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I have the following (new, I think) conjecture about the Mersenne prime numbers, where: [tex]M_q = 2^q - 1[/tex] with [tex]q[/tex] prime.

I've checked it up to q = 110503 (M29).

**Conjecture (Reix)**: [tex]\large \ order(3,M_q) = \frac {M_q - 1}{3^O}[/tex] where: [tex]\ \large O = 0,1,2[/tex] .

With [tex]I =[/tex] greatest [tex]i[/tex] such that [tex]M_q \equiv 1 \pmod{3^i}[/tex] , then we have: [tex]O \leq I[/tex] but no always: [tex]O = I[/tex] .

A longer description with experimental data is available at: ConjectureOrder3Mersenne.

Samuel Wagstaff was not aware of this conjecture and has no idea (yet) about how to prove it.

I need a proof...

Any idea ?

Tony