Quote:
Originally Posted by R.D. Silverman
You just changed your original statement.
Your original statement was that x^n = x is not true when n = 2^p1
is composite. Here x = 2^a + 2^b for a!= b
This is different from x^{n1} = 1 is not true when n = 2^p1 is
composite.

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.