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In number theory, octic reciprocity is a reciprocity law relating the residues of 8th powers modulo primes, analogous to the law of quadratic reciprocity, cubic reciprocity, and quartic reciprocity.
There is a rational reciprocity law for 8th powers, due to Williams. Define the symbol k to be +1 if x is a k-th power modulo the prime p and -1 otherwise. Let p and q be distinct primes congruent to 1 modulo 8, such that = = +1. Let p = a + b = c + 2d and q = A + B = C + 2D, with aA odd. Then
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