Mersenne Primes are primes that are one less than a power of 2. Small examples are 7 (one less than 8) and 31 (one less than 32). I've blogged before about Mersenne primes here and here. To recap, each Mersenne primes corresponds to an even perfect number. It is unknown whether there are infinitely many Mersenne primes. However, we do understand them well enough that we have a very fast test for whether a Mersenne number (a number one less than a power of 2) is prime. Thus, the largest primes known are generally Mersenne prime.
The newest Mersenne Prime is 2^42643801 - 1. This is in fact not the largest prime discovered because it is actually smaller than the previously discovered Mersenne prime. This prime, like most of the other Mersenne primes recently discovered, was discovered by the Great Internet Mersenne Prime Search. This project uses a distributed computing system to search for Mersenne primes. Volunteerrs install the GIMPS software on their machines and the program runs in the background, quietly searching for Mersenne primes. The program is set up to only use processing power that the computer is not otherwise using. Thus, the program has no negative impact on performance. This is another example of the triumphs of modern technology and mathematics.
One other thing to note: There are technical reasons to suspect that 2^n-1 will be more likely to be prime when n-1 has a lot of small prime factors. In this case, 42643800 factors into 2^3 * 3^3 * 5^2 * 53 * 149 so n-1 does in fact have many small prime factors in this example.
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