So I was looking around at some Numberphile videos before my Cambridge interview whilst searching for some extra, last-minute mathematical inspiration. I found out about these numbers called Mersenne primes and was struck by the strangeness of it all.
A prime is a number whose only prime factor above one is itself. They have always been elusive to us, and their distribution is an enigma to us. This post is about a specific class of numbers called Mersenne primes, which are tied to the so-called perfect numbers.
What is a Mersenne prime?
A Mersenne prime, named after the French monk and mathematician Marin Mersenne, is a prime in the form:
where p denotes a prime number.
This does not work for all prime numbers, as shown below:
We don’t know the pattern these are distributed in, and we don’t know how many there are, only that they get very far apart very fast. In fact, we have searched numbers up to tens of billions of digits long and have only found 51 Mersenne primes. They could carry on forever, or there could be a final Mersenne prime. We just don’t know. This is fascinating to me, and I can’t wait until, one day, we find a proof.
Mersenne primes and perfect numbers
Perfect numbers are numbers whose factors sum to double themselves, for example 2(6) = 1 + 2 + 3 + 6 . The first few are shown below:
Like Mersenne primes, we only know 51 perfect numbers, because the two properties are intrinsically linked. Each perfect number can be expressed in the form:
where (2^n) -1 is a Mersenne prime. This is the case for every Mersenne prime and perfect number:
and so on.
Therefore, the sum of the factors of a perfect number can be given by:
So, to find all the factors of a perfect number one only has to count in powers of two until they hit one power below the Mersenne prime, and then repeat the sequence now multiplied by the Mersenne prime. Since 1 is the only number which a prime number can be multiplied by and remain prime, and all powers of 2 are not Mersenne primes since 1 is the only number expressible as both 2 to a power and one less than 2 to a power, this means that each perfect number has only one Mersenne prime factor.
Numberphile attempted to use this as a proof, but I feel that it wasn’t so much a proof as an explanation of properties since there was no reasoning as to why a perfect number’s factors must be as they are. Maybe I’m looking at it the wrong way. Have a watch and see what you think:)
So I was intending to post this in December (but got way sidetracked – sorry) because last month, on the 7th, a computer found the 51st known Mersenne prime. The number is called M82589933 since that is p in 2^p – 1. The prime is 24, 862, 048 digits long and was discovered as a part of the Great Internet Mersenne Prime Search, which can be found at mersenne.org, where people around the world volunteer to search for and verify Mersenne primes using computer programmes. The online project has found 15 Mersenne primes, with hopefully more to come!
See you next time!