Jacobi symbol
You are encouraged to solve this task according to the task description, using any language you may know.
The Jacobi symbol is a multiplicative function that generalizes the Legendre symbol. Specifically, the Jacobi symbol (a | n) equals the product of the Legendre symbols (a | p_i)^(k_i), where n = p_1^(k_1)*p_2^(k_2)*...*p_i^(k_i) and the Legendre symbol (a | p) denotes the value of a ^ ((p-1)/2) (mod p)
- (a | p) ≡ 1 if a is a square (mod p)
- (a | p) ≡ -1 if a is not a square (mod p)
- (a | p) ≡ 0 if a ≡ 0
If n is prime, then the Jacobi symbol (a | n) equals the Legendre symbol (a | n).
- Task
Calculate the Jacobi symbol (a | n).
- Reference
11l
F jacobi(=a, =n)
I n <= 0
X ValueError(‘'n' must be a positive integer.’)
I n % 2 == 0
X ValueError(‘'n' must be odd.’)
a %= n
V result = 1
L a != 0
L a % 2 == 0
a /= 2
V n_mod_8 = n % 8
I n_mod_8 C (3, 5)
result = -result
(a, n) = (n, a)
I a % 4 == 3 & n % 4 == 3
result = -result
a %= n
I n == 1
R result
E
R 0
print(‘n\k|’, end' ‘’)
V kmax = 20
L(k) 0..kmax
print(‘#3’.format(k), end' ‘’)
print("\n----", end' ‘’)
L(k) 0..kmax
print(end' ‘---’)
print()
L(n) (1..21).step(2)
print(‘#<2 |’.format(n), end' ‘’)
L(k) 0..kmax
print(‘#3’.format(jacobi(k, n)), end' ‘’)
print()- Output:
n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ------------------------------------------------------------------- 1 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 | 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 5 | 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 7 | 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 9 | 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 11 | 0 1 -1 1 1 1 -1 -1 -1 1 -1 0 1 -1 1 1 1 -1 -1 -1 1 13 | 0 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 0 1 -1 1 1 -1 -1 -1 15 | 0 1 1 0 1 0 0 -1 1 0 0 -1 0 -1 -1 0 1 1 0 1 0 17 | 0 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 0 1 1 -1 19 | 0 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 -1 0 1 21 | 0 1 -1 0 1 1 0 0 -1 0 -1 -1 0 -1 0 0 1 1 0 -1 1
Action!
INCLUDE "D2:PRINTF.ACT" ;from the Action! Tool Kit
INT FUNC Jacobi(INT a,n)
INT res,tmp
IF a>=n THEN
a=a MOD n
FI
res=1
WHILE a
DO
WHILE (a&1)=0
DO
a=a RSH 1
tmp=n&7
IF tmp=3 OR tmp=5 THEN
res=-res
FI
OD
tmp=a a=n n=tmp
IF (a%3)=3 AND (n%3)=3 THEN
res=-res
FI
a=a MOD n
OD
IF n=1 THEN
RETURN (res)
FI
RETURN (0)
PROC PrintTable(INT maxK,maxN)
INT res,n,k
CHAR ARRAY t(10)
Put('n) Put(7) Put('k) Put(124)
FOR k=0 TO maxK
DO
StrI(k,t) PrintF("%3S",t)
OD
PutE()
Put(18) Put(18) Put(18) Put(19)
FOR k=0 TO 3*maxK+2
DO
Put(18)
OD
PutE()
FOR n=1 TO maxN STEP 2
DO
StrI(n,t) PrintF("%3S",t)
Put(124)
FOR k=0 TO maxK
DO
res=Jacobi(k,n)
StrI(res,t) PrintF("%3S",t)
OD
PutE()
OD
RETURN
PROC Main()
Put(125) PutE() ;clear the screen
PrintTable(10,39)
RETURN- Output:
Screenshot from Atari 8-bit computer
n\k│ 0 1 2 3 4 5 6 7 8 9 10 ───┼───────────────────────────────── 1│ 1 1 1 1 1 1 1 1 1 1 1 3│ 0 -1 1 0 -1 1 0 -1 1 0 -1 5│ 0 1 -1 1 1 0 1 -1 1 1 0 7│ 0 1 1 -1 1 -1 -1 0 1 1 -1 9│ 0 1 1 0 1 1 0 1 1 0 1 11│ 0 1 -1 1 1 1 -1 1 -1 1 -1 13│ 0 1 -1 -1 1 1 1 -1 -1 1 -1 15│ 0 1 1 0 1 0 0 1 1 0 0 17│ 0 1 1 1 1 -1 1 -1 1 1 -1 19│ 0 1 -1 -1 1 1 1 -1 -1 1 -1 21│ 0 1 -1 0 1 1 0 0 -1 0 -1 23│ 0 1 1 1 1 1 1 1 1 1 1 25│ 0 1 1 -1 1 0 -1 1 1 1 0 27│ 0 1 -1 0 1 -1 0 -1 -1 0 1 29│ 0 1 -1 1 1 1 -1 1 -1 1 -1 31│ 0 1 1 -1 1 1 -1 -1 1 1 1 33│ 0 1 1 0 1 1 0 -1 1 0 1 35│ 0 1 -1 1 1 0 -1 0 -1 1 0 37│ 0 1 -1 -1 1 -1 1 1 -1 1 1 39│ 0 1 1 0 1 1 0 1 1 0 1
Arturo
jacobi: function [n,k][
N: n % k
K: k
result: 1
while [N <> 0][
while [even? N][
N: shr N 1
if contains? [3 5] and K 7 ->
result: neg result
]
[N,K]: @[K,N]
if and? 3=and N 3 3=and K 3 ->
result: neg result
N: N % K
]
if K <> 1 ->
result: 0
return result
]
print ["" "k/n" "|"] ++ map to [:string] 1..20 'item -> pad.left item 2
print repeat "=" 67
loop range.step:2 1 21 'k [
print [
"" pad to :string k 3 "|" join.with:" " map to [:string] map 1..20 'n -> jacobi n k
'item -> pad.left item 2
]
]
- Output:
k/n | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 =================================================================== 1 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 | 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 5 | 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 7 | 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 9 | 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 11 | 1 -1 1 1 1 -1 -1 -1 1 -1 0 1 -1 1 1 1 -1 -1 -1 1 13 | 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 0 1 -1 1 1 -1 -1 -1 15 | 1 1 0 1 0 0 -1 1 0 0 -1 0 -1 -1 0 1 1 0 1 0 17 | 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 0 1 1 -1 19 | 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 -1 0 1 21 | 1 -1 0 1 1 0 0 -1 0 -1 -1 0 -1 0 0 1 1 0 -1 1
AutoHotkey
result := "n/k|"
loop 20
result .= SubStr(" " A_Index, -1) " "
l := StrLen(result)
result .= "`n"
loop % l
result .= "-"
result .= "`n"
loop 21
{
if !Mod(n := A_Index, 2)
continue
result .= SubStr(" " n, -1) " |"
loop 20
result .= SubStr(" " jacobi(a := A_Index, n), -1) " "
result .= "`n"
}
MsgBox, 262144, , % result
return
jacobi(a, n) {
a := Mod(a, n), t := 1
while (a != 0) {
while !Mod(a, 2)
a := a >> 1, r := Mod(n, 8), t := (r=3 || r=5) ? -t : t
r := n, n := a, a := r
if (Mod(a, 4)=3 && Mod(n, 4)=3)
t := -t
a := Mod(a, n)
}
return (n=1) ? t : 0
}
- Output:
n/k| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ---------------------------------------------------------------- 1 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 | 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 5 | 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 7 | 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 9 | 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 11 | 1 -1 1 1 1 -1 -1 -1 1 -1 0 1 -1 1 1 1 -1 -1 -1 1 13 | 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 0 1 -1 1 1 -1 -1 -1 15 | 1 1 0 1 0 0 -1 1 0 0 -1 0 -1 -1 0 1 1 0 1 0 17 | 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 0 1 1 -1 19 | 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 -1 0 1 21 | 1 -1 0 1 1 0 0 -1 0 -1 -1 0 -1 0 0 1 1 0 -1 1
AWK
# syntax: GAWK -f JACOBI_SYMBOL.AWK
BEGIN {
max_n = 29
max_a = max_n + 1
printf("n\\a")
for (i=1; i<=max_a; i++) {
printf("%3d",i)
underline = underline " --"
}
printf("\n---%s\n",underline)
for (n=1; n<=max_n; n+=2) {
printf("%3d",n)
for (a=1; a<=max_a; a++) {
printf("%3d",jacobi(a,n))
}
printf("\n")
}
exit(0)
}
function jacobi(a,n, result,tmp) {
if (n%2 == 0) {
print("error: 'n' must be a positive odd integer")
exit
}
a %= n
result = 1
while (a != 0) {
while (a%2 == 0) {
a /= 2
if (n%8 == 3 || n%8 == 5) {
result = -result
}
}
tmp = a
a = n
n = tmp
if (a%4 == 3 && n%4 == 3) {
result = -result
}
a %= n
}
if (n == 1) {
return(result)
}
return(0)
}
- Output:
