On the Number of ABC Solutions with Restricted Radical Sizes

Abstract

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 $a+b+c\ge 1$, it is conjectured that the number of solutions is asymptotically $N^{a+b+c-1\pm \epsilon }.$ We prove this conjecture as long as $a+b+c\ge 2.$