Is a Mersenne-Prime always of the form $3n + 1$?

Question

Examples are:

$M_3 = 7 = 3\times 2 + 1$

$M_5 = 31 = 3\times 10 + 1$

$M_7 = 127 = 3\times 42 + 1$

$M_{13} = 8191 = 3\times2730 + 1$

Answer 1

If $p$ is an odd prime, and more generally if it is an odd number, then $2^p-1\equiv (-1)^p-1\equiv -2\equiv 1\pmod{3}$.

The Mersenne prime $3$ is not of the form $3n+1$.

Answer 2

Every number of the form $2^{2n+1}-1$ is of such form, because $$2^{2n+1}-2\equiv2\cdot(4^n-1)\equiv2\cdot(1^n-1)\equiv0\ \ (\text{mod}\,3)$$ And because every prime except $2$ is odd, it's true for every Mersenne prime except the first.

Answer 3

$2^{2n+1} - 1 \equiv 1 \pmod 3$ 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.