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

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)root D is the fundamental unit epsilon(D) in the quadratic field Q(root D), and x(2) + y(2)(2)root D is either epsilon(D)(2) or E-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.