On the generalized Ramanujan-Nagell equation x2+bm=cn with a2+br=c2

Nobuhiro Terai; Saya Nakashiki; Yudai Suenaga

doi:10.55937/sut/1654320039

June 2022 On the generalized Ramanujan-Nagell equation

{x^2} + {b^m} = {c^n}

with

{a^2} + {b^r} = {c^2}

Nobuhiro Terai, Saya Nakashiki, Yudai Suenaga

Author Affiliations +

SUT J. Math. 58(1): 77-89 (June 2022). DOI: 10.55937/sut/1654320039

Abstract

Let $a$ , $b$ , $c$ be pairwise relatively prime positive integers such that ${a^2} + {b^r} = {c^2}$ with $r$ positive integer. Then we show that the equation ${x^2} + {b^m} = {c^n}$ has the positive integer solution $(x,m,n) = (a,r,2)$ only under some conditions. The proof is based on elementary methods and Zsigmondy’s theorem.

Acknowledgments

The authors would like to thank the referee for his valuable suggestions. The first author is supported by JSPS KAKENHI Grant Number 18K03247.

Citation

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Nobuhiro Terai. Saya Nakashiki. Yudai Suenaga. "On the generalized Ramanujan-Nagell equation ${x^2} + {b^m} = {c^n}$ with ${a^2} + {b^r} = {c^2}$ ." SUT J. Math. 58 (1) 77 - 89, June 2022. https://doi.org/10.55937/sut/1654320039