Counting Smooth Solutions to the Equation A+B=C

TL;DR

This paper investigates the distribution of primitive solutions to A+B=C with restricted prime factors, establishing their infinitude and asymptotic counts under the assumption of GRH.

Contribution

It proves the existence of infinitely many primitive solutions with bounded prime factors for K>8, providing an asymptotic formula under GRH.

Findings

Infinitely many solutions exist for K>8 under GRH.

Asymptotic formula for the count of solutions with bounded prime factors.

Results depend on the validity of the Generalized Riemann Hypothesis.

Abstract

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.