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 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 Let p = a<sup>2</sup> + b<sup>2</sup> = c<sup>2</sup> + 2d<sup>2</sup> and q = A<sup>2</sup> + B<sup>2</sup> = C<sup>2</sup> + 2D<sup>2</sup>, with aA odd. Then