The ABC conjecture where A and B are smooth

Mochizuki has already claimed to have proven the ABC-conjecture. But even if his claim turns out to be correct the proof will not be easy to understand. With that in mind I'm asking wether anything is known on the following weaker version of the ABC-conjecture that does not use Mochizuki's work.

Let $S = \{p_1,\ldots, p_k \}$ be a finite set of primes and call an integer $n$ to be $S$-smooth if all prime factors of $n$ are contained in $S$.

My main question now is: Fix a finite set of primes $S$ and an $\varepsilon >0$, then is the number of solutions to $$a+b=c, \quad c > \textrm{rad}(abc)^{1+\varepsilon}$$ with $a$ and $b$ coprime and $S$-smooth integers finite?

Note that since in this case we have $\textrm{rad}(ab)\leq \prod_{p \in S} p$ this question is equivalent to:

Fix a finite set of primes $S$ and an $\varepsilon >0$, then is the number of solutions to $$a+b=c, \quad c > \textrm{rad}(c)^{1+\varepsilon}$$ with $a$ and $b$ coprime and $S$-smooth integers finite?

The baby case where $\#S =1$ is also aready interesting to me. This comes down to the following, fix a prime $p$ and a $\varepsilon >0$. Then is there an $N$ such that for all $n>N$ one has that $p^n \pm 1 < rad(p^n \pm1)^{1+\varepsilon}$?

Note that I'm not requiring $c$ to be $S$-smooth, so it is not a consequence of the $S$-unit equation.

• I don't think the estimate you want is known (other than via full abc) even for $S = \{2\}$. – Felipe Voloch Oct 29 '15 at 15:46
• In my opinion, "friable" is a better term than "smooth". – Greg Martin Oct 29 '15 at 17:48
• As far as names, $S$-unit is unambiguous. – Felipe Voloch Oct 29 '15 at 18:46
• Is the mathematical opposite of friable called indurate? – kantelope Oct 29 '15 at 19:49
• Jeff Lagarias and I had talked quite a bit about this same question last year. He also told me not much is known about it. For others: arxiv.org/abs/0911.4147 Smooth solutions to the abc equation: the xyz Conjecture Jeffrey C. Lagarias, K. Soundararajan – Artur Jackson Apr 5 '17 at 7:34