Quote:
Originally Posted by Nick
Nice!

Not only that, this solves ( or at least heuristics) the first 3 all wihout direct invocation of fermat's little theorem.
1) noted above.
2)see formula from 1, M_m= 2^{mn}M_n+M_{mn} q divides the first term, and so q must divide into the second term in order to divide the LHS.
3) following from 2, if q divides other mersennes other than multiples, it follows there is no minimal p, this leads to the absurdity that 1/ q is integer.