(Lance posted on the search for Mersenne primes in 2006 after a new one was discovered. I will comment on his post later.)
A Mersenne Prime is a prime of the form \(2^n-1\) where n is a natural number. One can show that n must be a prime. There is a list of all known Mersenne primes here. A more complete list of large primes going back to 1456 (which is not prime) when they discovered that 8191 is prime, is here.)
New Mersenne primes were found in
1985
1992 (so a 7 year gap)
1994
1996
1997
1998
1999
2001
2003
2004
2005
2005 (wow, two in one year!)
2006
2008
2008 (wow, two in one year!)2009
2013 (so a 5 year gap)
2016
2017
2018
2024 (so a 6 year gap)
When will the next one be? Possibilities:
a) The techniques that got us to the one in 2024 are NEW and will be USED to get us some new ones soon.
b) The techniques that got us to the one in 2024 were the old techniques on their last legs. A new idea is needed either from number theory, algorithms, or hardware. So we may not find one for a while.
I do not know which of (a) or (b) is closer to the truth. It may be hard to tell.
There are two obstacles for making progress on numbers like this.
a) Technology, math, etc.
b) Sociology. Do enough people care about finding these numbers?
AND these two obstacles can interact to give you either
a) a death spiral: the problem is to hard so people don't want to work on it, and people don't want to work on it because the problem is to hard or
b) a live spiral: a breakthrough was made so lets all now really go for it!
I do not know how Mersenne primes do in this context.
In 2006 Lance's post raised the question Why Do We Care About Finding Large Primes?
a) We don't. Not that many people are working on this.
b) The actual primes aren't that interesting, but the techniques used to find them might be.
c) Why do people climb mountains?
Because they like getting frostbite!
Better off finding primes which you can do without risking life and limb.
d) As someone who has worked on proving primes infinite using Ramsey Theory I am reluctant to tell someone else that there work is not worthwhile. I blogged on using Ramsey Theory to Proof the number of primes is infinite here.
e) I leave the following as an open problem: Compare and contrast value and interest in finding the following:
Large Merseene Prime
Largest non-Mersenne Prime. Mersenne primes are easier to find, so most known large primes are Mersenne. Finding large non-Mersenne primes has been looked at, see here, but not much.
The Busy Beaver Function . Scott did a post on the new breakthrough here.
Small Universal Turing Machines. I did a blog post on that here. There is a cash prize for it!
Ramsey Numbers, in particular R(5). I did a blog post on the new breakthrough here.
