Note on the Exceptional Set in the ABC Conjecture

Abstract

Fix $\epsilon >0$, let $x>1$ be a large real number and let $\text{rad}(n)={\prod }_{p\mid n}p$ be the radical of an integer $n\ge 1$. A triple $(a,b,c)$, with $a+b=c$ and $\gcd (a,b,c)=1$, such that $c>(\text{rad}(abc){)}^{1+\epsilon }$, is called exceptional triple. Recent works have proved that the cardinality $\#\mathcal{E}(x)$ of set $\mathcal{E}$ of exceptional triples satisfies $\#\mathcal{E}(x)=O(x^{33/50})$. This note proves that the cardinality of the exceptional set $\mathcal{E}(x)$ of triples $(a,b,c)$ is an infinite set unconditionally for any $\epsilon <2/35$.

Sixteen Pages. Keywords: Diophantine equation; Integer inequality; Additive group; Multiplicative group; abc Conjecture