n\a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 --- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 5 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 7 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 9 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 11 1 -1 1 1 1 -1 -1 -1 1 -1 0 1 -1 1 1 1 -1 -1 -1 1 -1 0 1 -1 1 1 1 -1 -1 -1 13 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 0 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 0 1 -1 1 1 15 1 1 0 1 0 0 -1 1 0 0 -1 0 -1 -1 0 1 1 0 1 0 0 -1 1 0 0 -1 0 -1 -1 0 17 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 0 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 19 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 -1 0 1 -1 -1 1 1 1 1 -1 1 -1 1 21 1 -1 0 1 1 0 0 -1 0 -1 -1 0 -1 0 0 1 1 0 -1 1 0 1 -1 0 1 1 0 0 -1 0 23 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 0 1 1 1 1 -1 1 -1 25 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 27 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 29 1 -1 -1 1 1 1 1 -1 1 -1 -1 -1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 1 -1 -1 1 0 1
C
#include <stdlib.h>
#include <stdio.h>
#define SWAP(a, b) (((a) ^= (b)), ((b) ^= (a)), ((a) ^= (b)))
int jacobi(unsigned long a, unsigned long n) {
if (a >= n) a %= n;
int result = 1;
while (a) {
while ((a & 1) == 0) {
a >>= 1;
if ((n & 7) == 3 || (n & 7) == 5) result = -result;
}
SWAP(a, n);
if ((a & 3) == 3 && (n & 3) == 3) result = -result;
a %= n;
}
if (n == 1) return result;
return 0;
}
void print_table(unsigned kmax, unsigned nmax) {
printf("n\\k|");
for (int k = 0; k <= kmax; ++k) printf("%'3u", k);
printf("\n----");
for (int k = 0; k <= kmax; ++k) printf("---");
putchar('\n');
for (int n = 1; n <= nmax; n += 2) {
printf("%-2u |", n);
for (int k = 0; k <= kmax; ++k)
printf("%'3d", jacobi(k, n));
putchar('\n');
}
}
int main() {
print_table(20, 21);
return 0;
}
- Output:
n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ---+--------------------------------------------------------------- 1 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 | 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 5 | 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 7 | 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 9 | 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 11 | 0 1 -1 1 1 1 -1 -1 -1 1 -1 0 1 -1 1 1 1 -1 -1 -1 1 13 | 0 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 0 1 -1 1 1 -1 -1 -1 15 | 0 1 1 0 1 0 0 -1 1 0 0 -1 0 -1 -1 0 1 1 0 1 0 17 | 0 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 0 1 1 -1 19 | 0 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 -1 0 1 21 | 0 1 -1 0 1 1 0 0 -1 0 -1 -1 0 -1 0 0 1 1 0 -1 1
C++
#include <algorithm>
#include <cassert>
#include <iomanip>
#include <iostream>
int jacobi(int n, int k) {
assert(k > 0 && k % 2 == 1);
n %= k;
int t = 1;
while (n != 0) {
while (n % 2 == 0) {
n /= 2;
int r = k % 8;
if (r == 3 || r == 5)
t = -t;
}
std::swap(n, k);
if (n % 4 == 3 && k % 4 == 3)
t = -t;
n %= k;
}
return k == 1 ? t : 0;
}
void print_table(std::ostream& out, int kmax, int nmax) {
out << "n\\k|";
for (int k = 0; k <= kmax; ++k)
out << ' ' << std::setw(2) << k;
out << "\n----";
for (int k = 0; k <= kmax; ++k)
out << "---";
out << '\n';
for (int n = 1; n <= nmax; n += 2) {
out << std::setw(2) << n << " |";
for (int k = 0; k <= kmax; ++k)
out << ' ' << std::setw(2) << jacobi(k, n);
out << '\n';
}
}
int main() {
print_table(std::cout, 20, 21);
return 0;
}
- Output:
n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ------------------------------------------------------------------- 1 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 | 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 5 | 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 7 | 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 9 | 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 11 | 0 1 -1 1 1 1 -1 -1 -1 1 -1 0 1 -1 1 1 1 -1 -1 -1 1 13 | 0 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 0 1 -1 1 1 -1 -1 -1 15 | 0 1 1 0 1 0 0 -1 1 0 0 -1 0 -1 -1 0 1 1 0 1 0 17 | 0 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 0 1 1 -1 19 | 0 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 -1 0 1 21 | 0 1 -1 0 1 1 0 0 -1 0 -1 -1 0 -1 0 0 1 1 0 -1 1
Crystal
def jacobi(a, n)
raise ArgumentError.new "n must b positive and odd" if n < 1 || n.even?
res = 1
until (a %= n) == 0
while a.even?
a >>= 1
res = -res if [3, 5].includes? n % 8
end
a, n = n, a
res = -res if a % 4 == n % 4 == 3
end
n == 1 ? res : 0
end
puts "Jacobian symbols for jacobi(a, n)"
puts "n\\a 0 1 2 3 4 5 6 7 8 9 10"
puts "------------------------------------"
1.step(to: 17, by: 2) do |n|
printf("%2d ", n)
(0..10).each { |a| printf(" % 2d", jacobi(a, n)) }
puts
end
- Output:
Jacobian symbols for jacobi(a, n) n\a 0 1 2 3 4 5 6 7 8 9 10 ------------------------------------ 1 1 1 1 1 1 1 1 1 1 1 1 3 0 1 -1 0 1 -1 0 1 -1 0 1 5 0 1 -1 -1 1 0 1 -1 -1 1 0 7 0 1 1 -1 1 -1 -1 0 1 1 -1 9 0 1 1 0 1 1 0 1 1 0 1 11 0 1 -1 1 1 1 -1 -1 -1 1 -1 13 0 1 -1 1 1 -1 -1 -1 -1 1 1 15 0 1 1 0 1 0 0 -1 1 0 0 17 0 1 1 -1 1 -1 -1 -1 1 1 -1
Erlang
jacobi(_, N) when N =< 0 -> jacobi_domain_error;
jacobi(_, N) when (N band 1) =:= 0 -> jacobi_domain_error;
jacobi(A, N) when A < 0 ->
J2 = ja(-A, N),
case N band 3 of
1 -> J2;
3 -> -J2
end;
jacobi(A, N) -> ja(A, N).
ja(0, _) -> 0;
ja(1, _) -> 1;
ja(A, N) when A >= N -> ja(A rem N, N);
ja(A, N) when (A band 1) =:= 0 -> % A is even
J2 = ja(A bsr 1, N),
case N band 7 of
1 -> J2;
3 -> -J2;
5 -> -J2;
7 -> J2
end;
ja(A, N) -> % if we get here, A is odd, so we can flip it.
J2 = ja(N, A),
case (A band 3 =:= 3) and (N band 3 =:= 3) of
true -> -J2;
false -> J2
end.
F#
//Jacobi Symbol. Nigel Galloway: July 14th., 2020
let J n m=let rec J n m g=match n with
0->if m=1 then g else 0
|n when n%2=0->J(n/2) m (if m%8=3 || m%8=5 then -g else g)
|n->J (m%n) n (if m%4=3 && n%4=3 then -g else g)
J (n%m) m 1
printfn "n\m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30\n ----------------------------------------------------------------------------------------------------------------------"
[1..2..29]|>List.iter(fun m->printf "%3d" m; [1..30]|>List.iter(fun n->printf "%4d" (J n m)); printfn "")
- Output:
n\m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
----------------------------------------------------------------------------------------------------------------------
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
3 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0
5 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0
7 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1
9 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0
11 1 -1 1 1 1 -1 -1 -1 1 -1 0 1 -1 1 1 1 -1 -1 -1 1 -1 0 1 -1 1 1 1 -1 -1 -1
13 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 0 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 0 1 -1 1 1
15 1 1 0 1 0 0 -1 1 0 0 -1 0 -1 -1 0 1 1 0 1 0 0 -1 1 0 0 -1 0 -1 -1 0
17 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 0 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1
19 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 -1 0 1 -1 -1 1 1 1 1 -1 1 -1 1
21 1 -1 0 1 1 0 0 -1 0 -1 -1 0 -1 0 0 1 1 0 -1 1 0 1 -1 0 1 1 0 0 -1 0
23 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 0 1 1 1 1 -1 1 -1
25 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0
27 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0
29 1 -1 -1 1 1 1 1 -1 1 -1 -1 -1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 1 -1 -1 1 0 1
Factor
The jacobi word already exists in the math.extras vocabulary. See the implementation here.
