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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) statesthat for $\varepsilon > 0$, only finitely many $abc$ triples satisfy $c >R(abc)^{1+\varepsilon}$, where $R(n)$ denotes the radical of $n$. We examinethe patterns in squarefree factors of binary additive partitions of positiveintegers to elucidate the claim of the conjecture. With $abc$ hit referring toany $(a, b, c)$ triple satisfying $R(abc)<c$, we show an algorithm to generatehits forming infinite sequences within sets of equivalence classes of positiveintegers. Integer patterns in such sequences of hits are heuristicallyconsistent with the claim of the conjecture.

Squarefree integers and the $abc$ conjecture