A Generalization of Siegel's Theorem and Hall's Conjecture

Abstract

Consider an elliptic curve defined over the rational numbers andembedded in projective space. The rational points on the curveare viewed as integer vectors with coprime coordinates. Whatcan be said about the rational points for which the number ofprime factors dividing a fixed coordinate does not exceed a fixedbound? If the bound is zero, then Siegel's theorem guaranteesthat there are only finitely many such points. We consider, theoreticallyand computationally, two conjectures: one is a generalizationof Siegel's theorem, and the other is a refinement thatresonates with Hall's conjecture.