We now know that 2^74,207,281-1 is a prime and this is not only the largest prime of this form, a Mersenne prime, but the largest prime of any sort. Is this a discovery? After all, the number has been there all the time and it was always prime or it wasn't. Why do we do this?

If you think that the energy and computing power wasted on the Bitcoin is bad enough, you probably aren't going to like the idea of huge amounts of computing power being poured into discovering if a number is prime or not. The trouble is, if I'm being honest, I can't do much to convince you that the two things are very different. 

The Great Internet Mersenne Prime Search (GIMPS) was formed in January 1996 with the purpose of finding Mersenne primes. A Mersenne prime is of the form 2^p-1 and until recently we only knew of 48 of them. It can be shown that if p isn't itself a prime then 2^p-1 can't be a prime, so some definitions include the condition that p is a prime. Certainly if you are looking for Mersenne primes you only need to test values of p that are prime. 

It is also worth noting that, from a programming point of view, Mersenne primes are very simple. For example, in binary the first three are 11, 111, 11111 and so on. In general 2^p-1 is just a binary number with p ones. The fact that we have Mersenne primes for p=31, 61 and 127 is often a very useful fact when you need a big prime number quickly.

According to the announcement of the new prime:

"The new prime number, also known as M74207281, is calculated by multiplying together 74,207,281 twos then subtracting one. It is almost 5 million digits larger than the previous record prime number, in a special class of extremely rare prime numbers known as Mersenne primes. It is only the 49th known Mersenne prime ever discovered, each increasingly difficult to find. Mersenne primes were named for the French monk Marin Mersenne, who studied these numbers more than 350 years ago. GIMPS, founded in 1996, has discovered all 15 of the largest known Mersenne primes. Volunteers download a free program to search for these primes with a cash award offered to anyone lucky enough to compute a new prime. "

Marin Mersenne 1588-1648

The primality proof took 31 days of non-stop computing on a PC with an Intel I7-4790 CPU on a university computer volunteered by Curtis Cooper for the project.  The new prime has 22,338,618 digits and is the 49th Mersenne prime to be discovered and the fourth discovered by Curtis Cooper and eligible for the $3,000 GIMPS research discovery reward.

A twist in the story is that Cooper's computer reported the prime on September 17th 2015 but it went unnoticed until 

"... routine maintenance data-mined it. The official discovery date is the day a human took note of the result."

What is the point?

I guess if you have to ask you aren't going to understand the answer. You can try to justify it in terms of pushing computing forward:

 "... its global network of CPUs peaking at 450 trillion calculations per second remains the longest continuously-running 'grassroots supercomputing' project in Internet history."

You can even point to the recent incident in which the GIMPS software pushed an Intel Skylake CPU, the latest thing, to fall over thereby uncovering a design flaw in the chip. 

In fact, the whole point is to find out about Mersenne primes and their mysterious nature. There is something strange about the way that you can take 2 and multiply it by itself a few times to get what has to be a very factorable number and then taking one away from the result changes the pattern so much that we have a prime with no factors. 

What might surprise you is that not only are there larger Mersenne primes to discover, but there could well be smaller ones - we haven't explored all the possibilites. However, size does matter and:

"GIMPS' next major goal is to win the $150,000 award administered by the Electronic Frontier Foundation offered for finding a 100 million digit prime number."

UPDATE: Two more Mersenne Primes have been discovered see: Largest Prime Now Has Over 23 Million Digits (50th)

A New Mersenne Prime Discovery (51st)

More Information

GIMPS Project Discovers Largest Known Prime Number: 2^74,207,281-1

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