³ + 6x² - 4x + 1 5: x⁵ - 5x⁴ + 10x³ - 10x² + 5x - 1 6: x⁶ - 6x⁵ + 15x⁴ - 20x³ + 15x² - 6x + 1 7: x⁷ - 7x⁶ + 21x⁵ - 35x⁴ + 35x³ - 21x² + 7x - 1 8: x⁸ - 8x⁷ + 28x⁶ - 56x⁵ + 70x⁴ - 56x³ + 28x² - 8x + 1 9: x⁹ - 9x⁸ + 36x⁷ - 84x⁶ + 126x⁵ - 126x⁴ + 84x³ - 36x² + 9x - 1 10: x¹⁰ - 10x⁹ + 45x⁸ - 120x⁷ + 210x⁶ - 252x⁵ + 210x⁴ - 120x³ + 45x² - 10x + 1 11: x¹¹ - 11x¹⁰ + 55x⁹ - 165x⁸ + 330x⁷ - 462x⁶ + 462x⁵ - 330x⁴ + 165x³ - 55x² + 11x - 1 12: x¹² - 12x¹¹ + 66x¹⁰ - 220x⁹ + 495x⁸ - 792x⁷ + 924x⁶ - 792x⁵ + 495x⁴ - 220x³ + 66x² - 12x + 1 Primes up to 100: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
REXX
version 1
<lang rexx>/* 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' </lang>
- 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. <lang rexx>/*REXX program calculates primes via the Agrawal─Kayal─Saxena (AKS) primality test.*/ parse arg Z . /*obtain optional argument from the CL.*/ if Z== | Z=="," then Z= 200 /*Not specified? Then use the default.*/ OZ=Z; tell= Z<0; Z= abs(Z) /*Is Z negative? Then show expression.*/ numeric digits max(9, Z % 3) /*define a dynamic # of decimal digits.*/ call AKS /*invoke the AKS funtion for coef. bld.*/ if left(OZ,1)=='+' then do; say Z isAksp(); exit /*display if Z is or isn't a prime.*/
end /* [↑] call isAKSp if Z has leading +.*/
say; say "primes found:" # /*display the prime number list. */ say; if \datatype(#, 'W') then exit /* [↓] the digit length of a big coef.*/ say 'Found ' words(#) " primes and the largest coefficient has " length(@.pm.h) @dd exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ AKS: $.0= '-'; $.1= "+"; @. = 1 /*$.x: sign char; default coefficients.*/
q.= 1; q.1= 0; q.4= 0 /*sparse array for faster comparisons. */
#=; L= length(Z) /*define list of prime numbers (so far)*/
do p=3 for Z; pm=p - 1; pp=p + 1 /*PM & PP: used as a coding convenience*/
do m=2 for pp % 2 - 1; mm=m - 1 /*calculate coefficients for a power. */
@.p.m= @.pm.mm + @.pm.m; h=pp - m /*calculate left side of coefficients*/
@.p.h= @.p.m /* " right " " " */
end /*m*/ /* [↑] The M DO loop creates both */
end /*p*/ /* sides in the same loop. */
if tell then say '(x-1)^'right(0, L)": 1" /*possibly display the first expression*/
@dd= 'decimal digits.' /* [↓] test for primality by division.*/
do n=2 for Z; nh=n % 2; d= n - 1 /*create expressions; find the primes.*/
do k=3 to nh while @.n.k//d == 0 /*are coefficients divisible by N-1 ? */
end /*k*/ /* [↑] skip the 1st & 2nd coefficients*/
if k>nh then if q.d then #= # d /*add a number to the prime list. */
if \tell then iterate /*Don't tell? Don't show expressions.*/
y= '(x-1)^'right(d, L)":" /*define the 1st part of the expression*/
s=1 /*S: is the sign indicator (-1│+1).*/
do j=n for n-1 by -1 /*create the higher powers first. */
if j==2 then xp= 'x' /*if power=1, then don't show the power*/
else xp= 'x^' || j-1 /* ··· else show power with ^ */
if j==n then y=y xp /*no sign (+│-) for the 1st expression.*/
else y=y $.s || @.n.j'∙'xp /*build the expression with sign (+|-).*/
s= \s /*flip the sign for the next expression*/
