The $abc$ Conjecture Revisited

Abstract

We propose a new abc-type conjecture. We motivate the conjecture and illustrate its relevance through several applications. Our main result concerns the function $$W(x,y):=\sum _{j=1}^y\ddot{I}‰(x+j)(y\in \mathbb{N},x\in {\mathbb{Z}}_{\ge 0})$$ where $\ddot{I}‰(n)$ denotes the number of distinct prime divisors of $n$. The new conjecture implies that, for each fixed $y\in \mathbb{N}$, $$\underset{x\to \infty }{\mathrm{limsup}}\frac{W(x,y)\log \log x}{\log x}=1.$$