FreeBASIC
function gcdp( a as uinteger, b as uinteger ) as uinteger
if b = 0 then return a
return gcdp( b, a mod b )
end function
function gcd(a as integer, b as integer) as uinteger
return gcdp( abs(a), abs(b) )
end function
function jacobi( a as uinteger, n as uinteger ) as integer
if gcd(a, n)<>1 then return 0
if a = 1 then return 1
if a>n then return jacobi( a mod n, n )
if a mod 2 = 0 then
if n mod 8 = 1 or n mod 8 = 7 then
return jacobi(a/2, n)
else
return -jacobi(a/2, n)
end if
end if
dim as integer q = (-1)^((a-1)/2 * (n-1)/2)
return q/jacobi(n, a)
end function
'print a table
function padto( i as ubyte, j as integer ) as string
return wspace(i-len(str(j)))+str(j)
end function
dim as uinteger pn, k, prime(0 to 16) = {3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61}
dim as string outstr = " k "
for k = 1 to 36
outstr = outstr + padto(2, k)+" "
next k
print outstr
print " n"
for pn=0 to 16
outstr= " "+padto( 2, prime(pn) )+" "
for k = 1 to 36
outstr = outstr + padto(2, jacobi(k, prime(pn))) + " "
next k
print outstr
next pn- Output:
k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 n 3 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 5 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 7 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 11 1 -1 1 1 1 -1 -1 -1 1 -1 0 1 -1 1 1 1 -1 -1 -1 1 -1 0 1 -1 1 1 1 -1 -1 -1 1 -1 0 1 -1 1 13 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 0 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 0 1 -1 1 1 -1 -1 -1 -1 1 1 17 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 0 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 0 1 1 19 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 -1 0 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 23 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 0 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 29 1 -1 -1 1 1 1 1 -1 1 -1 -1 -1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 1 -1 -1 1 0 1 -1 -1 1 1 1 1 31 1 1 -1 1 1 -1 1 1 1 1 -1 -1 -1 1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 -1 -1 0 1 1 -1 1 1 37 1 -1 1 1 -1 -1 1 -1 1 1 1 1 -1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 -1 1 1 1 1 -1 1 -1 -1 1 1 -1 1 41 1 1 -1 1 1 -1 -1 1 1 1 -1 -1 -1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 -1 -1 -1 -1 1 1 1 -1 -1 1 43 1 -1 -1 1 -1 1 -1 -1 1 1 1 -1 1 1 1 1 1 -1 -1 -1 1 -1 1 1 1 -1 -1 -1 -1 -1 1 -1 -1 -1 1 1 47 1 1 1 1 -1 1 1 1 1 -1 -1 1 -1 1 -1 1 1 1 -1 -1 1 -1 -1 1 1 -1 1 1 -1 -1 -1 1 -1 1 -1 1 53 1 -1 -1 1 -1 1 1 -1 1 1 1 -1 1 -1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 1 59 1 -1 1 1 1 -1 1 -1 1 -1 -1 1 -1 -1 1 1 1 -1 1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 1 1 61 1 -1 1 1 1 -1 -1 -1 1 -1 -1 1 1 1 1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 -1 -1 -1 -1 -1 -1 1 -1 1
Go
The big.Jacobi function in the standard library (for 'big integers') returns the Jacobi symbol for given values of 'a' and 'n'.
This translates the Lua code in the above referenced Wikipedia article to Go (for 8 byte integers) and checks that it gives the same answers for a small table of values - which it does.
package main
import (
"fmt"
"log"
"math/big"
)
func jacobi(a, n uint64) int {
if n%2 == 0 {
log.Fatal("'n' must be a positive odd integer")
}
a %= n
result := 1
for a != 0 {
for a%2 == 0 {
a /= 2
nn := n % 8
if nn == 3 || nn == 5 {
result = -result
}
}
a, n = n, a
if a%4 == 3 && n%4 == 3 {
result = -result
}
a %= n
}
if n == 1 {
return result
}
return 0
}
func main() {
fmt.Println("Using hand-coded version:")
fmt.Println("n/a 0 1 2 3 4 5 6 7 8 9")
fmt.Println("---------------------------------")
for n := uint64(1); n <= 17; n += 2 {
fmt.Printf("%2d ", n)
for a := uint64(0); a <= 9; a++ {
fmt.Printf(" % d", jacobi(a, n))
}
fmt.Println()
}
ba, bn := new(big.Int), new(big.Int)
fmt.Println("\nUsing standard library function:")
fmt.Println("n/a 0 1 2 3 4 5 6 7 8 9")
fmt.Println("---------------------------------")
for n := uint64(1); n <= 17; n += 2 {
fmt.Printf("%2d ", n)
for a := uint64(0); a <= 9; a++ {
ba.SetUint64(a)
bn.SetUint64(n)
fmt.Printf(" % d", big.Jacobi(ba, bn))
}
fmt.Println()
}
}
- Output:
Using hand-coded version: n/a 0 1 2 3 4 5 6 7 8 9 --------------------------------- 1 1 1 1 1 1 1 1 1 1 1 3 0 1 -1 0 1 -1 0 1 -1 0 5 0 1 -1 -1 1 0 1 -1 -1 1 7 0 1 1 -1 1 -1 -1 0 1 1 9 0 1 1 0 1 1 0 1 1 0 11 0 1 -1 1 1 1 -1 -1 -1 1 13 0 1 -1 1 1 -1 -1 -1 -1 1 15 0 1 1 0 1 0 0 -1 1 0 17 0 1 1 -1 1 -1 -1 -1 1 1 Using standard library function: n/a 0 1 2 3 4 5 6 7 8 9 --------------------------------- 1 1 1 1 1 1 1 1 1 1 1 3 0 1 -1 0 1 -1 0 1 -1 0 5 0 1 -1 -1 1 0 1 -1 -1 1 7 0 1 1 -1 1 -1 -1 0 1 1 9 0 1 1 0 1 1 0 1 1 0 11 0 1 -1 1 1 1 -1 -1 -1 1 13 0 1 -1 1 1 -1 -1 -1 -1 1 15 0 1 1 0 1 0 0 -1 1 0 17 0 1 1 -1 1 -1 -1 -1 1 1
Haskell
jacobi :: Integer -> Integer -> Integer
jacobi 0 1 = 1
jacobi 0 _ = 0
jacobi a n =
let a_mod_n = rem a n
in if even a_mod_n
then case rem n 8 of
1 -> jacobi (div a_mod_n 2) n
3 -> negate $ jacobi (div a_mod_n 2) n
5 -> negate $ jacobi (div a_mod_n 2) n
7 -> jacobi (div a_mod_n 2) n
else if rem a_mod_n 4 == 3 && rem n 4 == 3
then negate $ jacobi n a_mod_n
else jacobi n a_mod_n
Or, expressing it slightly differently, and adding a tabulation:
import Data.Bool (bool)
import Data.List (replicate, transpose)
import Data.List.Split (chunksOf)
---------------------- JACOBI SYMBOL ---------------------
jacobi :: Int -> Int -> Int
jacobi = go
where
go 0 1 = 1
go 0 _ = 0
go x y
| even r =
plusMinus
(rem y 8 `elem` [3, 5])
(go (div r 2) y)
| otherwise = plusMinus (p r && p y) (go y r)
where
plusMinus = bool id negate
p = (3 ==) . flip rem 4
r = rem x y
--------------------------- TEST -------------------------
main :: IO ()
main = putStrLn $ jacobiTable 11 9
------------------------- DISPLAY ------------------------
jacobiTable :: Int -> Int -> String
jacobiTable nCols nRows =
let rowLabels = [1, 3 .. (2 * nRows)]
colLabels = [0 .. pred nCols]
in withColumnLabels ("" : fmap show colLabels) $
labelledRows (fmap show rowLabels) $
paddedCols $
chunksOf nRows $
uncurry jacobi
<$> ((,) <$> colLabels <*> rowLabels)
------------------- TABULATION FUNCTIONS -----------------
paddedCols ::
Show a =>
[[a]] ->
[[String]]
paddedCols cols =
let scols = fmap show <$> cols
w = maximum $ length <$> concat scols
in map (justifyRight w ' ') <$> scols
labelledRows :: [String] -> [[String]] -> [[String]]
labelledRows labels cols =
let w = maximum $ map length labels
in zipWith
(:)
((<> " ->") . justifyRight w ' ' <$> labels)
(transpose cols)
withColumnLabels :: [String] -> [[String]] -> String
withColumnLabels _ [] = ""
withColumnLabels labels rows@(x : _) =
let labelRow =
unwords $
zipWith
(`justifyRight` ' ')
(length <$> x)
labels
in unlines $
labelRow :
replicate (length labelRow) '-' : fmap unwords rows
justifyRight :: Int -> a -> [a] -> [a]
justifyRight n c = (drop . length) <*> (replicate n c <>)
- Output:
0 1 2 3 4 5 6 7 8 9 10 -------------------------------------- 1 -> 1 1 1 1 1 1 1 1 1 1 1 3 -> 0 1 -1 0 1 -1 0 1 -1 0 1 5 -> 0 1 -1 -1 1 0 1 -1 -1 1 0 7 -> 0 1 1 -1 1 -1 -1 0 1 1 -1 9 -> 0 1 1 0 1 1 0 1 1 0 1 11 -> 0 1 -1 1 1 1 -1 -1 -1 1 -1 13 -> 0 1 -1 1 1 -1 -1 -1 -1 1 1 15 -> 0 1 1 0 1 0 0 -1 1 0 0 17 -> 0 1 1 -1 1 -1 -1 -1 1 1 -1
J
NB. functionally equivalent translation of the Lua program found
NB. at https://en.wikipedia.org/wiki/Jacobi_symbol
jacobi=: {{
assert. (0<x) * 1=2|x
y=. x|y
t=. 1
while. y do.
