International Journal of Mathematics and Mathematical Sciences
Volume 5 (1982), Issue 3, Pages 565-584
doi:10.1155/S0161171282000532

A new formulation of the law of octic reciprocity for primes ≡±3(mod8) and its consequences

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.