In reading about this new largest know prime number, I learned that Mersenne prime numbers are of the form 2p – 1, where the exponent p is a prime number. For example, 25 – 1, is a Mersenne prime. This is because the exponent 5 is prime and 25 – 1 = 31, which is prime. Therefor, 31 is a Mersenne prime, since it can be expressed in the form 2p – 1.
Not all numbers of this form are Mersenne numbers. For example, 211 – 1 is not a Mersenne prime. It’s not even prime. Although, the exponent 11 is prime, 211 – 1 = 2047 which is not prime. 2047 has prime factors of 23 and 89.
Also not all prime numbers are one less than a power of two, just Mersenne primes. Mersenne primes are easier for the software to test for using the Lucas-Lehmer primality test, which is based on powers of two. Because the test is so remarkably simple, the computers can test numbers faster than with any other form of prime numbers. As a result — and thanks to G.I.M.P.S. — new prime numbers are usually Mersenne primes.
Visit the website, and download the software for free. You might become famous and make some money if your computer finds a new prime!