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?


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 $A$.

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  • $\begingroup$ (On the paragraph added today:) very interesting! -- What can be said about the constant $A$? $\endgroup$ – Stefan Kohl Aug 23 '14 at 21:18
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    $\begingroup$ @StefanKohl: This is not specified, but could be calculated with some effort. It would likely be quite large. $\endgroup$ – Lucia Aug 23 '14 at 22:26

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 $\kappa_0 \le 8$.

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  • $\begingroup$ Thanks. -- Very nice! -- Now it would certainly be interesting to find that $\kappa_0$ (and of course to prove the xyz conjecture), though i'd suppose this is difficult ... . $\endgroup$ – Stefan Kohl Sep 20 '13 at 12:25

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