Complete solutions of certain Lebesgue--Ramanujan--Nagell type  equations

Kalyan Chakraborty                <meta name=

2020/97/3-4 (5) — DOI: 10.5486/PMD.2020.8752 — pp. 339-352

Complete solutions of certain Lebesgue—Ramanujan—Nagell type equations

Authors: Kalyan Chakraborty, Azizul Hoque and Richa Sharma

Abstract:

It is well-known that for $p=1,2,3,7,11,19,43,67,163$, the class number of $\mathbb{Q}(\sqrt{-p})$ is one. We use this fact to determine all the solutions of $x^2+p^m=4y^n$ in non-negative integers $x,y,m$ and $n$.

Keywords: Diophantine equation, Lebesgue-Ramanujan-Nagell type equation, integer solution, Lucas sequences, primitive divisors

Mathematics Subject Classification: 11D61, 11D41, 11R29