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

Counting smooth solutions to the equation A + B = C