Smooth sums of coprime smooth integers Observe that for any $\epsilon > 0$ there are infinitely many triples of
$c^\epsilon$-smooth coprime positive integers $a$, $b$ and $c$ such
that $a + b = c$. -- Considering triples of the form $(2^n-1,1,2^n)$
and the factorizations of the polynomials $x^n-1 \in \mathbb{Z}[x]$ into
cyclotomic polynomials, this holds since the set of quotients $n/\varphi(n)$
for positive integers $n$ is unbounded.
How much can this obvious observation be improved, i.e. how much can the
smoothness bound $c^\epsilon$ be lowered such that there are still infinitely
many such triples which satisfy that bound?
Or to be more concrete: is there an $\epsilon > 0$ such that there are
infinitely many triples of $e^{(\ln c)^{1-\epsilon}}$-smooth coprime
positive integers $a$, $b$ and $c$ satisfying $a + b = c$?
-- And if yes, which is the supremum of the set of values of $\epsilon$
for which this holds?
 A: This is related to the $xyz$ conjecture
 and the $abc$ conjecture.  The $xyz$ is much stronger than your question.
Let $S(X,Y,Z)$ be largest prime factor of $ X Y Z$ and
$H(X,Y,Z)=\max(|X|,|Y|,|Z|)$.
From page (2):
$xyz$ conjecture (weak form-1). There exists a positive constant $\kappa_0$
such that the following hold.
(a) For each $\epsilon > 0$ there are only finitely many primitive solutions $(X, Y, Z)$
to the equation $X + Y = Z$ with
$$S(X, Y, Z) < (\log H(X, Y, Z))^{\kappa_0 - \epsilon}$$
(b) For each $\epsilon > 0$ there are infinitely many primitive solutions $(X, Y, Z)$
to to the equation $X + Y = Z$ with
$$S(X, Y, Z) < (\log H(X, Y, Z))^{\kappa_0 + \epsilon}$$
From page (7)
Assume GRH.
There are infinitely many primitive solutions $X,Y,Z$ to $X+Y+Z=0$ s.t.
$$S(X, Y, Z) < (\log H(X, Y, Z))^{8 + \epsilon}$$
and in particular $\kappa_0 \le 8$.
A: Balog and Sarkozy (Stud. Sci. Math. Hungarica 1984) showed that large $N$ may be written as $x+y+z$ where $x$, $y$, and $z$ are all $\exp(3\sqrt{\log N \log \log N})$ smooth.  An analogous result applies to $a+b=c$, answering your question with a bound of the form 
$\exp((\log c)^{1/2+\epsilon})$.  Much more is expected to be true.  For a discussion of this problem see the paper of Lagarias and Soundararajan (Proc. London Math. Soc; http://arxiv.org/abs/1102.4911 ) where it is shown on GRH that there are solutions to $a+b=c$ that are $(\log c)^{8+\epsilon}$ smooth.
Added:  Recent impressive work of Adam Harper obtains unconditional results of about the same quality as Lagarias and Soundararajan -- that is, solutions to $a+b=c$ with $a$, $b$ and $c$ being $(\log c)^A$ smooth for some constant $A$.
