Nate Silver's website had an article about it (written by Oliver Roeder) here
An article about why people do this is here
Lance posted about finding large primes in 2006 here
I'll just make some random comments
1) The prime is 277,232,917-1
2) The prime is not much bigger than the previous champion.
3) More generally, the graph (in Oliver Roeder's article) shows from 1600 to about 1951there was slow progress but since then there has been LOTS of progress. See the table in this article. I had wanted to say every year a new prime was found but, alas, not that simple a pattern. Even so, lots of new records.
4) I"ll list the reasons given for why people do this and my comments on them.
a) Tradition! (Note to self- write a novelty song to the tune of Fiddler-on-the-roof's Tradition about why people work on various mathematical things)
b) For the by-product of the quest. This one does make sense and I am sure has driven and helped test out some research. Reminds me of the spinoffs from going to the moon (see here). Contrary to what I heard as a kid, the orange-powder-drink Tang was not a spinoff. But there were many others. Of course- would these spinoffs have happened anyway? Hard to know.
c) People collect rare and Beautiful items. I don't really see this one. Finding a large prime doesn't make its yours, it belongs to the ages! And I don't think people get their names on the prime either. The only prime that has someone's name on it is the famous Grothendieck prime which is 57. Oh well. There are sets of primes with peoples names on them: Mersenne primes, Gaussian primes (which are subsets of Gaussian integers so maybe shouldn't count), Eisenstein primes, and
Sophie Germain primes. If you know of any other primes or set of primes named after someone, leave a comment please.
d) For the glory! Maybe, but given how briefly people hold the record, fame is fleeting.
e) To test the hardware. This one I didn't know! I'll quote the article as to why primes are good for this
Why are prime programs used this way? They are intensely CPU and bus bound. They are relatively short, give an easily checked answer (when run on a known prime they should output true after their billions of calculations). They can easily be run in the background while other "more important" tasks run, and they are usually easy to stop and restart.f) To learn more about their distribution. The prime number theorem was conjectured from data. We have so many primes now that I wonder if a few more really help formulate conjectures
g) For the money. The first person to get a ten-million digit prime gets $100,000. The first person to get a one billion digit prime gets $250,000. Wow! Except that the article must be a bit old since the $100,000 prize was claimed in 2009 (see here). Still, theres that one billion digit prize out there!
5) Mersenne primes are of the form 2^n-1. It is known that n must be prime for 2^n-1 to be prime (this is not hard). There are much faster primality testing algorithms for Mersenne primes than arb primes. But see next item.
6) While writing this blog post I looked up non-mersenne primes. It seems like the largest one is
10223*2^31172165 + 1 and was discovered in 2016.
But of more interest- there is no Wikipedia page on non-Mersenne primes, there are some outdated pages that don't have the most recent information. As the kids say, its not a thing.
8) I'll add one more reason why people work on this, but its more of a tautology: People work on finding large primes because they can!. By contrast, finding VDW numbers is hard and likely to not make much progress.
9) I think that the most reason advances have come from computing power and not number theory. (if this is incorrect let me know with a polite comment)
10) Have their been spinoffs in either number theory OR computing power lately?
11) I wonder if there will come a point where progress gets hard again and the graph of largest known primes flattens out. I tend to think yes, but hard to say when.