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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 $k$)?

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?

Note that by Lifting Exponent Lemma gives an estimate on radical for composite numbers n, so I hope some bounds on $rad(2^n \pm 1)$ exists.

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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 $k$)?

As an example If abc-conjecture is true in the form $max(|a|,|b|,|c|) \leq rad(abc) ^2$ then $$rad(2^n \pm 1) \geq (1/4) 2^{n/2}$$ 2^{n/2 - 1}$$I wonder if this estimate is proven (or perhaps conjectured) by anyone? Are there any nontrivial results here? Note that by Lifting Exponent Lemma gives an estimate on radical for composite numbers n, so I hope some bounds on rad(2^n \pm 1) exists. 2 added 165 characters in body; deleted 2 characters in body; added 6 characters in body Not sure if this belongs to MO or not. Are there any lower bound on radical of 2^n \pm 1 (Recall that radical of integer rad(k) is a product of primes which divide integer k).k)? As an example If abc-conjecture is true in the form max(|a|,|b|,|c|) \leq rad(abc) ^2  then$$rad(2^n \pm 1) \leq geq (1/4) 2^{n/2} . I wonder if this estimate is proven (or perhaps conjectured) by anyone? Are there any nontrivial results here?

Note that by Lifting Exponent Lemma gives an estimate on radical for composite numbers n, so I hope some bounds on $rad(2^n \pm 1)$ exists.

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