Smooth solutions to the abc equation: the xyz Conjecture

Details Smooth solutions to the abc equation: the xyz Conjecture Smooth solutions to the abc equation: the xyz Conjecture

2009-11-21

arxiv.org/abs/0911.4147v4

J. Theor. Nombres Bordeaux 23 (2011), No. 1, 209--234

Mathematics

Number Theory

21 pages, presented at 26th Journees Arithmetiques, 2009; v2 added new examples 1.2, updated references; v3 changed title, mor

Scientific paper

This paper studies integer solutions to the ABC equation A+B+C=0 in which none of A, B, C has a large prime factor. Set H(A,B, C)= max(|A|,|B|,|C|) and set the smoothness S(A, B, C) to be the largest prime factor of ABC. We consider primitive solutions (gcd(A, B, C)=1) having smoothness no larger than a fixed power p of log H. Assuming the abc Conjecture we show that there are finitely many solutions if p<1. We discuss a conditional result, showing that the Generalized Riemann Hypothesis (GRH) implies there are infinitely many primitive solutions when p>8. We sketch some details of the proof of the latter result.

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