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 ≤ a < b, the famous abc conjecture (Masser and Oesterlè, 1985) states that for ε > 0, only finitely many abc triples satisfy c > R(abc) 1+ε , 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.