Archiv der Mathematik

, Volume 73, Issue 2, pp 119–125

A note on a theorem of Ljunggren and the Diophantine equations x2–kxy2 + y4 = 1, 4

Authors

  • Gary Walsh
    • Department of Mathematics, University of Ottawa, 585 King Edward, Ottawa, Ontario K1N-6N5, Canada

DOI: 10.1007/s000130050376

Cite this article as:
Walsh, G. Arch. Math. (1999) 73: 119. doi:10.1007/s000130050376

Abstract.

Let D denote a positive nonsquare integer. Ljunggren has shown that there are at most two solutions in positive integers (x, y) to the Diophantine equation x2Dy4 = 1, and that if two such solutions (x1, y1), (x2, y2) exist, with x1x2, then \(x_1+y_1^2\sqrt {D}\) is the fundamental unit \(\epsilon _{D}\) in the quadratic field \({\Bbb Q}(\sqrt {D})\), and \(x_2+y_2^2\sqrt {D}\) is either \(\epsilon _{D}^2\) or \(\epsilon _{D}^4\). The purpose of this note is twofold. Using a recent result of Cohn, we generalize Ljunggren’s theorem. We then use this generalization to completely solve the Diophantine equations x2kxy2 + y4 = 1, 4.

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Copyright information

© Birkhäuser Verlag, Basel 1999