Three term recurrence and residue completeness

TL;DR

This paper investigates the residue completeness of three-term recurrence sequences, specifically Pell and Pell-Lucas numbers, modulo m, establishing precise conditions under which these sequences are residue complete.

Contribution

It provides a complete characterization of when Pell and Pell-Lucas numbers are residue complete modulo m, a novel result in recurrence sequence analysis.

Findings

Pell numbers are residue complete modulo m if and only if m is 2, a power of 3, or a power of 5.

Pell-Lucas numbers are residue complete modulo m if and only if m is a power of 3.

The paper offers new insights into the modular behavior of these recurrence sequences.

Abstract

We study the three term recurrence modulo m. In particular, we prove that Pell numbers modulo m is residue complete if and only m is 2, a power of 3, or a power of 5. Pell-Lucas numbers modulo m is residue complete if and only if m is a power of 3.