e=. (|.#:y) i.1
y=. <.y%2^e
t=. t*_1^(*/3 = 4|x,y)+(2|e)*(8|x) e.3 5
'x y'=. y, y|x
end.
t*x=1
}}"0
k=: 1 2 p. i. 30 n=: #\ k k jacobi table n +------+--------------------------------------------------------------------------------------+ |jacobi|1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30| +------+--------------------------------------------------------------------------------------+ | 1 |1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1| | 3 |1 _1 0 1 _1 0 1 _1 0 1 _1 0 1 _1 0 1 _1 0 1 _1 0 1 _1 0 1 _1 0 1 _1 0| | 5 |1 _1 _1 1 0 1 _1 _1 1 0 1 _1 _1 1 0 1 _1 _1 1 0 1 _1 _1 1 0 1 _1 _1 1 0| | 7 |1 1 _1 1 _1 _1 0 1 1 _1 1 _1 _1 0 1 1 _1 1 _1 _1 0 1 1 _1 1 _1 _1 0 1 1| | 9 |1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0| |11 |1 _1 1 1 1 _1 _1 _1 1 _1 0 1 _1 1 1 1 _1 _1 _1 1 _1 0 1 _1 1 1 1 _1 _1 _1| |13 |1 _1 1 1 _1 _1 _1 _1 1 1 _1 1 0 1 _1 1 1 _1 _1 _1 _1 1 1 _1 1 0 1 _1 1 1| |15 |1 1 0 1 0 0 _1 1 0 0 _1 0 _1 _1 0 1 1 0 1 0 0 _1 1 0 0 _1 0 _1 _1 0| |17 |1 1 _1 1 _1 _1 _1 1 1 _1 _1 _1 1 _1 1 1 0 1 1 _1 1 _1 _1 _1 1 1 _1 _1 _1 1| |19 |1 _1 _1 1 1 1 1 _1 1 _1 1 _1 _1 _1 _1 1 1 _1 0 1 _1 _1 1 1 1 1 _1 1 _1 1| |21 |1 _1 0 1 1 0 0 _1 0 _1 _1 0 _1 0 0 1 1 0 _1 1 0 1 _1 0 1 1 0 0 _1 0| |23 |1 1 1 1 _1 1 _1 1 1 _1 _1 1 1 _1 _1 1 _1 1 _1 _1 _1 _1 0 1 1 1 1 _1 1 _1| |25 |1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0| |27 |1 _1 0 1 _1 0 1 _1 0 1 _1 0 1 _1 0 1 _1 0 1 _1 0 1 _1 0 1 _1 0 1 _1 0| |29 |1 _1 _1 1 1 1 1 _1 1 _1 _1 _1 1 _1 _1 1 _1 _1 _1 1 _1 1 1 1 1 _1 _1 1 0 1| |31 |1 1 _1 1 1 _1 1 1 1 1 _1 _1 _1 1 _1 1 _1 1 1 1 _1 _1 _1 _1 1 _1 _1 1 _1 _1| |33 |1 1 0 1 _1 0 _1 1 0 _1 0 0 _1 _1 0 1 1 0 _1 _1 0 0 _1 0 1 _1 0 _1 1 0| |35 |1 _1 1 1 0 _1 0 _1 1 0 1 1 1 0 0 1 1 _1 _1 0 0 _1 _1 _1 0 _1 1 0 1 0| |37 |1 _1 1 1 _1 _1 1 _1 1 1 1 1 _1 _1 _1 1 _1 _1 _1 _1 1 _1 _1 _1 1 1 1 1 _1 1| |39 |1 1 0 1 1 0 _1 1 0 1 1 0 0 _1 0 1 _1 0 _1 1 0 1 _1 0 1 0 0 _1 _1 0| |41 |1 1 _1 1 1 _1 _1 1 1 1 _1 _1 _1 _1 _1 1 _1 1 _1 1 1 _1 1 _1 1 _1 _1 _1 _1 _1| |43 |1 _1 _1 1 _1 1 _1 _1 1 1 1 _1 1 1 1 1 1 _1 _1 _1 1 _1 1 1 1 _1 _1 _1 _1 _1| |45 |1 _1 0 1 0 0 _1 _1 0 0 1 0 _1 1 0 1 _1 0 1 0 0 _1 _1 0 0 1 0 _1 1 0| |47 |1 1 1 1 _1 1 1 1 1 _1 _1 1 _1 1 _1 1 1 1 _1 _1 1 _1 _1 1 1 _1 1 1 _1 _1| |49 |1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 1 1| |51 |1 _1 0 1 1 0 _1 _1 0 _1 1 0 1 1 0 1 0 0 1 1 0 _1 1 0 1 _1 0 _1 1 0| |53 |1 _1 _1 1 _1 1 1 _1 1 1 1 _1 1 _1 1 1 1 _1 _1 _1 _1 _1 _1 1 1 _1 _1 1 1 _1| |55 |1 1 _1 1 0 _1 1 1 1 0 0 _1 1 1 0 1 1 1 _1 0 _1 0 _1 _1 0 1 _1 1 _1 0| |57 |1 1 0 1 _1 0 1 1 0 _1 _1 0 _1 1 0 1 _1 0 0 _1 0 _1 _1 0 1 _1 0 1 1 0| |59 |1 _1 1 1 1 _1 1 _1 1 _1 _1 1 _1 _1 1 1 1 _1 1 1 1 1 _1 _1 1 1 1 1 1 _1| +------+--------------------------------------------------------------------------------------+
Java
public class JacobiSymbol {
public static void main(String[] args) {
int max = 30;
System.out.printf("n\\k ");
for ( int k = 1 ; k <= max ; k++ ) {
System.out.printf("%2d ", k);
}
System.out.printf("%n");
for ( int n = 1 ; n <= max ; n += 2 ) {
System.out.printf("%2d ", n);
for ( int k = 1 ; k <= max ; k++ ) {
System.out.printf("%2d ", jacobiSymbol(k, n));
}
System.out.printf("%n");
}
}
// Compute (k n), where k is numerator
private static int jacobiSymbol(int k, int n) {
if ( k < 0 || n % 2 == 0 ) {
throw new IllegalArgumentException("Invalid value. k = " + k + ", n = " + n);
}
k %= n;
int jacobi = 1;
while ( k > 0 ) {
while ( k % 2 == 0 ) {
k /= 2;
int r = n % 8;
if ( r == 3 || r == 5 ) {
jacobi = -jacobi;
}
}
int temp = n;
n = k;
k = temp;
if ( k % 4 == 3 && n % 4 == 3 ) {
jacobi = -jacobi;
}
k %= n;
}
if ( n == 1 ) {
return jacobi;
}
return 0;
}
}
- Output:
