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April 10, 2015 By

A note on the Diophantine equation $x^2-dy^4=1$ with prime discriminant

C. R. Math. Rep. Acad. Sci. Canada Vol. 27, (2), 2005 pp. 54–57

June 30, 2005

D. Poulakis, Department of Mathematics, Aristotle University of Thessaloniki, University Campus, 541 24 Thessaloniki, Greece; email: poulakis@ccf.auth.gr

P.G. Walsh, Department of Mathematics, University of Ottawa. 585 King Edward St., Ottawa, Ontario K1N-6N5; email: gwalsh@mathstat.uottawa.ca

Abstract/Résumé:

Ljunggren proved that for a nonsquare positive integer \(d\), the quartic Diophantine equation \(X^2-dY^4=1\) has at most two solutions in positive integers, and gave precise information on the location of these solutions in the case that two such solutions actually do exist. Inspired by recent work of P. Samuel, we show that in the case that \(d>3\) is prime, there is at most one positive integer solution to \(X^2-dY^4=1\), and that it arises from the fundamental solution of the Pell equation \(X^2-dY^2=1\).

Ljunggren a montré que pour un nombre entier positif de nonsquare \(d\), l’équation \(X^2-dY^4=1\) a au plus deux solutions dans des nombres entiers positifs, et a fourni l’information précise sur l’endroit de ces solutions dans le cas que deux telles solutions réellement existent. Inspirer par les travaux récents de P. Samuel, nous montrons cela dans le cas que \(d>3\) est une nombre premier, il y a au plus une solution positive de nombre entier \(X^2-dY^4=1\), et qu’elle résulte de la solution fondamentale de l’équation de Pell \(X^2-dY^2=1\).

Keywords:
AMS Subject Classification: Higher degree equations; Fermat's equation 11D41

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