Is a Mersenne prime always of the form $4n + 3$?

Question

Is a Mersenne prime always of the form $4n + 3$?

Examples:

$M_3 = 7 = 4 \times 1 + 3$

$M_5 = 31 = 4 \times 7 + 3$

$M_7 = 127 = 4 \times 31 + 3$

$M_{13} = 8191 = 4 \times 2047 + 3$

Answer 1

Yes, since $n \ge 2, 2^n-1 \equiv -1 \pmod 4$.