Estimate on radical of $2^n \pm 1$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T10:31:33Z http://mathoverflow.net/feeds/question/119005 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/119005/estimate-on-radical-of-2n-pm-1 Estimate on radical of $2^n \pm 1$ Ostap Chervak 2013-01-15T17:32:07Z 2013-01-16T20:45:40Z <p>Not sure if this belongs to MO or not. Are there any lower bound on radical of $2^n \pm 1$ (Recall that radical of integer $rad(k)$ is a product of primes which divide integer $k$)?</p> <p>As an example If 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?</p>