%0 Journal Article
%T A new formulation of the law of octic reciprocity for primes  ¡é ¡ë     ¡À3(mod8) and its consequences
%A Richard H. Hudson
%A Kenneth S. Williams
%J International Journal of Mathematics and Mathematical Sciences
%D 1982
%I Hindawi Publishing Corporation
%R 10.1155/s0161171282000532
%X Let p and q be odd primes with q ¡é ¡ë     ¡À3(mod8), p ¡é ¡ë  1(mod8)=a2+b2=c2+d2 and with the signs of a and c chosen so that a ¡é ¡ë  c ¡é ¡ë  1(mod4). In this paper we show step-by-step how to easily obtain for large q necessary and sufficient criteria to have ( ¡é   ¡¯1(q ¡é   ¡¯1)/2q(p ¡é   ¡¯1)/8 ¡é ¡ë  (a ¡é   ¡¯b)d/ac)j(modp) for j=1, ¡é ? |,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 ¡é ¡ë  3(mod8) and significant modification is needed to obtain similar results for q ¡é ¡ë     ¡À1(mod8). Only recently the author has completely resolved the case q ¡é ¡ë  5(mod8), j=1, ¡é ? |,8 and a sketch of the method appears in the closing section of this paper.
%K quartic and octic residuacity criteria
%K A.E. Western's formula
%K binary quadratic forms.
%U http://dx.doi.org/10.1155/S0161171282000532