On the Number of ABC Solutions with Restricted Radical Sizes

TL;DR

This paper investigates a variant of the ABC Conjecture by counting solutions with restricted radical sizes, proving the conjecture for cases where the sum of exponents is at least 2.

Contribution

It proves the conjecture that the number of solutions grows as N^{a+b+c-1} when a+b+c ≥ 2, extending previous bounds.

Findings

Confirmed the growth rate for solutions when a+b+c ≥ 2

Established bounds on the number of solutions based on radical restrictions

Extended understanding of solution distribution in the ABC problem

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