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