Generalizing a Pair of Diophantine Equations
Abstract
For coprime integers a and b, it is known that exactly one of the two Diophantine equations ax+by = ((a−1)(b−1))/2 and 1+ax+by = ((a−1)(b−1))/2 admits a nonnegative integer solution, and that this solution is unique. We first generalize this result by replacing the right-hand side with an arbitrary integer m and its complement ab−a−b−m. This framework enables us to study the existence and uniqueness of nonnegative integer solutions to ax+by = ((a−1)(b−1))/k and 1+ax+by = ((a−1)(b−1))/k, where k is a fixed positive integer. We then obtain explicit results when a and b are consecutive Fibonacci numbers. Finally, we examine the original pair of equations in several particular settings, including when b≡ ±1mod a, when b is replaced by a higher power, and when the parameters are squared.
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BibTeX
@misc{chu2026generalizing,
title = {{Generalizing a Pair of Diophantine Equations}},
author = {Hung Viet Chu and Dongho Kim and Theodore Koch and Steven J. Miller and Anh Viet Nguyen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08728},
eprint = {2609.08728},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08728},
}
RIS
TY - UNPB
TI - Generalizing a Pair of Diophantine Equations
AU - Hung Viet Chu
AU - Dongho Kim
AU - Theodore Koch
AU - Steven J. Miller
AU - Anh Viet Nguyen
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08728
UR - https://arxiv.org/abs/2609.08728
AB - For coprime integers a and b, it is known that exactly one of the two Diophantine equations ax+by = ((a−1)(b−1))/2 and 1+ax+by = ((a−1)(b−1))/2 admits a nonnegative integer solution, and that this solution is unique. We first generalize this result by replacing the right-hand side with an arbitrary integer m and its complement ab−a−b−m. This framework enables us to study the existence and uniqueness of nonnegative integer solutions to ax+by = ((a−1)(b−1))/k and 1+ax+by = ((a−1)(b−1))/k, where k is a fixed positive integer. We then obtain explicit results when a and b are consecutive Fibonacci numbers. Finally, we examine the original pair of equations in several particular settings, including when b≡ ±1mod a, when b is replaced by a higher power, and when the parameters are squared.
ER -