Let ${\rm rad}(n)$ denote the [radical](https://en.wikipedia.org/wiki/Radical_of_an_integer) 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](https://en.wikipedia.org/wiki/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 $0.72594$.



A converse of the abc conjecture?
[Robert, Stewart and Tenenbaum](http://iecl.univ-lorraine.fr/~Gerald.Tenenbaum/PUBLIC/PPP/abc.pdf) 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](http://perso.univ-st-etienne.fr/rool6510/Nxy.pdf) which gives a detailed analysis of the number of integers up to $x$ with radical at most $y$. 