n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 5 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 7 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 9 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 11 1 -1 1 1 1 -1 -1 -1 1 -1 0 1 -1 1 1 1 -1 -1 -1 1 -1 0 1 -1 1 1 1 -1 -1 -1 13 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 0 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 0 1 -1 1 1 15 1 1 0 1 0 0 -1 1 0 0 -1 0 -1 -1 0 1 1 0 1 0 0 -1 1 0 0 -1 0 -1 -1 0 17 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 0 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 19 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 -1 0 1 -1 -1 1 1 1 1 -1 1 -1 1 21 1 -1 0 1 1 0 0 -1 0 -1 -1 0 -1 0 0 1 1 0 -1 1 0 1 -1 0 1 1 0 0 -1 0 23 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 0 1 1 1 1 -1 1 -1 25 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 27 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 29 1 -1 -1 1 1 1 1 -1 1 -1 -1 -1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 1 -1 -1 1 0 1
jq
def lpad($len): tostring | ($len - length) as $l | (" " * $l)[:$l] + .;
def rpad($len): tostring | ($len - length) as $l | . + (" " * $l)[:$l];
def jacobi(a; n):
{a: (a % n), n: n, result: 1}
| until(.a == 0;
until( .a % 2 != 0;
.a /= 2
| if (.n % 8) | IN(3, 5) then .result *= -1 else . end )
| {a: .n, n: .a, result} # swap .a and .n
| (.n % 4) as $nmod4
| if (.a % 4) == $nmod4 and $nmod4 == 3 then .result *= -1 else . end
| .a = .a % .n )
| if .n == 1 then .result else 0 end ;
" Table of jacobi(a; n)",
"n\\k 1 2 3 4 5 6 7 8 9 10 11 12",
"_____________________________________________________",
(range( 1; 32; 2) as $n
| "\($n|rpad(3))" + reduce range(1; 13) as $a (""; . + (jacobi($a; $n) | lpad(4) ))
)- Output:
Table of jacobi(a, n) n\k 1 2 3 4 5 6 7 8 9 10 11 12 _____________________________________________________ 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 -1 0 1 -1 0 1 -1 0 1 -1 0 5 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 7 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 9 1 1 0 1 1 0 1 1 0 1 1 0 11 1 -1 1 1 1 -1 -1 -1 1 -1 0 1 13 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 15 1 1 0 1 0 0 -1 1 0 0 -1 0 17 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 19 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 21 1 -1 0 1 1 0 0 -1 0 -1 -1 0 23 1 1 1 1 -1 1 -1 1 1 -1 -1 1 25 1 1 1 1 0 1 1 1 1 0 1 1 27 1 -1 0 1 -1 0 1 -1 0 1 -1 0 29 1 -1 -1 1 1 1 1 -1 1 -1 -1 -1 31 1 1 -1 1 1 -1 1 1 1 1 -1 -1
Julia
function jacobi(a, n)
a %= n
result = 1
while a != 0
while iseven(a)
a ÷= 2
((n % 8) in [3, 5]) && (result *= -1)
end
a, n = n, a
(a % 4 == n % 4 == 3) && (result *= -1)
a %= n
end
return n == 1 ? result : 0
end
print(" Table of jacobi(a, n) for a 1 to 12, n 1 to 31\n 1 2 3 4 5 6 7 8",
" 9 10 11 12\nn\n_____________________________________________________")
for n in 1:2:31
print("\n", rpad(n, 3))
for a in 1:11
print(lpad(jacobi(a, n), 4))
end
end
- Output:
Table of jacobi(a, n) for a 1 to 12, n 1 to 31 1 2 3 4 5 6 7 8 9 10 11 12 n _____________________________________________________ 1 1 1 1 1 1 1 1 1 1 1 1 3 1 -1 0 1 -1 0 1 -1 0 1 -1 5 1 -1 -1 1 0 1 -1 -1 1 0 1 7 1 1 -1 1 -1 -1 0 1 1 -1 1 9 1 1 0 1 1 0 1 1 0 1 1 11 1 -1 1 1 1 -1 -1 -1 1 -1 0 13 1 -1 1 1 -1 -1 -1 -1 1 1 -1 15 1 1 0 1 0 0 -1 1 0 0 -1 17 1 1 -1 1 -1 -1 -1 1 1 -1 -1 19 1 -1 -1 1 1 1 1 -1 1 -1 1 21 1 -1 0 1 1 0 0 -1 0 -1 -1 23 1 1 1 1 -1 1 -1 1 1 -1 -1 25 1 1 1 1 0 1 1 1 1 0 1 27 1 -1 0 1 -1 0 1 -1 0 1 -1 29 1 -1 -1 1 1 1 1 -1 1 -1 -1 31 1 1 -1 1 1 -1 1 1 1 1 -1
Kotlin
fun jacobi(A: Int, N: Int): Int {
assert(N > 0 && N and 1 == 1)
var a = A % N
var n = N
var result = 1
while (a != 0) {
var aMod4 = a and 3
while (aMod4 == 0) { // remove factors of four
a = a shr 2
aMod4 = a and 3
}
if (aMod4 == 2) { // if even
a = a shr 1 // remove factor 2 and possibly change sign
if ((n and 7).let { it == 3 || it == 5 })
result = -result
aMod4 = a and 3
}
if (aMod4 == 3 && n and 3 == 3)
result = -result
a = (n % a).also { n = a }
}
return if (n == 1) result else 0
}
Mathematica / Wolfram Language
TableForm[Table[JacobiSymbol[n, k], {n, 1, 17, 2}, {k, 16}],
TableHeadings -> {ReplacePart[Range[1, 17, 2], 1 -> "n=1"],
ReplacePart[Range[16], 1 -> "k=1"]}]
- Output:
Produces a nicely typeset table.
Nim
Translation of the Lua program from Wikipedia page.
