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I am tasked with proving that all Mersenne composites (that is, composite numbers of the form $2^n -1$) are either always Carmichael numbers or never are.

Running some tests, I have found some Mersenne composites that are not Carmichael, so the "always" can't be true. Thus, I'm trying to prove they can never be Carmichael numbers. That is, for a given Mersenne number $m$, there exists an $a$ coprime with $m$ such that $a^{m-1} \not \equiv 1 \text{ mod m}$ or a $b$ such such that $b^{m} \not \equiv b \text{ mod m}$

I don't even know where to start. How should I go about it?

Rararat
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  • Have you ever heard of Korselt's criterion? – Jonah Oct 15 '23 at 03:05
  • @Jonah I have heard that $n$ had to be odd to be Carmichael (which all Mersenne are, right?). I didn't know it had a name, thanks! Then I suppose I just need to prove that Mersenne numbers are never square free or that there exists some prime factor $p$ such that $p-1$ does not divide $n-1$ – Rararat Oct 15 '23 at 03:15

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