Congruence ABC implies ABC

Details Congruence ABC implies ABC Congruence ABC implies ABC

1999-09-17

arxiv.org/abs/math/9909098v1

Mathematics

Number Theory

4 pages; to appear, Indag. Math

Scientific paper

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.

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