Congruence ABC implies ABC

Abstract

The ABC conjecture of Masser and Oesterle&#39; 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&#39; 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 &#34;congruence ABC conjecture&#34; that ABC holds for all (a,b,c) with N|abc implies the full ABC conjecture.