Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On a Pair of Diophantine Equations

Sujith Uthsara Kalansuriya Arachchi, Hung Viet Chu, Jiasen Liu, Qitong Luan, Rukshan Marasinghe, Steven J. Miller

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.

On a Pair of Diophantine Equations

Abstract

For relatively prime natural numbers and , we study the two equations and , 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 . We then use the criteria to study the sequence of equations used by the pair from several special sequences . Finally, fixing , we investigate the periodicity of the sequence of equations used by the pair as increases.
Paper Structure (7 sections, 10 theorems, 75 equations)

This paper contains 7 sections, 10 theorems, 75 equations.

Table of Contents

  1. Introduction
  2. On $\Gamma(a,b)$ and Preliminary Results
  3. The Sequence $\Delta((a_n)_n)$ for Some Special $(a_n)_n$
  4. Sequences Whose $\Delta$ Has a Periodic Form
  5. The $k$th-Power Sequence
  6. Arithmetic Progression
  7. Shifted Geometric Progression

Key Result

theorem thmcountertheorem

Let $a,b\in \mathbb{N}$. If $a$ divides $b$ or $b$ divides $a$, then $\Gamma(a, b) = 1$. Otherwise,

Theorems & Definitions (23)

  • theorem thmcountertheorem
  • remark thmcounterremark
  • theorem thmcountertheorem
  • remark thmcounterremark
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • proof : Proof of Theorem \ref{['m1']}
  • corollary thmcountercorollary
  • ...and 13 more