23/01/2019

Mersenne prime numbers, aiming at infinity.

Reproduction of Marcelo Viana's column in Folha de S.Paulo.

About 20 years ago, I was working at IMPA when I noticed my computer was slow. I ran a diagnostic and found another user's process. The title, "mersenne," explained almost everything: I already knew that an international effort was underway to find a new Mersenne prime number.

This task used to be entrusted to supercomputers, but the promoters of the initiative had transformed it into a collaborative effort: pieces of the calculation were run on thousands of computers around the world. Among them was mine, as I had just discovered.

My colleague apologized, said he didn't think it would bother me, and stopped the calculation immediately. But he couldn't resist asking, "What do you have to do on the computer that's more interesting than finding a cousin with a million digits?"

Marin Mersenne (1588–1648) was a French scholar-monk with an unusually diverse range of scientific interests. He has been described as “the center of the world of science and mathematics in the first half of the 17th century.” Today he is remembered primarily for Mersenne numbers, those that have the form 2 ^{^n} – 1 for some integer n.

It is not difficult to show that 2 ^{^n} – 1 can only be prime if n is prime, but the converse is false: 11 is prime and yet 2 ^{^11} – 1 = 2047 is not (this was already observed by Hudalricus Regius in 1536). We do not know if there is an infinite number of Mersenne primes, and many other questions remain unsolved.

On the other hand, there are much more effective methods for testing the primality of ²ⁿ – 1 than for other numbers. For this reason, the record for the largest known prime has long been held by a Mersenne prime.

This record has just been broken with the discovery of the 51st Mersenne prime, which corresponds to an=82,589,933 and has 24,862,048 digits! Once again, this was the result of a collective effort called the Great Internet Search for Mersenne Primes (GIMPS). The decisive step was achieved on December 7th by Patrick Laroche, an IT professional working in Florida, USA.

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