The Mersenne numbers are the numbers of the form "2^p - 1". Several centuries ago, some people used to think that these numbers were always primes if "p" itself is a prime. While it's true for the first few values, the general claim is, of course, false. Primality of "p" is a necessary condition for primality of "2^p - 1", but not a sufficient one. (It's necessary because if "p=qr", then "2^q - 1" and "2^r - 1" are divisors of "2^p - 1".)In fact, only 41 Mersenne primes are officially known. The highest one, "2^24,036,583 - 1", is also the greatest officially known prime integer in the world. It was found in May 2004. Usually, the exponent roughly doubles if you want to find a new Mersenne prime. However, the Mersenne prime numbers look surprisingly dense for the exponents between 20 million and 26 million.
An international network of computers GIMPS whose website is at
is looking for new Mersenne primes. You may join. One can see that the main page of this server announces that a new record prime may have been found. It would be the 42nd known Mersenne prime number. I predict that the exponent "p" will be
- "p = 25,964,951"
Note that if you find the first Mersenne prime with at least 10 million digits, you will win at least 1/2 of the $100k award from the EFF foundation. It takes roughly one month for a 3GHz computer to test (using the Lucas-Lehmer test) a number like "2^34,362,227 - 1" which is how a candidate for a 10-million-digit prime number looks like.
Note added later: Of course that the word "predict" was partially a joke. I knew that the exponent would be what I wrote. Congratulations to Dr. Martin Nowak from Germany (by the way, it's almost a Czech name) to the discovery of the 42nd known Mersenne prime.