Determination of a unique solution of the quadratic partition for primes p ≡ 1 (mod 7)

Budh Singh Nashier and A. R. Rajwade

Vol. 72 (1977), No. 2, 513–521

DOI: 10.2140/pjm.1977.72.513

Abstract

Let p be a rational prime ≡ 1 (mod 7). Williams shows that a certain triple of a Diophantine system of quadratic equations has exactly six nontrivial solutions. We obtain here a congruence condition which uniquely fixes one of these six solutions. Further if 2 is not a seventh power residue (mod p) then we obtain a congruence (mod p) for 2^(p−1)∕7 in terms of the above uniquely fixed solution.

Mathematical Subject Classification

Primary: 10B05, 10B05

Milestones

Received: 4 April 1977

Published: 1 October 1977