There’s a New Prime! And It Looks Like…Wait…What?

A new prime has been discovered. It’s really long. Over 22 million digits. And the number has just been sitting on a computer in the middle of Missouri unnoticed since September. But that’s not the crazy thing about it. The crazy thing is that it’s equal to 2^74,207,281-1. Why should something like that be a prime, and how on earth did we find it?

If you printed out the new prime it would take over 15 reams of paper. Written out end to end, the number would be over 22 kilometers long. Photo courtesy of Pawpaw67 via Flickr.

Turns out, primes of that form have been a hot topic for some time. Let’s go back to the beginning of the story. Since humans figured out how to factor numbers, humans have been looking for primes — remember, those are the numbers greater than 1 which are only divisible by 1 and themselves. In the 14th and 15th century people were very interested in finding a simple rule that you could apply to numbers to guarantee that you would get a prime, some sort of prime generating function.

In the 16th century, monk slash mathematician Marin Mersenne started to consider numbers of the form 2ⁿ-1. He noticed that often, when you put a number in place of n you get a prime back. If you try out a few values of n, it seems to work: 2²-1=3, 2³-1=7, and 2⁵-1=31. Today, if a number of that form turns out to be a prime, we call it a Mersenne Prime. So, is it true that if we plug any number n into that rule, we get a prime back?

Nope. He (and probably anybody who’s taken a course in elementary number theory!) can see that if n is composite — so if n=ab for some integers a and b — then of course both 2^a-1 and 2^b-1 divide 2ⁿ-1, so there’s no way it’s prime.

But it’s not enough just to insist that n is prime, since it doesn’t take too much heavy lifting to see that 2¹¹-1=23×89.

Mersenne knew it wasn’t so simple, but he put together a list of n values that he thought would give back primes. Since he was living in the 16th century, and only had a quill pen and some parchment to do his calculations, quite a few turned out to be wrong. Quite notably, as was the subject of the best This American Life Prologue of all time, Frank Nelson Cole found an error in one of Mersenne’s calculations.

Nevertheless, due to significant advances in computing power and the Lucas-Lehmer test for primeness, searching through Mersenne’s numbers has proven to be a reasonably successful way to look for primes.

So successful, in fact, that we just found a new Mersenne prime this week! Needless to say, my Twitter feed has been blowing up. This discovery has gotten a great deal of media attention, some that’s tons of fun and some that’s…well…it’s what you might expect from math press. But my co-blogger Evelyn gave a beautiful recap of the search for the prime on Slate.com.

This is exciting news, not in that it changes the face of mathematics or unlocks some great mystery, but that it’s just so cool and satisfying this to come up with concrete examples of theoretical concepts. It’s ties such a beautiful thread through history to know that we are chipping away at the same simple ideas that have occupied minds for so many centuries.