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

Research paper by Gary Walsh

Indexed on: 01 Aug '99Published on: 01 Aug '99Published in: Archiv der Mathematik

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 x2–Dy4 = 1, and that if two such solutions (x1, y1), (x2, y2) exist, with x1≤x2, 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 x2–kxy2 + y4 = 1, 4.