Squarefree integers and the $abc$ conjecture

TL;DR

This paper investigates the patterns of squarefree factors in integer partitions to explore the $abc$ conjecture, providing an algorithm to generate infinite sequences of triples that support the conjecture's claims.

Contribution

It introduces an algorithm to generate infinite sequences of $abc$ triples with specific properties, offering heuristic evidence related to the $abc$ conjecture.

Findings

Sequences of $abc$ hits are heuristically consistent with the conjecture.

An algorithm for generating infinite $abc$ triples is proposed.

Patterns in squarefree factors support the conjecture's claims.

Abstract

For coprime positive integers $a, b, c$, where $a+b=c$, $\gcd(a,b,c)=1$ and $1\leq a < b$, the famous $abc$ conjecture (Masser and Oesterl\`e, 1985) states that for $\varepsilon > 0$, only finitely many $abc$ triples satisfy $c > R(abc)^{1+\varepsilon}$, where $R(n)$ denotes the radical of $n$. We examine the patterns in squarefree factors of binary additive partitions of positive integers to elucidate the claim of the conjecture. With $abc$ hit referring to any $(a, b, c)$ triple satisfying $R(abc)<c$, we show an algorithm to generate hits forming infinite sequences within sets of equivalence classes of positive integers. Integer patterns in such sequences of hits are heuristically consistent with the claim of the conjecture.