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

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

2008-03-05

arxiv.org/abs/0803.0700v1

Mathematics

Number Theory

11 pages, 5 tables

Scientific paper

Consider an elliptic curve, defined over the rational numbers, and embedded in projective space. The rational points on the curve are viewed as integer vectors with coprime coordinates. What can be said about a rational point if a bound is placed upon the number of prime factors dividing a fixed coordinate? If the bound is zero, then Siegel's Theorem guarantees that there are only finitely many such points. We consider, theoretically and computationally, two conjectures: one is a generalization of Siegel's Theorem and the other is a refinement which resonates with Hall's conjecture.

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