# nt.number theory – Asymptotic of rad(abc) in the abc conjecture

The abc conjecture famously predicts that, given any $$epsilon>0$$, for all but finitely many positive coprime integers $$a,b,c$$ with $$a+b=c$$, the radical $$rad(abc)$$ (i.e., the product of all prime factors of $$abc$$ without multiplicity) is at least $$c^{1-epsilon}$$. My question is, are there any quantitative predictions about what happens if one increases the exponent from $$1-epsilon$$ to something larger than $$1$$? Obviously, there are then examples such that the statement in the above form is not true any more, but how frequent are these? To be explicit, given any $$d in (1,2)$$ ($$d>2$$ might start to be nonsensical, since $$abc$$ would often be no larger than $$O(c^2)$$), what is a reasonable expected upper bound for the asymptotic proportion (depending on $$N$$) of coprime triples $$(a,b,c)$$, with $$a+b=cle N$$ such that $$rad(abc) < c^{d-epsilon}$$?

abc conjecture predicts $$O(1)$$ for $$d=1$$, but is there any guess for an asymptotic upper bound $$N^{f(d)}$$, with $$f:(1,2)to (0,2)$$?

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