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arXiv · 2509.01781

A Pair of Diophantine Equations and Fibonacci-Like Sequences

Hung Viet Chu·Rishabh Gulecha·Sicheng Guo·Nathanael Johnson·Steven J. Miller·Yeju Shin
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Abstract

Given two relatively prime numbers $a$ and $b$, it is known that exactly one of the two Diophantine equations has a nonnegative integral solution $(x,y)$: $$ ax + by \ =\ \frac{(a-1)(b-1)}{2}\quad \mbox{ and }\quad 1 + ax + by \ =\ \frac{(a-1)(b-1)}{2}. $$ Furthermore, the solution is unique. This paper surveys recent results on finding the solution and determining which equation is used when $a$ and $b$ are taken from certain sequences. We contribute to the literature by finding $(x,y)$ when $a$ and $b$ are consecutive terms of sequences having the Fibonacci recurrence and arbitrary initial terms.

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Hung Viet Chu, Rishabh Gulecha, Sicheng Guo, Nathanael Johnson, Steven J. Miller, Yeju Shin. 2025-09-01. A Pair of Diophantine Equations and Fibonacci-Like Sequences. https://arxiv.org/abs/2509.01781

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