Some very large numbers that have been shown to be prime as a result of advances in computer techniques are:

2⁸⁶²⁴³ — 1
2²¹⁷⁰¹ — 1
2²³²⁰⁹ — 1
2⁴⁴⁴⁹⁷ — 1

Such huge numbers are central to modern encryption techniques that rely on the fact that it is a very difficult task, even for a powerful computer, to factorise large numbers; but it is simple to multiply two primes together. Cryptographers select two very large primes and raise the numbers in their message to a power which is also prime, utilising modulo arithmetic, thus creating huge problems for potential code breakers, even if they have access to the most powerful computers. The recipient of the message needs only to be told the original two prime numbers, but the code breaker needs to factorise the product, a task that could take even a powerful computer millions of years. In our age of electronic communication the need for such encryption techniques has vastly increased in importance and use.

Marin Mersenne

There is a useful biography of Marin Mersenne at http://www.scruznet.com/~luke/mersenne.htm. An interesting project is the Great Mersenne Internet Prime Search or GIMPS (http://www.mersenne.org/prime.htm). This project brings together thousands of enthusiasts from around the world, who donate spare time on their computers to checking potential Mersenne primes. This sort of problem used to be undertaken by supercomputers, but modern desktop machines are almost as quick as the monster Crays of a decade ago. By pooling the resources of thousands of small computers, astonishing power can be achieved. Read their FAQ (http://www.mersenne.org/faq.htm) for details on how to sign up.