Abstract

A method for determining which natural numbers satisfy reciprocity is given. The method is applicable to quadratic, cubic, quintic, and in general “prime” reciprocity. The method is also applicable to biquadratic reciprocity. The even powers of a primitive root of a prime are quadratic residues and the odd powers are quadratic nonresidues. This is generalized to cubic residues and nonresidues, etc. Let n denote the “degree” of prime reciprocity (2 for quadratic reciprocity, 3 for cubic reciprocity, 5 for quintic reciprocity, etc.). The residues and nonresidues are determined for the degree 2n and applied to the degree of n. For example, the residues and nonresidues for biquadratic reciprocity are used to analyze quadratic reciprocity. For a degree of 2n, there are 2 groups of residues of the same size and 2n – 2 groups of nonresidues all the same size as each of the two groups of residues. Each of the 2n groups is mapped to certain differences modulo p of the sorted least residues of one of the groups of nonresidues. This is a one-to-one transformation since it does not change the elements of a group. When certain counts associated with the differences are not distinct, groups are effectively merged together. The number of distinct difference counts will be referred to as the “degrees of freedom”. For quadratic reciprocity, there are either 1 or 2 degrees of freedom. For quintic reciprocity, there are up to 5 degrees of freedom and as few as 2 degrees of freedom. This transformation is useful for identifying properties of the residues and nonresidues. Also, reciprocity is not entirely restricted to primes. Reciprocity is interpreted as being a collection of finite commutative groups.

Keywords: Quadratic reciprocity law, Supplemental reciprocity laws, Perron’s theorem, Gaussian sum