end /*j*/ /* [↑] the sign (now) is either 0 │ 1,*/
say y $.s'1' /*just show the first N expressions, */
end /*n*/ /* [↑] ··· but only for negative Z. */
if #== then #= "none"; return # /*if null, return "none"; else return #*/
/*──────────────────────────────────────────────────────────────────────────────────────*/ isAKSp: if z==word(#,words(#)) then return ' is a prime.'; else return " isn't a prime."</lang>
- 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:
<lang ruby>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(',')</lang>
Or without the dependency:
<lang ruby>def x_minus_1_to_the(p)
p.times.inject([1]) do |ex, _|
([0] + ex).zip(ex + [0]).map { |x,y| x - y }
end
end
def prime?(p)
return false if p < 2
coeff = x_minus_1_to_the(p)[1..p/2] # only need half of coeff terms
coeff.all?{ |n| n%p == 0 }
end
8.times do |n|
puts "(x-1)^#{n} = " +
x_minus_1_to_the(n).map.with_index { |c, p|
p.zero? ? c.to_s :
(c < 0 ? " - " : " + ") + (c.abs == 1 ? "x" : "#{c.abs}x") + (p == 1 ? "" : "^#{p}")
}.join
end
puts "\nPrimes below 50:", 50.times.select {|n| prime? n}.join(',')</lang>
- 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
<lang rust>fn aks_coefficients(k: usize) -> Vec<i64> {
let mut coefficients = vec![0i64; k + 1];
coefficients[0] = 1;
for i in 1..(k + 1) {
coefficients[i] = -(1..i).fold(coefficients[0], |prev, j|{
let old = coefficients[j];
coefficients[j] = old - prev;
old
});
}
coefficients
}
fn is_prime(p: usize) -> bool {
if p < 2 {
false
} else {
let c = aks_coefficients(p);
(1..p / 2 + 1).all(|i| c[i] % p as i64 == 0)
}
}
fn main() {
for i in 0..8 {
println!("{}: {:?}", i, aks_coefficients(i));
}
for i in (1..=50).filter(|&i| is_prime(i)) {
print!("{} ", i);
}
}</lang>
- 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.
<lang rust> 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() }) } } </lang>
Scala
<lang 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 " ")</lang>
- 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
<lang 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)
</lang>
Scilab
<lang> 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) </lang>
- 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
<lang 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;</lang>
- Output:
# p: (x-1)^p for small p 0: +1 1: -1 +1x^1 2: +1 -2x^1 +1x^2 3: -1 +3x^1 -3x^2 +1x^3 4: +1 -4x^1 +6x^2 -4x^3 +1x^4 5: -1 +5x^1 -10x^2 +10x^3 -5x^4 +1x^5 6: +1 -6x^1 +15x^2 -20x^3 +15x^4 -6x^5 +1x^6 7: -1 +7x^1 -21x^2 +35x^3 -35x^4 +21x^5 -7x^6 +1x^7 8: +1 -8x^1 +28x^2 -56x^3 +70x^4 -56x^5 +28x^6 -8x^7 +1x^8 9: -1 +9x^1 -36x^2 +84x^3 -126x^4 +126x^5 -84x^6 +36x^7 -9x^8 +1x^9 10: +1 -10x^1 +45x^2 -120x^3 +210x^4 -252x^5 +210x^6 -120x^7 +45x^8 -10x^9 +1x^10 11: -1 +11x^1 -55x^2 +165x^3 -330x^4 +462x^5 -462x^6 +330x^7 -165x^8 +55x^9 -11x^10 +1x^11 # small primes using the aks test 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61
Sidef
<lang ruby>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(' ')}]"</lang>
- 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.
