A new formulation of the law of octic reciprocity for primes ≡±3(mod8) and its consequences - Richard H. Hudson; Kenneth S. Williams (International Journal of Mathematics and Mathematical Sciences)
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.