Elementary results for certain Diophantine equations (Q558138)

In a series of papers (1942--1954) W. Ljunggren proved that for a given positive integer \(a\), the Diophantine equations \(x^4-ay^2=1\) and \(x^2-ay^4=1\) have at most two solutions in pairs of positive integers \((x,y)\). The proofs of Ljunggren used complicated studies of units of certain quartic number fields. The author considers the problem: determine the number of such solutions. For example, using only elementary methods he solves completely this problem for the family of equations \(x^4-ay^2=1\) when \(a\) is an odd prime number or twice an odd prime number. The paper also contains many other results on similar equations and also on \(x^2-ay^4=1\).