A note on a theorem of Ljunggren and the Diophantine equations x 2 - kxy 2 + y 4 = 1, 4

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 x 2–Dy 4 = 1, and that if two such solutions (x 1, y 1), (x 2, y 2) exist, with x 1≤x 2, 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 x 2–kxy 2 + y 4 = 1, 4.