The Diophantine equation $aX^{4} - bY^{2} = 1$
Abstract
Description
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.
20 pages, To appear in Journal fur die Reine und Angewandte Mathematik (Crelle's Journal)
20 pages, To appear in Journal fur die Reine und Angewandte Mathematik (Crelle's Journal)