The number of equations c=a+b satisfying the abc-conjecture

TL;DR

This paper investigates the number of solutions to the equation c=a+b under certain radical-based inequalities, providing asymptotic formulas and analogues related to the abc-conjecture, advancing understanding of radical inequalities in number theory.

Contribution

It establishes asymptotic estimates for the count of solutions to c=a+b satisfying specific radical inequalities, including an analogue related to the abc-conjecture.

Findings

Asymptotic formula for N(c) as c approaches infinity

Proof of an analogue of the abc-conjecture inequality without a constant factor

Quantitative bounds on the number of solutions to c=a+b under radical constraints

Abstract

We prove that for a positive integer $c$ and any given $\varepsilon$, $0<\varepsilon<1$, the number $N(c)$ of equations $c=a+b$, $a<b$, with positive coprime integers $a$ and $b$, which satisfy the inequality $$c < R(c)^{\frac{\varepsilon}{1+\varepsilon}}R(a)^{\frac{1}{1+\varepsilon}}R(b)^{\frac{1}{1+\varepsilon}},$$ where R(n) is the radical of $n$, is for $c\to\infty$ $$N(c)=(1-\varepsilon)\frac{\phi(c)}{2}+O\Bigl(\frac{\phi(c)}{2}\Bigr).$$ An analogue for the abc-conjecture inequality $c<R(abc)^{1+\varepsilon}$ (without a constant factor) will also be proved.