Complete solutions of certain Lebesgue-Ramanujan-Nagell type equations

TL;DR

This paper completely solves certain Lebesgue-Ramanujan-Nagell type equations by leveraging class number properties of quadratic fields, providing explicit solutions for specific cases.

Contribution

It determines all solutions to the equation x^2 + p^m = 4 y^n for particular primes p with class number one, extending known results.

Findings

Explicit solutions for the equations with specified primes p.

Complete characterization of solutions for these Lebesgue-Ramanujan-Nagell type equations.

Utilization of class number one property to solve Diophantine equations.

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 $n$.