Octic reciprocity

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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.

There is a rational reciprocity law for 8th powers, due to Williams. Define the symbol (x|p)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 (p|q) = (q|p) = +1. Let p = a2 + b2 = c2 + 2d2 and q = A2 + B2 = C2 + 2D2, with aA odd. Then

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