A new formulation of the law of octic reciprocity for primes ≡ ± 3 ( mod 8 ) and its consequences

Let p and q be odd primes with q ≡ ± 3 ( mod 8 ) , p ≡ 1 ( mod 8 ) = a 2 + b 2 = c 2 + d 2 and with the signs of a and c chosen so that a ≡ c ≡ 1 ( mod 4 ) . In this paper we show step-by-step how to easily obtain for large q necessary and sufficient criteria to have ( − 1 ( q − 1 ) / 2 q ( p − 1 ) / 8 ≡ ( a − b ) d / a c ) j ( mod p ) 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 ( mod 8 ) and significant modification is needed to obtain similar results for q ≡ ± 1 ( mod 8 ) . Only recently the author has completely resolved the case q ≡ 5 ( mod 8 ) , 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 ( mod 8 ) of earlier authors. In particular, the largest prime ≡ 3 ( mod 8 ) 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 / a c ) j ( mod p ) 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 ∑ γ i , j where the γ i , 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 .