The Diophantine equation aX4-bY2=1

In a series of papers over nearly forty years, Ljunggren derived remarkably sharp bounds for the number of solutions to various quartic Diophantine equations, particularly those of the shape aX4-bY2=±1, typically via a sophisticated application of Skolem's p-adic method. More recent results along these lines are well surveyed in a paper of Walsh. For general a and b, however, there is no absolute upper bound for the number of integral solutions to aX4-bY2=1 available in the literature. Computations and assorted heuristics suggest the following conjecture of Walsh: For any positive integers a and b, the equation aX4-bY2=1 has at most two solutions in positive integers X and Y. In this talk, we will appeal to a classical result of Thue from the theory of Diophantine approximation to deduce the following result: For any positive integers a and b, the equation aX4-bY2=1 has at most three solutions in positive integers X and Y.