This post is about the Mersenne primes. Mersenne primes are prime numbers of the form . A bit of trivia: This formula for generating prime numbers is known for a few centuries, however it continues to be interesting even today. It is easy to see that this formula generates primes, for example for n=2 we get 2, n=3 we get 7. However, for n=4 we get 16-1=15 and this number is composite. Generally, the result might be prime only if n is prime.

I am not going to write much about the properties of the Mersenne primes, it is easy to find this information on the net. What I want is to write why those primes are interesting to me and perhaps I will manage to interest you as well.

Firstly, it was proven by Euclid that all the numbers of the form are perfect if is prime. Later is was proven that all the even perfect numbers are of this form. It is still unknown if there exists an odd perfect number. Despite all the attempts to solve this problem it stands for over 2000 years.... The Mersenne formula therefore can be seen as an connection between two fascinating problems in modern mathematics - the Riemann Hypothesis and the odd perfect number problem.

Secondly, there is a project on the internet whose goal is to find large Mersenne primes. This project is a typical example of distributed computer network - you download their program, and it uses your computer idle cycles to try and factor numbers produced by the formula. They even have a prize offered, but it is pointless to participate only to try and win the prize in my opinion. The last prime they found was found in 2006... However if you want to put your computer idle cycles to some use, this is the place to go to. Since my computer is now staying idle most of the time so I am starting to consider this use.

Thirdly, there is an interesting theorem which happens to concern them. This theorem (proven by Gauss) says that it is possible to divide a circle into p equal segments using square roots (that is, only square roots are required to find the solution - points in a plane), only and only if p is a Merssene prime. To be precise, Gauss grooved only one part of this theorem - he showed that if p is Merssene prime than the construction is possible. The second part was proofed latter. From this we see that there is also a connection between these primes and geometry.

I am sure it is possible to think about more interesting properties of Mersenne primes, but those are the properties I know about and find interesting. What do you find interesting in them?

Grothendieck seminar at the ENS

21 hours ago

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