Sunday, February 15, 2009

How large can Mersenne primes get?

Long time ago I mentioned about the discovery of 43rd Mersenne Prime number. (Any number of the form 2^n - 1 where n is a prime is a mersenne number. If that number is prime, it is called a mersenne prime) Recently (in Aug/Sep last year) UCLA mathematicians discovered the 45th and a week after the 46th by Germans.

Why do people bother finding them? It is a grand computational challenge! (thus considered to be a great discovery which brings enormous recognition from peers) and can advance the field of computing by dealing with the enormous computational complexity. Since it is so hard to prove whether a Mersenne number is a prime number (primality testing is computationally expensive for large numbers), researchers have been developing various distributed/parallel architectures to attack the problem. As a result, it has led to the advancement of our understanding on such architectures. These numbers are also useful for cryptographic protocols. For example, Mersenne primes are used in PRNG's.

Can there be larger Mersenne primes than the 46th one? It is still uncertain whether there are infinitely many of them. But scientists believe there are.

I know a Mersenne prime uniquely identify a perfect number. But, apart from the applications where prime numbers are useful, are there other practical applications of Mersenne primes?

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