A classical theorem of Siegel asserts that the set of S-integral points of an algebraic curve C over a number field is finite unless C has genus 0 and at most two points at infinity. In this paper we give necessary and sufficient conditions for C to have infinitely many S-integral points.

Let A âˆˆ {k2(k2l2 + 1), 4k2(k2(2l âˆ’ 1)2 + 1)}, where k and l are positive integers, and let B be a non-zero square-free integer such that |B| < âˆš A. In this paper we determine all the possible integer solutions of the equation y2 = Ax4 + B by using terms of Lucas sequences of the form mx2.