<s>[INST] Is a Mersenne-Prime always of the form $3n + 1$? Examples are:
$M_3 = 7 = 3\times 2 + 1$
$M_5 = 31 = 3\times 10 + 1$
$M_7 = 127 = 3\times 42 + 1$
$M_{13} = 8191 = 3\times2730 + 1$
 [/INST] $2^{2n+1} - 1 \equiv 1 \pmod 3$ is true for all odd numbers $2n + 1$.
$2^{2n+1} \equiv 2 \pmod 3$
$2^{2n} \equiv 1 \pmod 3$
$4^n \equiv 1^n \equiv 1 \pmod 3$
Alternatively, you could show the cyclic pattern of the powers of two $\pmod 3$: $1, 2, 1, 2, 1, 2, \dots$, so it's clear that $2^k \equiv 1$ for $k$ even and $2$ for $k$ odd.
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