template isOdd(n: int): bool = (n and 1) != 0
template isEven(n: int): bool = (n and 1) == 0
func jacobi(n, k: int): range[-1..1] =
assert k > 0 and k.isOdd
var n = n mod k
var k = k
result = 1
while n != 0:
while n.isEven:
n = n shr 1
if (k and 7) in [3, 5]:
result = -result
swap n, k
if (n and 3) == 3 and (k and 3) == 3:
result = -result
n = n mod k
if k != 1: result = 0
when isMainModule:
import strutils
stdout.write "n/k|"
for n in 1..20:
stdout.write align($n, 3)
echo '\n' & repeat("—", 64)
for k in countup(1, 21, 2):
stdout.write align($k, 2), " |"
for n in 1..20:
stdout.write align($jacobi(n, k), 3)
echo ""
- Output:
n/k| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ———————————————————————————————————————————————————————————————— 1 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 | 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 5 | 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 7 | 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 9 | 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 11 | 1 -1 1 1 1 -1 -1 -1 1 -1 0 1 -1 1 1 1 -1 -1 -1 1 13 | 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 0 1 -1 1 1 -1 -1 -1 15 | 1 1 0 1 0 0 -1 1 0 0 -1 0 -1 -1 0 1 1 0 1 0 17 | 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 0 1 1 -1 19 | 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 -1 0 1 21 | 1 -1 0 1 1 0 0 -1 0 -1 -1 0 -1 0 0 1 1 0 -1 1
Perl
use strict;
use warnings;
sub J {
my($k,$n) = @_;
$k %= $n;
my $jacobi = 1;
while ($k) {
while (0 == $k % 2) {
$k = int $k / 2;
$jacobi *= -1 if $n%8 == 3 or $n%8 == 5;
}
($k, $n) = ($n, $k);
$jacobi *= -1 if $n%4 == 3 and $k%4 == 3;
$k %= $n;
}
$n == 1 ? $jacobi : 0
}
my $maxa = 1 + (my $maxn = 29);
print 'n\k';
printf '%4d', $_ for 1..$maxa;
print "\n";
print ' ' . '-' x (4 * $maxa) . "\n";
for my $n (1..$maxn) {
next if 0 == $n % 2;
printf '%3d', $n;
printf '%4d', J($_, $n) for 1..$maxa;
print "\n"
}
- Output:
n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 ------------------------------------------------------------------------------------------------------------------------ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 5 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 7 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 9 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 11 1 -1 1 1 1 -1 -1 -1 1 -1 0 1 -1 1 1 1 -1 -1 -1 1 -1 0 1 -1 1 1 1 -1 -1 -1 13 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 0 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 0 1 -1 1 1 15 1 1 0 1 0 0 -1 1 0 0 -1 0 -1 -1 0 1 1 0 1 0 0 -1 1 0 0 -1 0 -1 -1 0 17 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 0 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 19 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 -1 0 1 -1 -1 1 1 1 1 -1 1 -1 1 21 1 -1 0 1 1 0 0 -1 0 -1 -1 0 -1 0 0 1 1 0 -1 1 0 1 -1 0 1 1 0 0 -1 0 23 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 0 1 1 1 1 -1 1 -1 25 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 27 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 29 1 -1 -1 1 1 1 1 -1 1 -1 -1 -1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 1 -1 -1 1 0 1
Phix
with javascript_semantics function jacobi(integer a, n) atom result = 1 a = remainder(a,n) while a!=0 do while remainder(a,2)==0 do a /= 2 if find(remainder(n,8),{3,5}) then result *= -1 end if end while {a, n} = {n, a} if remainder(a,4)==3 and remainder(n,4)==3 then result *= -1 end if a = remainder(a,n) end while return iff(n==1 ? result : 0) end function printf(1,"n\\a 0 1 2 3 4 5 6 7 8 9 10 11\n") printf(1," ________________________________________________\n") for n=1 to 31 by 2 do printf(1,"%3d", n) for a=0 to 11 do printf(1,"%4d",jacobi(a, n)) end for printf(1,"\n") end for
- Output:
n\a 0 1 2 3 4 5 6 7 8 9 10 11 ________________________________________________ 1 1 1 1 1 1 1 1 1 1 1 1 1 3 0 1 -1 0 1 -1 0 1 -1 0 1 -1 5 0 1 -1 -1 1 0 1 -1 -1 1 0 1 7 0 1 1 -1 1 -1 -1 0 1 1 -1 1 9 0 1 1 0 1 1 0 1 1 0 1 1 11 0 1 -1 1 1 1 -1 -1 -1 1 -1 0 13 0 1 -1 1 1 -1 -1 -1 -1 1 1 -1 15 0 1 1 0 1 0 0 -1 1 0 0 -1 17 0 1 1 -1 1 -1 -1 -1 1 1 -1 -1 19 0 1 -1 -1 1 1 1 1 -1 1 -1 1 21 0 1 -1 0 1 1 0 0 -1 0 -1 -1 23 0 1 1 1 1 -1 1 -1 1 1 -1 -1 25 0 1 1 1 1 0 1 1 1 1 0 1 27 0 1 -1 0 1 -1 0 1 -1 0 1 -1 29 0 1 -1 -1 1 1 1 1 -1 1 -1 -1 31 0 1 1 -1 1 1 -1 1 1 1 1 -1
Python
def jacobi(a, n):
if n <= 0:
raise ValueError("'n' must be a positive integer.")
if n % 2 == 0:
raise ValueError("'n' must be odd.")
a %= n
result = 1
while a != 0:
while a % 2 == 0:
a /= 2
n_mod_8 = n % 8
if n_mod_8 in (3, 5):
result = -result
a, n = n, a
if a % 4 == 3 and n % 4 == 3:
result = -result
a %= n
if n == 1:
return result
else:
return 0
Raku
(formerly Perl 6)
# Jacobi function
sub infix:<J> (Int $k is copy, Int $n is copy where * % 2) {
$k %= $n;
my $jacobi = 1;
while $k {
while $k %% 2 {
$k div= 2;
$jacobi *= -1 if $n % 8 == 3 | 5;
}
($k, $n) = $n, $k;
$jacobi *= -1 if 3 == $n%4 & $k%4;
$k %= $n;
}
$n == 1 ?? $jacobi !! 0
}
# Testing
my $maxa = 30;
my $maxn = 29;
say 'n\k ', (1..$maxa).fmt: '%3d';
say ' ', '-' x 4 * $maxa;
for 1,*+2 … $maxn -> $n {
print $n.fmt: '%3d';
for 1..$maxa -> $k {
print ($k J $n).fmt: '%4d';
}
print "\n";
}
- Output:
n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 ------------------------------------------------------------------------------------------------------------------------ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 5 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 7 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 9 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 11 1 -1 1 1 1 -1 -1 -1 1 -1 0 1 -1 1 1 1 -1 -1 -1 1 -1 0 1 -1 1 1 1 -1 -1 -1 13 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 0 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 0 1 -1 1 1 15 1 1 0 1 0 0 -1 1 0 0 -1 0 -1 -1 0 1 1 0 1 0 0 -1 1 0 0 -1 0 -1 -1 0 17 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 0 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 19 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 -1 0 1 -1 -1 1 1 1 1 -1 1 -1 1 21 1 -1 0 1 1 0 0 -1 0 -1 -1 0 -1 0 0 1 1 0 -1 1 0 1 -1 0 1 1 0 0 -1 0 23 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 1 -1 1 -1 -1 -1 -1 0 1 1 1 1 -1 1 -1 25 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 27 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 29 1 -1 -1 1 1 1 1 -1 1 -1 -1 -1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 1 1 -1 -1 1 0 1
REXX
A little extra code was added to make a prettier grid.
/*REXX pgm computes/displays the Jacobi symbol, the # of rows & columns can be specified*/
parse arg rows cols . /*obtain optional arguments from the CL*/
if rows='' | rows=="," then rows= 17 /*Not specified? Then use the default.*/
if cols='' | cols=="," then cols= 16 /* " " " " " " */
call hdrs /*display the (two) headers to the term*/
do r=1 by 2 to rows; _= right(r, 3) /*build odd (numbered) rows of a table.*/
do c=0 to cols /* [↓] build a column for a table row.*/
_= _ ! right(jacobi(c, r), 2); != '│' /*reset grid end char.*/
end /*c*/
say _ '║'; != '║' /*display a table row; reset grid glyph*/
end /*r*/
say translate(@.2, '╩╧╝', "╬╤╗") /*display the bottom of the grid border*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
hdrs: @.1= 'n/a ║'; do c=0 to cols; @.1= @.1 || right(c, 3)" "; end
L= length(@.1); @.1= left(@.1, L - 1) ; say @.1
@.2= '════╬'; do c=0 to cols; @.2= @.2 || "════╤" ; end
L= length(@.2); @.2= left(@.2, L - 1)"╗" ; say @.2
!= '║' ; return /*define an external grid border glyph.*/
/*──────────────────────────────────────────────────────────────────────────────────────*/
jacobi: procedure; parse arg a,n; er= '***error***'; $ = 1 /*define result.*/
if n//2==0 then do; say er n " must be a positive odd integer."; exit 13
end
a= a // n /*obtain A modulus N */
do while a\==0 /*perform while A isn't zero. */
do while a//2==0; a= a % 2 /*divide A (as a integer) by 2 */
if n//8==3 | n//8==5 then $= -$ /*use N mod 8 */
end /*while a//2==0*/
parse value a n with n a /*swap values of variables: A N */
if a//4==3 & n//4==3 then $= -$ /* A mod 4, N mod 4. Both ≡ 3 ?*/
a= a // n /*obtain A modulus N */
end /*while a\==0*/
if n==1 then return $
return 0
- output when using the default inputs:
n/a ║ 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ════╬════╤════╤════╤════╤════╤════╤════╤════╤════╤════╤════╤════╤════╤════╤════╤════╤════╗ 1 ║ 1 │ 1 │ 1 │ 1 │ 1 │ 1 │ 1 │ 1 │ 1 │ 1 │ 1 │ 1 │ 1 │ 1 │ 1 │ 1 │ 1 ║ 3 ║ 0 │ 1 │ -1 │ 0 │ 1 │ -1 │ 0 │ 1 │ -1 │ 0 │ 1 │ -1 │ 0 │ 1 │ -1 │ 0 │ 1 ║ 5 ║ 0 │ 1 │ -1 │ -1 │ 1 │ 0 │ 1 │ -1 │ -1 │ 1 │ 0 │ 1 │ -1 │ -1 │ 1 │ 0 │ 1 ║ 7 ║ 0 │ 1 │ 1 │ -1 │ 1 │ -1 │ -1 │ 0 │ 1 │ 1 │ -1 │ 1 │ -1 │ -1 │ 0 │ 1 │ 1 ║ 9 ║ 0 │ 1 │ 1 │ 0 │ 1 │ 1 │ 0 │ 1 │ 1 │ 0 │ 1 │ 1 │ 0 │ 1 │ 1 │ 0 │ 1 ║ 11 ║ 0 │ 1 │ -1 │ 1 │ 1 │ 1 │ -1 │ -1 │ -1 │ 1 │ -1 │ 0 │ 1 │ -1 │ 1 │ 1 │ 1 ║ 13 ║ 0 │ 1 │ -1 │ 1 │ 1 │ -1 │ -1 │ -1 │ -1 │ 1 │ 1 │ -1 │ 1 │ 0 │ 1 │ -1 │ 1 ║ 15 ║ 0 │ 1 │ 1 │ 0 │ 1 │ 0 │ 0 │ -1 │ 1 │ 0 │ 0 │ -1 │ 0 │ -1 │ -1 │ 0 │ 1 ║ 17 ║ 0 │ 1 │ 1 │ -1 │ 1 │ -1 │ -1 │ -1 │ 1 │ 1 │ -1 │ -1 │ -1 │ 1 │ -1 │ 1 │ 1 ║ ════╩════╧════╧════╧════╧════╧════╧════╧════╧════╧════╧════╧════╧════╧════╧════╧════╧════╝
Ruby
def jacobi(a, n)
raise ArgumentError.new "n must b positive and odd" if n < 1 || n.even?
