A note on a theorem of Ljunggren and the diophantine equations \(x^2-kxy^2+y^4=1,4\)
Let \(D\) denote a positive nonsquare integer. W. Ljunggren has shown that there are at most two solutions in positive integers \((x,y)\) to the diophantine equation \(x^2-Dy^2=1\), and that if two 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 \(\varepsilon_D\) in the quadratic field \(\mathbb{Q}(\sqrt D)\), and \(x_2+y^2_2\sqrt D\) is either \(\varepsilon^2_D\) or \(\varepsilon^4_D\). The purpose of this note is twofold: Using a result of \textit{J. H. E. Cohn} [Acta Arith. 78, 401-403 (1997; Zbl 0870.11018)], Ljunggren's theorem is generalized. This generalization is then used to completely solve the diophantine equations in the title.
scientific article; zbMATH DE number 1346254