Congruence ABC implies ABC
Number Theory
2007-05-23 v1
Abstract
The ABC conjecture of Masser and Oesterle' states that if (a,b,c) are coprime integers with a + b + c = 0, then sup(|a|,|b|,|c|) < c_e (rad(abc))^{1+e} for any e > 0. Oesterle' has observed that if the ABC conjecture holds for all (a,b,c) with 16 | abc, then the full ABC conjecture holds. We extend that result to show that, for every integer N, the "congruence ABC conjecture" that ABC holds for all (a,b,c) with N|abc implies the full ABC conjecture.
Keywords
Cite
@article{arxiv.math/9909098,
title = {Congruence ABC implies ABC},
author = {Jordan S. Ellenberg},
journal= {arXiv preprint arXiv:math/9909098},
year = {2007}
}
Comments
4 pages; to appear, Indag. Math