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.