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

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