Prime numbers have always been a central point of mathematical studies as they yield (at least for now) lots of stuff that we don’t understand (like factoring and it’s distribution among natural numbers - read the Wikipedia link bellow if you want to find out more). Prime numbers also play a big role paper in our world, as strange as it seems, our unability to fully understand primes is the basis of our digital security as we know it today. Finding new prime numbers nowadays is a task that no longer can be done by humans (as far as I’m aware of) and a computer takes at least a couple of days to find a new one.
From Wikipedia:
In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. There exists an infinitude of prime numbers, as demonstrated by Euclid in about 300 B.C….
The study of prime numbers is part of number theory, the branch of mathematics which encompasses the study of natural numbers. Prime numbers have been the subject of intense research, yet some fundamental questions, such as the Riemann hypothesis and the Goldbach conjecture, have been unresolved for more than a century. The problem of modelling the distribution of prime numbers is a popular subject of investigation for number theorists: when looking at individual numbers, the primes seem to be randomly distributed, but the “global” distribution of primes follows well-defined laws.
Anyway, this post is not just about primes, it is about a specific type of primes: Mersenne primes.
From Wikipedia (again):
In mathematics, a Mersenne number is a number that is one less than a power of two.
- Mn = 2n − 1.
A Mersenne prime is a Mersenne number that is a prime number. For this it is necessary, but not sufficient, that also the exponent n be prime. Many mathematicians prefer the definition of a Mersenne number where n has to be a prime number.
There is something called GIMPS (Greatest Internet Mersenne Prime Search), it is a distributed computing effort to find new Mersenne primes. Mersenne primes currently dominate the top of the biggest primes list for two important reasons: the grow very fast and, the Lucas-Lehmer primality test is the fastest algorithm for testing the primality of a number. You can join GIMPS and if you are lucky enough to find the 45th Mersenne prime (the next one, having more than 10 million digits) you will win $100,000 - a prize from the EFF (Electronic Frontier Foundation).
According to my math the 45th Mersenne prime should be near M37153927 and have about 11,184,447 digits - I’m currently iterating M35262373.
I also believe that there is a medium probability that M2147483647 is a Mersenne prime with about 646 million digits, but the primality test of such a large number would take years to complete with my current available computer resources.
UPDATE: The 45th (M37156667) and 46th (M43112609) Mersenne primes were finally found by GIMPS having 11,185,272 and 12,978,189 digits respectively, the previously discovered prime was found almost 2 years ago on September 4, 2006. If you check the 45th Mersenne prime against my prediction above (or at Archive.org) you should see that I was off by 2740 numbers, or more importantly, by approximately 148 primes (37153927/log(37153927) - 37156667/log(37156667)). Still, I would say it was a pretty good guess. 
PS: Prime numbers are the key to understanding the Universe.
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