A remark on rational octic reciprocity (Q1609957)

The author presents a new variant of an octic reciprocity law for primes \(p,q\) congruent to \(1\bmod 8\). It depends on the representations \(p=a^2-2b^2\), \(q=c^2-2d^2\), contrary to \(a^2+2b^2\), \(c^2+2d^2\) of a previous result [e.g., \textit{K. S. Williams}, Pac. J. Math. 63, 563-570 (1976; Zbl 0311.10004)]. Note that there are infinitely many choices of \(a,b,c,d\). Two proofs are given, one similar to the proof by Williams [op. cit.] and the other following an idea of \textit{C. Hélou} [Proc. Am. Math. Soc. 108, 861-866 (1990; Zbl 0701.11002)].