Counting Smooth Solutions to the Equation A+B=C

Details Counting Smooth Solutions to the Equation A+B=C Counting Smooth Solutions to the Equation A+B=C

2011-02-24

arxiv.org/abs/1102.4911v2

Mathematics

Number Theory

35 pages latex; v2 corrected misprints

Scientific paper

This paper studies integer solutions to the Diophantine equation A+B=C in which none of A, B, C have a large prime factor. We set H(A, B,C) = max(|A|, |B|, |C|), and consider primitive solutions (gcd}(A, B, C)=1) having no prime factor p larger than (log H(A, B,C))^K, for a given finite K. On the assumption that the Generalized Riemann hypothesis (GRH) holds, we show that for any K > 8 there are infinitely many such primitive solutions having no prime factor larger than (log H(A, B, C))^K. We obtain in this range an asymptotic formula for the number of such suitably weighted primitive solutions.

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