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

## Authors

DOI: 10.1007/s000130050376

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

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## 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.