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Let ${\rm rad}(n)$ denote the radical of a positive integer $n$, i.e. the product of its distinct prime divisors. Given positive integers $a$ and $b$, the triple $(a,b,a+b)$ is called an abc triple if $a$ and $b$ are coprime and ${\rm rad}(ab(a+b)) < a+b$. The quality of an abc triple $(a,b,a+b)$ is defined as the quantity $$ q(a,b,a+b) \ := \ \frac{\log{(a+b)}}{\log{({\rm rad}(ab(a+b)))}}, $$ and the abc conjecture asserts that for any $\epsilon > 0$ there are only finitely many abc triples of quality greater than $1+\epsilon$.

Now let the smoothness of an abc triple $(a,b,a+b)$ be $$ s(a,b,a+b) \ := \ \frac{\log{(\log{(\frac{a+b}{{\rm rad}(ab(a+b))})})}}{\log{(\log{(a+b)})}}. $$ Clearly the smoothness of any abc triple is strictly smaller than $1$.

Question: Is it true that for any $\epsilon > 0$ there are abc triples with smoothness $\geq 1-\epsilon$?

Examples:

  • The triple $(1, 8, 9)$ has quality $1.22629$ and smoothness $-1.14676$.

  • The triple $(3, 125, 128)$ has quality $1.42657$ and smoothness $0.23562$.

  • The triple $(10, 2187, 2197)$ has quality $1.28975$ and smoothness $0.26825$.

  • The triple $(1, 2400, 2401)$ has quality $1.45567$ and smoothness $0.43400$.

  • The triple $(1, 4374, 4375)$ has quality $1.56789$ and smoothness $0.52238$.

  • The triple $(343, 59049, 59392)$ has quality $1.54708$ and smoothness $0.56635$.

  • The triple $(3200, 4823609, 4826809)$ has quality $1.46192$ and smoothness $0.57855$.

  • The triple $(2, 6436341, 6436343)$ has quality $1.62991$ and smoothness $0.65457$.

  • The triple $(283, 8251953125, 8251953408)$ has quality $1.58076$ and smoothness $0.67991$.

  • The triple $(125, 11174240024064, 11174240024189)$ has quality $1.53671$ and smoothness $0.690851$.

  • The triple $(24833, 5020969537415167, 5020969537440000)$ has quality $1.62349$ and smoothness $0.73326$.

  • The triple $(8654525279998779296875, 229727166528260169448321941, 229735821053540168227618816)$

    has quality $1.48053$ and smoothness $0.72594$.

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Robert, Stewart and Tenenbaum have put forward a refined version of the abc conjecture (other variants are due to Granville, Baker, van Frankenhuijsen, ...) which states that if $a+b=c$ with $a$, $b$, $c$ positive, and if $k$ denotes the radical of $abc$ then (for an $abc$-triple with $k\le c$) $$ \frac{c}{k} \le \exp \Big(A \sqrt{\frac{\log k}{\log \log k}}\Big) \le \exp\Big( A \sqrt{\frac{\log c}{\log \log c}}\Big), $$ for some constant $A$ and all $c$ large enough. In fact they even conjecture that $A> 4\sqrt{3}$ is enough, and that this constant is best possible. This conjecture implies that $$ s(a,b,c) = \frac{\log \log (c/k)}{\log \log c} \le \frac 12, $$ if $c$ is large enough. The conjecture of Robert, Stewart and Tenenbaum is based on the work of Robert and Tenenbaum which gives a detailed analysis of the number of integers up to $x$ with radical at most $y$.

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  • $\begingroup$ Thank you, that is interesting! -- I guess proving this refined conjecture would still seem out of reach even if Mochizuki's proof of the abc conjecture is correct(?) $\endgroup$ – Stefan Kohl Feb 23 '16 at 10:31
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    $\begingroup$ Well if you assume everything in Mochizuki, Vesselin Dimitrov has an arXiv preprint with strong consequences. $\endgroup$ – Lucia Feb 23 '16 at 11:21

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