<lang stata>mata function pol(n) { a=J(1,n+1,1) r=1 s=1 for (k=0; k<n; k++) { s=-s r=(r*(n-k))/(k+1) a[k+2]=r*s } return(a) }
for (n=0; n<=7; n++) mm_matlist(pol(n))
1
+-------------+
1 | 1 |
+-------------+
1 2
+-------------------------+
1 | 1 -1 |
+-------------------------+
1 2 3
+-------------------------------------+
1 | 1 -2 1 |
+-------------------------------------+
1 2 3 4
+-------------------------------------------------+
1 | 1 -3 3 -1 |
+-------------------------------------------------+
1 2 3 4 5
+-------------------------------------------------------------+
1 | 1 -4 6 -4 1 |
+-------------------------------------------------------------+
1 2 3 4 5 6
+-------------------------------------------------------------------------+
1 | 1 -5 10 -10 5 -1 |
+-------------------------------------------------------------------------+
1 2 3 4 5 6 7
+-------------------------------------------------------------------------------------+
1 | 1 -6 15 -20 15 -6 1 |
+-------------------------------------------------------------------------------------+
1 2 3 4 5 6 7 8
+-------------------------------------------------------------------------------------------------+
1 | 1 -7 21 -35 35 -21 7 -1 |
+-------------------------------------------------------------------------------------------------+
function isprime(n) {
a=pol(n)
for (k=2; k<=n; k++) {
if (mod(a[k],n)) return(0)
}
return(1)
}
for (n=2; n<=50; n++) { if (isprime(n)) printf("%f ",n) }
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 end</lang>
Swift
<lang 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) ")
}
} </lang>
- 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. <lang tcl>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 ,]"</lang>
- Output:
(x-1)^0 = +1x^0 (x-1)^1 = +1x^1 -1x^0 (x-1)^2 = +1x^2 -2x^1 +1x^0 (x-1)^3 = +1x^3 -3x^2 +3x^1 -1x^0 (x-1)^4 = +1x^4 -4x^3 +6x^2 -4x^1 +1x^0 (x-1)^5 = +1x^5 -5x^4 +10x^3 -10x^2 +5x^1 -1x^0 (x-1)^6 = +1x^6 -6x^5 +15x^4 -20x^3 +15x^2 -6x^1 +1x^0 (x-1)^7 = +1x^7 -7x^6 +21x^5 -35x^4 +35x^3 -21x^2 +7x^1 -1x^0 primes under 35: 2,3,5,7,11,13,17,19,23,29,31 primes under 50: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47
Transd
<lang scheme>
- lang transd
MainModule: {
poly: (λ n Long()
(with v Vector<Long>([1])
(for i in Range(n) do
(append v (/ (* (get v -1) (- (- n i))) (to-Long (+ i 1))))
)
(reverse v)
(ret v)
)
),
aks_test: (λ n Long() -> Bool()
(if (< n 2) (ret false))
(with v (poly n)
(set-el v 0 (+ (get v 0) 1))
(ret (not (any Range(in: v 0 -1) (λ (mod @it n)))))
)
),
_start: (λ (with v Vector<Long>()
(for p in Seq( 12 ) do
(set v (poly p))
(textout "(x - 1)^" p " = ")
(for i in v do (textout :sign i "x^" @idx " "))
(textout "\n")
)
(textout "\nList of primes in 2-62 interval:\n")
(for p in Seq( 2 62 ) do
(if (aks_test p) (textout p " "))
)
))
}</lang>
- Output:
(x - 1)^0 = +1x^0 (x - 1)^1 = -1x^0 +1x^1 (x - 1)^2 = +1x^0 -2x^1 +1x^2 (x - 1)^3 = -1x^0 +3x^1 -3x^2 +1x^3 (x - 1)^4 = +1x^0 -4x^1 +6x^2 -4x^3 +1x^4 (x - 1)^5 = -1x^0 +5x^1 -10x^2 +10x^3 -5x^4 +1x^5 (x - 1)^6 = +1x^0 -6x^1 +15x^2 -20x^3 +15x^4 -6x^5 +1x^6 (x - 1)^7 = -1x^0 +7x^1 -21x^2 +35x^3 -35x^4 +21x^5 -7x^6 +1x^7 (x - 1)^8 = +1x^0 -8x^1 +28x^2 -56x^3 +70x^4 -56x^5 +28x^6 -8x^7 +1x^8 (x - 1)^9 = -1x^0 +9x^1 -36x^2 +84x^3 -126x^4 +126x^5 -84x^6 +36x^7 -9x^8 +1x^9 (x - 1)^10 = +1x^0 -10x^1 +45x^2 -120x^3 +210x^4 -252x^5 +210x^6 -120x^7 +45x^8 -10x^9 +1x^10 (x - 1)^11 = -1x^0 +11x^1 -55x^2 +165x^3 -330x^4 +462x^5 -462x^6 +330x^7 -165x^8 +55x^9 -11x^10 +1x^11 List of primes in 2-62 interval: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61
uBasic/4tH
<lang>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</lang>
- 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
<lang vb> '-- Does not work for primes above 97, which is actually beyond the original task anyway. '-- Translated from the C version, just about everything is (working) out-by-1, what fun. '-- This updated VBA version utilizes the Decimal datatype to handle numbers requiring '-- more than 32 bits. Const MAX = 99 Dim c(MAX + 1) As Variant Private Sub coef(n As Integer) '-- out-by-1, ie coef(1)==^0, coef(2)==^1, coef(3)==^2 etc.
