Quote:
Originally Posted by pacionet
Do some math analysis exist on the residues of composite Mersenne candidates? Can the residue values suggest something of next Mersenne?

Theoretically, if we could store the other divisor of S_n we could start a test at the nearest mersenne prime exponent. It gets a bit harder with nonzero residues though. There are properties a residue has to have if the number is composite, these are at best useful for checking factors slower than trial factoring currently though.