The Diophantine equation $a{X}^4-b{Y}^2=1$

Abstract

As an application of the method of Thue-Siegel, we will resolve a conjecture of Walsh to the effect that the Diophantine equation $a{X}^4-b{Y}^2=1$, 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.

20 pages, To appear in Journal fur die Reine und Angewandte Mathematik (Crelle's Journal)

References in corpus (1)

• On squares in Lucas sequences