Solving quadratic congruences

Abstract

This paper deals with the problem of determining all solutions, in positive integers, of the congruence systems X2 ≡1 (mod Y) Y2 ≡1 (mod X) and X2 ≡1 (mod Y) Y2 ≡1 (mod Z) Z2 ≡1 (mod X) For this purpose, we firstly solve the system of equations in two variables. This system and its solutions are well known; the aim of this work is to point out that these solutions have an inner structure in the sense that each solution is uniquely given by consecutive elements of a particular kind of Lucas sequence. Secondly and as for cases of system of equations in three variables, particular families of solutions are shown identifying ways to define triplets (x,y,z) presented as solutions of the system of equations. The inner structure of the set of all solutions of the three-variable quadratic system is also a wide problem to study.