The Diophantine equation $aX^{4} - bY^{2} = 1$

TL;DR

This paper proves that for fixed positive integers a and b, the Diophantine equation aX^4 - bY^2 = 1 has at most two solutions in positive integers, confirming a conjecture of Walsh using Thue-Siegel methods.

Contribution

It establishes a sharp upper bound of two solutions for the equation, resolving Walsh's conjecture with a novel application of Thue-Siegel techniques.

Findings

The equation has at most two positive integer solutions for fixed a and b.

The result confirms Walsh's conjecture and is sharp due to existing infinite solution pairs.

The method applies Thue-Siegel techniques to a specific class of Diophantine equations.

Abstract

As an application of the method of Thue-Siegel, we will resolve a conjecture of Walsh to the effect that the Diophantine equation $aX^{4} - bY^2=1$, for fixed positive integers $a$ and $b$, possesses at most two solutions in positive integers $X$ and $Y$. Since there are infinitely many pairs $(a,b)$ for which two such solutions exist, this result is sharp.