FWIW, all Mersenne Prime Exponents > 3 are congruent to 1, 5, 7, 11, 13, 17, 19 or 23 (mod 24), which happens to be the automorphism group for 24 (8 total self-mappings). This follows from the fact that all Mersenne Prime exponents are prime and therefore follow the form n = 6x +/- 1 for n >3.
- AC