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Definition

For any integer a and any positive odd integer n, the Jacobi symbol (a / n) is defined as the product of the Legendre symbols corresponding to the prime factors of n:

where

is the prime factorization of n.

The Legendre symbol (a / p) is defined for all integers a and all odd primes p by

Following the normal convention for the empty product, (a / 1) = 1.

When the lower argument is an odd prime, the Jacobi symbol is equal to the Legendre symbol.