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

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

2009-03-10

arxiv.org/abs/0903.1742v1

Mathematics

Number Theory

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

Scientific paper

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.

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