The Speculist

Most things, the bigger they are, the easier they are to find. This is not true with prime numbers.

Many years ago, commentor and guest blogger (at Winds of Change and other places) M. Simon had a TRS80 Model 3 off in a corner running the sieve. If he'd had it on battery backup for when he moved it probably would have gotten out into new prime territory by now.

The problems of finding all the Mersenne primes are surprisingly complex. A key factor is that when such a prime is found, it's often not known whether there is a smaller Mersenne prime that just hasn't been discovered yet. GIMPS is the group organizing the global Mersenne prime search. In addition, they usually check every potential Mersenne prime twice to verify that some computational error (apparently these are surprisingly common!) didn't generate a false negative the first time around.

Here's the status page. What's key is that all prime exponents (a Mersenne prime is of the form 2^p -1 where p is prime) have been checked that are less than 10,412,700. All prime exponents less than 7,614,100 have been double checked. They aren't completely sure even of the place of the second highest Mersenne prime (need to double check around 70,000 possible exponents to be sure that no missing Mersenne prime is below it)!

It probably is a Mersenne prime that was found, simply because it's a known form, and there are some shortcut tests that can be used to verify it.

Because of that, it's probable that this is *not* the next prime up, although it may be the next Mersenne prime up. There are probably lots of primes that haven't been found in between the Mersenne primes that have been.

wheels,

As far as I know, being of the form 2^p - 1 for some prime p, is as good as it gets. To my knowledge, there's no work out there that shows certain prime exponents are more likely to yield Mersenne primes than other prime exponents.