c(n) = CDec(1) 'converts c(n) from Variant to Decimal, a 12 byte data type
For i = n - 1 To 2 Step -1
c(i) = c(i) + c(i - 1)
Next i
End Sub Private Function is_prime(fn As Variant) As Boolean
fn = CDec(fn)
Call coef(fn + 1) '-- (I said it was out-by-1)
For i = 2 To fn - 1 '-- (technically "to n" is more correct)
If c(i) - fn * Int(c(i) / fn) <> 0 Then 'c(i) Mod fn <> 0 Then --Mod works upto 32 bit numbers
is_prime = False: Exit Function
End If
Next i
is_prime = True
End Function Private Sub show(n As Integer) '-- (As per coef, this is (working) out-by-1)
Dim ci As Variant
For i = n To 1 Step -1
ci = c(i)
If ci = 1 Then
If (n - i) Mod 2 = 0 Then
If i = 1 Then
If n = 1 Then
ci = "1"
Else
ci = "+1"
End If
Else
ci = ""
End If
Else
ci = "-1"
End If
Else
If (n - i) Mod 2 = 0 Then
ci = "+" & ci
Else
ci = "-" & ci
End If
End If
If i = 1 Then '-- ie ^0
Debug.Print ci
Else
If i = 2 Then '-- ie ^1
Debug.Print ci & "x";
Else
Debug.Print ci & "x^" & i - 1;
End If
End If
Next i
End Sub Public Sub AKS_test_for_primes()
Dim n As Integer
For n = 1 To 10 '-- (0 to 9 really)
coef n
Debug.Print "(x-1)^" & n - 1 & " = ";
show n
Next n
Debug.Print "primes (<="; MAX; "):"
coef 2 '-- (needed to reset c, if we want to avoid saying 1 is prime...)
For n = 2 To MAX
If is_prime(n) Then
Debug.Print n;
End If
Next n
End Sub
</lang>
- 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
Wren
<lang ecmascript>var bc = Fn.new { |p|
var c = List.filled(p+1, 0)
var r = 1
var half = (p/2).floor
for (i in 0..half) {
c[i] = r
c[p-i] = r
r = (r * (p-i) / (i+1)).floor
}
var j = p - 1
while (j >= 0) {
c[j] = -c[j]
j = j - 2
}
return c
}
var e = "²³⁴⁵⁶⁷".codePoints.toList
var pp = Fn.new { |c|
if (c.count == 1) return "%(c[0])"
var p = c.count - 1
var s = ""
if (c[p] != 1) s = "%(c[p])"
if (p == 0) return s
for (i in p..1) {
s = s + "x"
if (i != 1) s = s + String.fromCodePoint(e[i-2])
var d = c[i-1]
s = s + ((d < 0) ? " - %(-d)" : " + %(d)")
}
return s
}
var aks = Fn.new { |p|
var c = bc.call(p)
c[p] = c[p] - 1
c[0] = c[0] + 1
for (d in c) {
if (d%p != 0) return false
}
return true
}
for (p in 0..7) System.print("%(p): %(pp.call(bc.call(p)))") System.print("\nAll primes under 50:") for (p in 2..49) if (aks.call(p)) System.write("%(p) ") System.print()</lang>
- Output:
0: 1 1: x - 1 2: x² - 2x + 1 3: x³ - 3x² + 3x - 1 4: x⁴ - 4x³ + 6x² - 4x + 1 5: x⁵ - 5x⁴ + 10x³ - 10x² + 5x - 1 6: x⁶ - 6x⁵ + 15x⁴ - 20x³ + 15x² - 6x + 1 7: x⁷ - 7x⁶ + 21x⁵ - 35x⁴ + 35x³ - 21x² + 7x - 1 All primes under 50: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Yabasic
<lang Yabasic>// Does not work for primes above 53, which is actually beyond the original task anyway. // Translated from the C version, just about everything is (working) out-by-1, what fun.
dim c(100)
sub coef(n) local i // out-by-1, ie coef(1)==^0, coef(2)==^1, coef(3)==^2 etc.
c(n) = 1
for i = n-1 to 2 step -1
c(i) = c(i) + c(i-1)
next
end sub
sub is_prime(n) local i
coef(n+1) // (I said it was out-by-1)
for i = 2 to n-1 // (technically "to n" is more correct)
if int(c(i)/n) <> c(i)/n then
return 0
end if
next
return 1
end sub
sub show(n) // (As per coef, this is (working) out-by-1) local ci, ci$, i
for i = n to 1 step -1
ci = c(i)
if ci = 1 then
if mod(n-i, 2) = 0 then
if i = 1 then
if n = 1 then
ci$ = "1"
else
ci$ = "+1"
end if
else
ci$ = ""
end if
else
ci$ = "-1"
end if
else
if mod(n-i, 2) = 0 then
ci$ = "+" + str$(ci)
else
ci$ = "-" + str$(ci)
end if
end if
if i = 1 then // ie ^0
print ci$;
elsif i=2 then // ie ^1
print ci$, "x";
else
print ci$, "x^", i-1;
end if
next
end sub
sub AKS_test_for_primes() local n
for n = 1 to 10 // (0 to 9 really)
coef(n)
print "(x-1)^", n-1, " = ";
show(n)
print
next
print "\nprimes (<=53): ";
c(2) = 1 // (this manages "", which is all that call did anyway...)
