On the Number of ABC Solutions with Restricted Radical Sizes

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We consider a variant of the ABC Conjecture, attempting to count the number of solutions to A+B+C=0, in relatively prime integers A,B,C each of absolute value less than N with r(A)<|A|ᵃ, r(B)<|B|ᵇ, r(C)<|C|ᶜ. The ABC Conjecture is equivalent to the statement that for a+b+c<1, the number of solutions is bounded independently of N. If a+b+c ≥ 1, it is conjectured that the number of solutions is asymptotically Nᵃ⁺ᵇ⁺ᶜ⁻¹ ± ε. We prove this conjecture as long as a+b+c ≥ 2.