res = 1
until (a %= n) == 0
while a.even?
a >>= 1
res = -res if [3, 5].include? n % 8
end
a, n = n, a
res = -res if [a % 4, n % 4] == [3, 3]
end
n == 1 ? res : 0
end
puts "Jacobian symbols for jacobi(a, n)"
puts "n\\a 0 1 2 3 4 5 6 7 8 9 10"
puts "------------------------------------"
1.step(to: 17, by: 2) do |n|
printf("%2d ", n)
(0..10).each { |a| printf(" % 2d", jacobi(a, n)) }
puts
end
- Output:
Jacobian symbols for jacobi(a, n) n\a 0 1 2 3 4 5 6 7 8 9 10 ------------------------------------ 1 1 1 1 1 1 1 1 1 1 1 1 3 0 1 -1 0 1 -1 0 1 -1 0 1 5 0 1 -1 -1 1 0 1 -1 -1 1 0 7 0 1 1 -1 1 -1 -1 0 1 1 -1 9 0 1 1 0 1 1 0 1 1 0 1 11 0 1 -1 1 1 1 -1 -1 -1 1 -1 13 0 1 -1 1 1 -1 -1 -1 -1 1 1 15 0 1 1 0 1 0 0 -1 1 0 0 17 0 1 1 -1 1 -1 -1 -1 1 1 -1
Rust
fn jacobi(mut n: i32, mut k: i32) -> i32 {
assert!(k > 0 && k % 2 == 1);
n %= k;
let mut t = 1;
while n != 0 {
while n % 2 == 0 {
n /= 2;
let r = k % 8;
if r == 3 || r == 5 {
t = -t;
}
}
std::mem::swap(&mut n, &mut k);
if n % 4 == 3 && k % 4 == 3 {
t = -t;
}
n %= k;
}
if k == 1 {
t
} else {
0
}
}
fn print_table(kmax: i32, nmax: i32) {
print!("n\\k|");
for k in 0..=kmax {
print!(" {:2}", k);
}
print!("\n----");
for _ in 0..=kmax {
print!("---");
}
println!();
for n in (1..=nmax).step_by(2) {
print!("{:2} |", n);
for k in 0..=kmax {
print!(" {:2}", jacobi(k, n));
}
println!();
}
}
fn main() {
print_table(20, 21);
}
- Output:
n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ------------------------------------------------------------------- 1 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 | 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 5 | 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 7 | 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 9 | 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 11 | 0 1 -1 1 1 1 -1 -1 -1 1 -1 0 1 -1 1 1 1 -1 -1 -1 1 13 | 0 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 0 1 -1 1 1 -1 -1 -1 15 | 0 1 1 0 1 0 0 -1 1 0 0 -1 0 -1 -1 0 1 1 0 1 0 17 | 0 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 0 1 1 -1 19 | 0 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 -1 0 1 21 | 0 1 -1 0 1 1 0 0 -1 0 -1 -1 0 -1 0 0 1 1 0 -1 1
Scala
def jacobi(a_p: Int, n_p: Int): Int =
{
var a = a_p
var n = n_p
if (n <= 0) return -1
if (n % 2 == 0) return -1
a %= n
var result = 1
while (a != 0) {
while (a % 2 == 0) {
a /= 2
if (n % 8 == 3 || n % 8 == 5) result = -result
}
val t = a
a = n
n = t
if (a % 4 == 3 && n % 4 == 3) result = -result
a %= n
}
if (n != 1) result = 0
result
}
def main(args: Array[String]): Unit =
{
for {
a <- 0 until 11
n <- 1 until 31 by 2
} yield println("n = " + n + ", a = " + a + ": " + jacobi(a, n))
}
- output:
n = 1, a = 0: 1 n = 3, a = 0: 0 n = 5, a = 0: 0 n = 7, a = 0: 0 n = 9, a = 0: 0 n = 1, a = 1: 1 n = 3, a = 1: 1 n = 5, a = 1: 1 n = 7, a = 1: 1 n = 9, a = 1: 1 n = 1, a = 2: 1 n = 3, a = 2: -1 n = 5, a = 2: -1 n = 7, a = 2: 1 n = 9, a = 2: 1 n = 1, a = 3: 1 n = 3, a = 3: 0 n = 5, a = 3: -1 n = 7, a = 3: -1 n = 9, a = 3: 0 n = 1, a = 4: 1 n = 3, a = 4: 1 n = 5, a = 4: 1 n = 7, a = 4: 1 n = 9, a = 4: 1 n = 1, a = 5: 1 n = 3, a = 5: -1 n = 5, a = 5: 0 n = 7, a = 5: -1 n = 9, a = 5: 1 n = 1, a = 6: 1 n = 3, a = 6: 0 n = 5, a = 6: 1 n = 7, a = 6: -1 n = 9, a = 6: 0 n = 1, a = 7: 1 n = 3, a = 7: 1 n = 5, a = 7: -1 n = 7, a = 7: 0 n = 9, a = 7: 1 n = 1, a = 8: 1 n = 3, a = 8: -1 n = 5, a = 8: -1 n = 7, a = 8: 1 n = 9, a = 8: 1 n = 1, a = 9: 1 n = 3, a = 9: 0 n = 5, a = 9: 1 n = 7, a = 9: 1 n = 9, a = 9: 0 n = 1, a = 10: 1 n = 3, a = 10: 1 n = 5, a = 10: 0 n = 7, a = 10: -1 n = 9, a = 10: 1
Scheme
(define jacobi (lambda (a n)
(let ((a-mod-n (modulo a n)))
(if (zero? a-mod-n)
(if (= n 1)
1
0)
(if (even? a-mod-n)
(case (modulo n 8)
((3 5) (- (jacobi (/ a-mod-n 2) n)))
((1 7) (jacobi (/ a-mod-n 2) n)))
(if (and (= (modulo a-mod-n 4) 3) (= (modulo n 4) 3))
(- (jacobi n a-mod-n))
(jacobi n a-mod-n)))))))
Sidef
Also built-in as kronecker(n,k).
