Jacobi symbol
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
Factor
The jacobi word already exists in the math.extras vocabulary. See the implementation here.
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. <lang go>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()
}
}</lang>
- 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
Julia
<lang 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 == 3) && (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
</lang>
- 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
Perl 6
<lang perl6># 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";
}</lang>
- 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
Python
<lang python>def jacobi(a, n):
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</lang>
Scheme
<lang 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)))))))</lang>
zkl
<lang 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
}</lang> <lang zkl>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();
}</lang>
GNU Multiple Precision Arithmetic Library
<lang zkl>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();
}</lang>
- 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