On the number of ABC solutions with restricted radical sizes

We establish upper and lower bounds for the number of solutions to A+B=C in relatively prime integers A,B,C with max⁡(|A|,|B|,|C|)≤N and Rad(A)≤|A|a,Rad(B)≤|B|b,Rad(C)≤|C|c, valid when 00. In particular, these bounds match up to Nϵ factors so long as a+b+c≥2, verifying a conjecture or Mazur in this parameter range.