TITLE: Estimate on radical of $2^n \pm 1$
QUESTION [19 upvotes]: Not sure if this belongs to MO or not. 
Are there any lower bound on radical of $2^n \pm 1$?
We recall that radical of an integer $rad(k)$ is a product of primes which divide $k$.
As an example, if the abc-conjecture is true in the form $max(|a|,|b|,|c|) \leq rad(abc) ^2 $  then 
$$rad(2^n \pm 1) \geq 2^{n/2 - 1}.$$
I wonder if this estimate is proven (or perhaps conjectured) by anyone? Are there any nontrivial results here?

REPLY [5 votes]: The paper arXiv:1409.2974v1 proves a lemma regarding $2^L-1$. 
Let $n$ be a positive integer, and define $L = \text{lcm}[1, 2, \dots n]$ and $t = \lfloor \frac{\log(n)}{\log(2)} \rfloor$. Then $L*\text{rad}(2^L-1) \le 2^t(2^L-1)$.