func jacobi(n, k) {
assert(k > 0, "#{k} must be positive")
assert(k.is_odd, "#{k} must be odd")
var t = 1
while (n %= k) {
var v = n.valuation(2)
t *= (-1)**v if (k%8 ~~ [3,5])
n >>= v
(n,k) = (k,n)
t = -t if ([n%4, k%4] == [3,3])
}
k==1 ? t : 0
}
for n in (0..50), k in (0..50) {
assert_eq(jacobi(n, 2*k + 1), kronecker(n, 2*k + 1))
}
Swift
import Foundation
func jacobi(a: Int, n: Int) -> Int {
var a = a % n
var n = n
var res = 1
while a != 0 {
while a & 1 == 0 {
a >>= 1
if n % 8 == 3 || n % 8 == 5 {
res = -res
}
}
(a, n) = (n, a)
if a % 4 == 3 && n % 4 == 3 {
res = -res
}
a %= n
}
return n == 1 ? res : 0
}
print("n/a 0 1 2 3 4 5 6 7 8 9")
print("---------------------------------")
for n in stride(from: 1, through: 17, by: 2) {
print(String(format: "%2d", n), terminator: "")
for a in 0..<10 {
print(String(format: " % d", jacobi(a: a, n: n)), terminator: "")
}
print()
}
- Output:
n/a 0 1 2 3 4 5 6 7 8 9 --------------------------------- 1 1 1 1 1 1 1 1 1 1 1 3 0 1 -1 0 1 -1 0 1 -1 0 5 0 1 -1 -1 1 0 1 -1 -1 1 7 0 1 1 -1 1 -1 -1 0 1 1 9 0 1 1 0 1 1 0 1 1 0 11 0 1 -1 1 1 1 -1 -1 -1 1 13 0 1 -1 1 1 -1 -1 -1 -1 1 15 0 1 1 0 1 0 0 -1 1 0 17 0 1 1 -1 1 -1 -1 -1 1 1
V (Vlang)
fn jacobi(aa u64, na u64) ?int {
mut a := aa
mut n := na
if n%2 == 0 {
return error("'n' must be a positive odd integer")
}
a %= n
mut result := 1
for a != 0 {
for a%2 == 0 {
a /= 2
nn := n % 8
if nn == 3 || nn == 5 {
result = -result
}
}
a, n = n, a
if a%4 == 3 && n%4 == 3 {
result = -result
}
a %= n
}
if n == 1 {
return result
}
return 0
}
fn main() {
println("Using hand-coded version:")
println("n/a 0 1 2 3 4 5 6 7 8 9")
println("---------------------------------")
for n := u64(1); n <= 17; n += 2 {
print("${n:2} ")
for a := u64(0); a <= 9; a++ {
t := jacobi(a, n)?
print(" ${t:2}")
}
println('')
}
}- Output:
n/a 0 1 2 3 4 5 6 7 8 9 --------------------------------- 1 1 1 1 1 1 1 1 1 1 1 3 0 1 -1 0 1 -1 0 1 -1 0 5 0 1 -1 -1 1 0 1 -1 -1 1 7 0 1 1 -1 1 -1 -1 0 1 1 9 0 1 1 0 1 1 0 1 1 0 11 0 1 -1 1 1 1 -1 -1 -1 1 13 0 1 -1 1 1 -1 -1 -1 -1 1 15 0 1 1 0 1 0 0 -1 1 0 17 0 1 1 -1 1 -1 -1 -1 1 1
Wren
import "/fmt" for Fmt
var jacobi = Fn.new { |a, n|
if (!n.isInteger || n <= 0 || n%2 == 0) {
Fiber.abort("The 'n' parameter must be an odd positive integer.")
}
a = a % n
var result = 1
while (a != 0) {
while (a%2 == 0) {
a = a / 2
var nm8 = n % 8
if ([3, 5].contains(nm8)) result = -result
}
var t = a
a = n
n = t
if (a%4 == 3 && n%4 == 3) result = -result
a = a % n
}
return (n == 1) ? result : 0
}
System.print("Table of jacobi(a, n):")
System.print("n/a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15")
System.print("---------------------------------------------------------------")
var n = 1
while (n < 31) {
System.write(Fmt.d(3, n))
for (a in 1..15) System.write(Fmt.d(4, jacobi.call(a, n)))
System.print()
n = n + 2
}- Output:
Table of jacobi(a, n): n/a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 --------------------------------------------------------------- 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 5 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 7 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 9 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 11 1 -1 1 1 1 -1 -1 -1 1 -1 0 1 -1 1 1 13 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 0 1 -1 15 1 1 0 1 0 0 -1 1 0 0 -1 0 -1 -1 0 17 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 19 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 21 1 -1 0 1 1 0 0 -1 0 -1 -1 0 -1 0 0 23 1 1 1 1 -1 1 -1 1 1 -1 -1 1 1 -1 -1 25 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0 27 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 29 1 -1 -1 1 1 1 1 -1 1 -1 -1 -1 1 -1 -1
XPL0
func Jacobi(A, N);
int A, N, Result, T;
[if A >= N then A:= rem(A/N);
Result:= 1;
while A do
[while (A&1) = 0 do
[A:= A >> 1;
if (N&7) = 3 or (N&7) = 5 then Result:= -Result;
];
T:= A; A:= N; N:= T;
if (A&3) = 3 and (N&3) = 3 then Result:= -Result;
A:= rem(A/N);
];
if N = 1 then return Result;
return 0;
];
proc PrintTable(KMax, NMax);
int KMax, NMax, K, N;
[Text(0, "N\K|");
Format(3, 0);
for K:= 0 to KMax do RlOut(0, float(K));
CrLf(0);
Text(0, "----");
for K:= 0 to KMax do Text(0, "---");
CrLf(0);
for N:= 1 to NMax do
[Format(2, 0);
RlOut(0, float(N));
Text(0, " |");
Format(3, 0);
for K:= 0 to KMax do
RlOut(0, float(Jacobi(K, N)));
CrLf(0);
N:= N+1;
];
];
PrintTable(20, 21);- Output:
N\K| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ------------------------------------------------------------------- 1 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 | 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 0 1 -1 5 | 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 1 -1 -1 1 0 7 | 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 0 1 1 -1 1 -1 -1 9 | 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 0 1 1 11 | 0 1 -1 1 1 1 -1 -1 -1 1 -1 0 1 -1 1 1 1 -1 -1 -1 1 13 | 0 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 0 1 -1 1 1 -1 -1 -1 15 | 0 1 1 0 1 0 0 -1 1 0 0 -1 0 -1 -1 0 1 1 0 1 0 17 | 0 1 1 -1 1 -1 -1 -1 1 1 -1 -1 -1 1 -1 1 1 0 1 1 -1 19 | 0 1 -1 -1 1 1 1 1 -1 1 -1 1 -1 -1 -1 -1 1 1 -1 0 1 21 | 0 1 -1 0 1 1 0 0 -1 0 -1 -1 0 -1 0 0 1 1 0 -1 1
zkl
fcn jacobi(a,n){
if(n.isEven or n<1)
throw(Exception.ValueError("'n' must be a positive odd integer"));
a=a%n; result,t := 1,0;
while(a!=0){
while(a.isEven){
a/=2; n_mod_8:=n%8;
if(n_mod_8==3 or n_mod_8==5) result=-result;
}
t,a,n = a,n,t;
if(a%4==3 and n%4==3) result=-result;
a=a%n;
}
if(n==1) result else 0
}println("Using hand-coded version:");
println("n/a 0 1 2 3 4 5 6 7 8 9");
println("---------------------------------");
foreach n in ([1..17,2]){
print("%2d ".fmt(n));
foreach a in (10){ print(" % d".fmt(jacobi(a,n))) }
println();
}var [const] BI=Import.lib("zklBigNum"); // libGMP
println("\nUsing BigInt library function:");
println("n/a 0 1 2 3 4 5 6 7 8 9");
println("---------------------------------");
foreach n in ([1..17,2]){
print("%2d ".fmt(n));
foreach a in (10){ print(" % d".fmt(BI(a).jacobi(n))) }
println();
}- Output:
Using hand-coded version: n/a 0 1 2 3 4 5 6 7 8 9 --------------------------------- 1 1 1 1 1 1 1 1 1 1 1 3 0 1 -1 0 1 -1 0 1 -1 0 5 0 1 -1 -1 1 0 1 -1 -1 1 7 0 1 1 -1 1 -1 -1 0 1 1 9 0 1 1 0 1 1 0 1 1 0 11 0 1 -1 1 1 1 -1 -1 -1 1 13 0 1 -1 1 1 -1 -1 -1 -1 1 15 0 1 1 0 1 0 0 -1 1 0 17 0 1 1 -1 1 -1 -1 -1 1 1 Using BigInt library function: n/a 0 1 2 3 4 5 6 7 8 9 --------------------------------- 1 1 1 1 1 1 1 1 1 1 1 3 0 1 -1 0 1 -1 0 1 -1 0 5 0 1 -1 -1 1 0 1 -1 -1 1 7 0 1 1 -1 1 -1 -1 0 1 1 9 0 1 1 0 1 1 0 1 1 0 11 0 1 -1 1 1 1 -1 -1 -1 1 13 0 1 -1 1 1 -1 -1 -1 -1 1 15 0 1 1 0 1 0 0 -1 1 0 17 0 1 1 -1 1 -1 -1 -1 1 1
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