On a Pair of Diophantine Equations

TL;DR

This paper investigates when certain Diophantine equations derived from cyclotomic polynomials have solutions, providing criteria to identify the relevant equation for given pairs and analyzing the periodicity of related sequences.

Contribution

It introduces criteria to determine which of two related equations applies to a pair of numbers and studies the behavior of sequences generated by these criteria, including their periodicity.

Findings

Exactly one of the two equations has a nonnegative solution for coprime pairs.

Criteria to identify which equation applies to a given pair.

Analysis of periodicity in sequences derived from these equations.

Abstract

For relatively prime natural numbers $a$ and $b$, we study the two equations $ax+by = (a-1)(b-1)/2$ and $ax+by+1= (a-1)(b-1)/2$, which arise from the study of cyclotomic polynomials. Previous work showed that exactly one equation has a nonnegative solution, and the solution is unique. Our first result gives criteria to determine which equation is used for a given pair $(a,b)$. We then use the criteria to study the sequence of equations used by the pair $(a_n/\gcd{(a_n, a_{n+1})}, a_{n+1}/\gcd{(a_n, a_{n+1})})$ from several special sequences $(a_n)_{n\geq 1}$. Finally, fixing $k \in \mathbb{N}$, we investigate the periodicity of the sequence of equations used by the pair $(k/\gcd{(k, n)}, n/\gcd{(k, n)})$ as $n$ increases.