2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/222819As 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)Number Theory11D25, 11D41, 11B39, 11J25text