On the Number of ABC Solutions with Restricted Radical Sizes

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 $r(A)<\mathrm{\mid }A{\mathrm{\mid }}^a,r(B)<\mathrm{\mid }B{\mathrm{\mid }}^b,r(C)<\mathrm{\mid }C{\mathrm{\mid }}^c\mathrm{.}$ The ABC Conjecture is equivalent to the statement that for $a+b+c<1$, the number of solutions is bounded independently of $N$. If $a+b+c\ge 1$, it is conjectured that the number of solutions is asymptotically $N^{a+b+c-1\pm ϵ}\mathrm{.}$ We prove this conjecture as long as $a+b+c\ge 2.$