On an elementary method for solving $Ax^4-By^2=1$

TL;DR

This paper investigates a new elementary method for solving specific quartic equations, providing a straightforward solution for Bumby's equation and proposing a conjecture that could extend to a broader class of similar equations.

Contribution

It introduces a simple elementary approach to solving certain quartic equations and formulates a conjecture that may generalize the method to an infinite family of equations.

Findings

Elementary solution for Bumby's equation $3X^4-2Y^2=1$

Computational and theoretical validation of the method

Proposes a conjecture for broader applicability

Abstract

A new method for solving quartic equations due to Luo and Lin is investigated both computationally and theoretically. As a result, a completely straightforward elementary method is given for solving Bumby's equation $3X^4-2Y^2=1$, along with a conjecture, which if resolved, would enable a similar proof for a possibly infinite family of similar equations.