The number of solutions to $y^2=px(Ax^2+2)$

TL;DR

This paper establishes an upper bound on the positive solutions to a specific quadratic Diophantine equation and confirms a related conjecture in certain cases, advancing understanding in number theory.

Contribution

It provides a new bound for solutions to the equation and proves Togbé's conjecture in specific instances, improving previous results.

Findings

Bound on the number of positive solutions established

Conjecture of Togbé proved in particular cases

Enhanced understanding of solutions to the equation

Abstract

In this paper, we find a bound for the number of the positive solutions to the titled equation, improving a result of Togb\'e. As a consequence, we prove a conjecture of Togb\'e in a few cases.