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