On the Number of ABC Solutions with Restricted Radical Sizes
نویسنده
چکیده
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) ≤ N,Rad(B) ≤ N,Rad(C) ≤ N, valid when 0 < a, b, c ≤ 1. The lower bound is Ω(Na+b+c−1(logN)−2), and the upper bound is of the form O(Na+b+c−1+ + N ), for any fixed > 0. In particular, these bounds match up to N factors so long as a+ b+ c ≥ 2, verifying a conjecture or Mazur in this parameter range.
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