Copyright © 1982 Richard H. Hudson and Kenneth S. Williams. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let p and q be odd primes with q≡±3(mod8), p≡1(mod8)=a2+b2=c2+d2 and with the signs of a and c chosen so that a≡c≡1(mod4). In this paper we show step-by-step how to easily obtain for large q necessary and sufficient criteria to have (−1(q−1)/2q(p−1)/8≡(a−b)d/ac)j(modp) for j=1,…,8 (the cases with j odd have been treated only recently [3] in connection with the sign ambiguity in Jacobsthal sums of order 4. This is accomplished by breaking the formula of A.E. Western into three distinct parts involving two polynomials and a Legendre symbol; the latter condition restricts the validity of the method presented in section 2 to primes q≡3(mod8) and significant modification is needed to obtain similar results for q≡±1(mod8). Only recently the author has completely resolved the case q≡5(mod8), j=1,…,8 and a sketch of the method appears in the closing section of this paper.
Our formulation of the law of octic reciprocity makes possible a considerable extension of the results for q≡±3(mod8) of earlier authors. In particular, the largest prime ≡3(mod8) treated to date is q=19, by von Lienen [6] when j=4 or 8 and by Hudson and Williams [3] when j=1,2,3,5,6, or 7. For q=19 there are 200 distinct choices relating a,b,c,d which are equivalent to (−q)(p−1)/8≡((a−b)d/ac)j(modp) for one of j=1,…,8. We give explicit results in this paper for primes as large as q=83 where there are 3528 distinct choices.
This paper makes several other minor contributions including a computationally efficient version of Gosset's [2] formulation of Gauss' law of quartic reciprocity, observations on sums ∑γi,j where the γi,j's are the defining parameters for the distinct choices mentioned above, and proof that the results of von Lienen [6] may not only be appreciably abbreviated, but may be put into a form remarkably similar to the case in which q is a quadratic residue but a quartic non-residue of p.
An important contribution of the paper consists in showing how to use Theorems
1 and 3 of [3], in conjunction with Theorem 4 of this paper, to reduce from (q+1/4)2 to (q−1)/2 the number of cases which must be considered to obtain the criteria in Theorems 2 and 3.