The Intricacies of Pairwise Modular Multiplicative Inverse in Lucas Numbers

Abstract

Let (p,q) be a pair of relatively prime integers greater than 1. The pairwise modular multiplicative inverse (PMMI) of (p,q) is defined as the unique pair of positive integers (p′, q′) such that p p′ ≡ 1 (mod q), p′ \u3c q, qq′ ≡ 1 (mod p), q′ \u3c p. In this paper, we determine all pairs of Lucas numbers such that their PMMIs are pairs of Lucas numbers