  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.
