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It's clear that Mersenne primes can't end in $9$, since $2^n$ can't end in $0$, but $2^n$ can end in $4$ and $2^{n}-1$ would end in 3. From the list at http://mathworld.wolfram.com/MersennePrime.html though, there aren't any known Mersenne primes that end in $3$ besides $3$. Is that a coincidence or is it impossible for a Mersenne prime to end in $3$?

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    $\begingroup$ Great question! $\endgroup$
    – 6005
    Aug 13, 2015 at 18:13

1 Answer 1

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We have $2^n\equiv 4\pmod{ 10}$ iff $n\equiv 2\pmod 4$, especially $n$ is even. As $n$ itself needs to be prime, the only candidate is with $n=2$ and in that case $2^n-1=3$ gives us the only Mersenne prime ending in $3$.

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