Squarefree integers and the $abc$ conjecture
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 Oesterlè, 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.
 Publication:

arXiv eprints
 Pub Date:
 September 2021
 arXiv:
 arXiv:2109.10226
 Bibcode:
 2021arXiv210910226B
 Keywords:

 Mathematics  General Mathematics;
 11A41 (Primary);
 11D45 (Secondary)
 EPrint:
 20 pages