Quote:
Originally Posted by R.D. Silverman
Note that 34 = 2^1 + 2^5 and 34^255 = 34 mod 255. Enough said?
Not enough? Note that 65 = 2^0 + 2^6 and 65^510 = 1 mod 511.
Still not enough? 72 = 2^3 + 2^6 and 72^510 = 1 mod 511.
I am sure that one could find additional counterexamples.
There was no reason to believe that an integer with Hamming weight 2
could not be a false witness to Mersenne numbers.

OK. my fault. I forgotten to mention that mersenne Numbers are Numbers of the Form n = 2^{p}1 with:
The variable p is prime !!!!!
If the mersenne number
with
is also prime then this
Mersenne Number is called Mersenne Prime.
Your example with mod 511 >
9 is not prime. > 511 is not a mersenne number.
And 255 =
> 8 is also not prime > 255 is not a mersenne number