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Show that if $p=2^a -1$ is prime then $a$ is prime.

I found this problem very hard to solve and I only managed to show that $a$ must be odd; if $a$ was even we could rewrite $p$ as $$p=2^{2k}-1=(2^k-1)(2^k+1)$$ with $k$ different than $2$. Other than this I couldn't do anything else.

Thomas Andrews
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PunkZebra
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  • So many duplicates of this one that even Google finds one (without being able to distinguish between $2^n-1$ and $2n-1$). Tut, tut. – Jyrki Lahtonen Nov 12 '16 at 13:42

1 Answers1

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Prove the contrapositive:

If $a$ is not prime, then $2^a-1$ is not prime.

Write $a=bc$ with $b,c>1$ and $u=2^b$. Then $2^a-1=u^c-1=(u-1)(\cdots)=(2^b-1)(\cdots)$.

lhf
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