A NEW FORMULATION OF THE LAW OF OCTIC RECIPROCITY FOR PRIMES + 3 (MOD 8) AND ITS CONSEQUENCES

2 b2ABSTRACT. Let p and q be odd primes with q---+3 (rood 8), p i (mod 8) a + 2 d2c + and with the signs of a and c chosen so that a c i (rood 4). In this paper we show step-by-step how to easily obtain for large q necessary and sufficient criteria to have (-l(q-1)/2 (p-1)/8q (a-b)d/ac)J(mod p) for j i 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 sec-tion 2 to primes q 3 (mod 8) and significant modification is needed to obtain slm-ilar results for q--+l (rood 8). Only recently the author has completely resolved the case q 5 (mod 8), j I, 8 and a sketch of the method appears in the clos-ing 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