for n = 2 to 53
if is_prime(n) then
print " ", n;
end if
next
print
end sub
AKS_test_for_primes()</lang>
Zig
Uses Zig's compile-time interpreter to create Pascal's triangle at compile-time. <lang Zig> const std = @import("std"); const assert = std.debug.assert; const stdout = std.io.getStdOut().writer();
pub fn main() !void {
var i: u6 = 0;
while (i < 8) : (i += 1)
try showBinomial(i);
try stdout.print("\nThe primes upto 50 (via AKS) are: ", .{});
i = 2;
while (i <= 50) : (i += 1) if (aksPrime(i))
try stdout.print("{} ", .{i});
try stdout.print("\n", .{});
}
fn showBinomial(n: u6) !void {
const row = binomial(n).?;
var sign: u8 = '+';
var exp = row.len;
try stdout.print("(x - 1)^{} =", .{n});
for (row) |coef| {
try stdout.print(" ", .{});
if (exp != row.len)
try stdout.print("{c} ", .{sign});
exp -= 1;
if (coef != 1 or exp == 0)
try stdout.print("{}", .{coef});
if (exp >= 1) {
try stdout.print("x", .{});
if (exp > 1)
try stdout.print("^{}", .{exp});
}
sign = if (sign == '+') '-' else '+';
}
try stdout.print("\n", .{});
}
fn aksPrime(n: u6) bool {
return for (binomial(n).?) |coef| {
if (coef > 1 and coef % n != 0)
break false;
} else true;
}
pub fn binomial(n: u32) ?[]const u64 {
if (n >= rmax)
return null
else {
const k = n * (n + 1) / 2;
return pascal[k .. k + n + 1];
}
}
const rmax = 68;
const pascal = build: {
@setEvalBranchQuota(100_000);
var coefficients: [(rmax * (rmax + 1)) / 2]u64 = undefined;
coefficients[0] = 1;
var j: u32 = 0;
var k: u32 = 1;
var n: u32 = 1;
while (n < rmax) : (n += 1) {
var prev = coefficients[j .. j + n];
var next = coefficients[k .. k + n + 1];
next[0] = 1;
var i: u32 = 1;
while (i < n) : (i += 1)
next[i] = prev[i] + prev[i - 1];
next[i] = 1;
j = k;
k += n + 1;
}
break :build coefficients;
}; </lang>
- Output:
$ zig run aks.zig (x - 1)^0 = 1 (x - 1)^1 = x - 1 (x - 1)^2 = x^2 - 2x + 1 (x - 1)^3 = x^3 - 3x^2 + 3x - 1 (x - 1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1 (x - 1)^5 = x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1 (x - 1)^6 = x^6 - 6x^5 + 15x^4 - 20x^3 + 15x^2 - 6x + 1 (x - 1)^7 = x^7 - 7x^6 + 21x^5 - 35x^4 + 35x^3 - 21x^2 + 7x - 1 The primes upto 50 (via AKS) are: 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
zkl
<lang zkl>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(*));</lang>
- Output:
# p: (x-1)^p for small p 0: +1 1: -1 +1x^1 2: +1 -2x^1 +1x^2 3: -1 +3x^1 -3x^2 +1x^3 4: +1 -4x^1 +6x^2 -4x^3 +1x^4 5: -1 +5x^1 -10x^2 +10x^3 -5x^4 +1x^5 6: +1 -6x^1 +15x^2 -20x^3 +15x^4 -6x^5 +1x^6 7: -1 +7x^1 -21x^2 +35x^3 -35x^4 +21x^5 -7x^6 +1x^7 8: +1 -8x^1 +28x^2 -56x^3 +70x^4 -56x^5 +28x^6 -8x^7 +1x^8 9: -1 +9x^1 -36x^2 +84x^3 -126x^4 +126x^5 -84x^6 +36x^7 -9x^8 +1x^9 10: +1 -10x^1 +45x^2 -120x^3 +210x^4 -252x^5 +210x^6 -120x^7 +45x^8 -10x^9 +1x^10 11: -1 +11x^1 -55x^2 +165x^3 -330x^4 +462x^5 -462x^6 +330x^7 -165x^8 +55x^9 -11x^10 +1x^11 # small primes using the aks test 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)
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