# Kronecker-Legendre Symbol x Over y

Kron(x, y) = Kronecker-Legendre symbol x over y.

See: Kronecker Symbol -- From MathWorld
And: Wolfram Mathematica Documentation -- Number Theory Functions, KroneckerSymbol[n, m]

Here are some notes from my program XICalc - Extra Precision Integer Calculator http://www.oocities.org/hjsmithh/download.html#XICalc :

The Kronecker symbol is an extension of the Jacobi symbol to all integers. It is variously written as (x/y) or (x|y) or (x vinculum y). Some values are:

```
y =: -9 -8 -7 -6 -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9
x = -9:  0  -  +  0  -  -  0  -  -  0  +  +  0  +  +  0  -  +  0
x = -8:  -  0  +  0  +  0  -  0  -  0  +  0  +  0  -  0  -  0  +
x = -7:  -  -  0  +  +  -  +  -  -  0  +  +  -  +  -  -  0  +  +
x = -6:  0  0  -  0  -  0  0  0  -  0  +  0  0  0  +  0  +  0  0
x = -5:  -  +  -  +  0  -  -  +  -  0  +  -  +  +  0  -  +  -  +
x = -4:  -  0  +  0  -  0  +  0  -  0  +  0  -  0  +  0  -  0  +
x = -3:  0  +  -  0  +  -  0  +  -  0  +  -  0  +  -  0  +  -  0
x = -2:  -  0  +  0  +  0  -  0  -  0  +  0  +  0  -  0  -  0  +
x = -1:  -  -  +  +  -  -  +  -  -  +  +  +  -  +  +  -  -  +  +
x =  0:  0  0  0  0  0  0  0  0  +  0  +  0  0  0  0  0  0  0  0
x =  1:  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +  +
x =  2:  +  0  +  0  -  0  -  0  +  0  +  0  -  0  -  0  +  0  +
x =  3:  0  -  -  0  -  +  0  -  +  0  +  -  0  +  -  0  -  -  0
x =  4:  +  0  +  0  +  0  +  0  +  0  +  0  +  0  +  0  +  0  +
x =  5:  +  -  -  +  0  +  -  -  +  0  +  -  -  +  0  +  -  -  +
x =  6:  0  0  -  0  +  0  0  0  +  0  +  0  0  0  +  0  -  0  0
x =  7:  +  +  0  +  -  +  +  +  +  0  +  +  +  +  -  +  0  +  +
x =  8:  +  0  +  0  -  0  -  0  +  0  +  0  -  0  -  0  +  0  +
x =  9:  0  +  +  0  +  +  0  +  +  0  +  +  0  +  +  0  +  +  0
```
where + is for +1 and − is for −1. For example Kron(3, 7) = −1, Kron(7, 3) = +1. This function is used internally by the Adleman function for prime number testing, and is part of the prime factorization by Elliptic Curve Method (ECM).

The algorithm for Kron(x, y) can be found in the C# source file PrimeFA.cs in my program XICalc. See download link above.