Math Enthusiast Finds The Largest Known Prime Number

IRA FLATOW: This is Science Friday. I’m Ira Flatow. Let’s go back to grade school for a moment, OK? Do you remember learning about prime numbers, those numbers that can only be divided by themselves or 1? So 2, 3, 5, 7, 11, and so on are prime numbers. The number 12, for example, wouldn’t be prime because you can divide it by other numbers like 2 and 3. And as you count up and up, prime numbers become more sparse. They’re spread out.

And if you follow math at all, you’ll know that math geeks are always competing to find the largest prime number. And just recently, an engineer discovered the largest one so far, and you won’t believe how ginormous it is. Here with the story is Jack Murtagh, math writer and columnist for Scientific American based in Jersey City, New Jersey. Jack, welcome to Science Friday.

JACK MURTAGH: Thanks so much for having me.

IRA FLATOW: All right. I’m going to give you a drum roll if you can hear that tapping it out. I want you to reveal just how large that new prime number is.

JACK MURTAGH: So it’s unfathomably large. It’s 41 million digits long. It’s totally beyond human comprehension. So that’s the number of digits, not the size of the number. So for reference, I mean, a trillion is a decently big number, but that’s only 13 digits long. And I mean, estimates for the number of atoms in the observable universe is something like 80 digits long. So 41 million digits is just too big to comprehend.

IRA FLATOW: Wow. Tell me, as someone who watches this go on all the time with math nerds, what’s going on here with prime numbers being so cool in the first place, and people are trying to get the largest one?

JACK MURTAGH: Yeah. So we care about primes. I mean, they’re often called the building blocks of the numbers, and that’s because every whole number can be broken down into the product of primes. So if you want to understand numbers, then you need to understand primes, much in the same way that a physicist who wants to understand matter needs to understand atoms. It’s sort of like the indivisible unit that underlies everything.

And so they crop up all the time in pure math. They’re like main characters in pure math, but they’re even useful in practice, too. So, for example, primes play a critical role in widely used encryption algorithms that keep our digital transactions secure, so primes are everywhere. Primes are definitely cool. I will say this new 41 million digit prime is not likely to find real world application anytime soon. This is more a feather in the cap for the folks who care about this kind of thing.

IRA FLATOW: Right. And there are a lot of those folks.

JACK MURTAGH: Yes.

IRA FLATOW: And you notice that when you talk about primes, they get pretty hard to find. The higher you go, the more uncommon they are.

JACK MURTAGH: Exactly. And not only are they more uncommon, but for an enormous number, it’s very time consuming to check whether or not it’s prime. Some numbers are easy. OK, if I give you a huge even number, then you know it’s not going to be prime because even numbers are divisible by 2, and that’s not just one in itself. But if I give you some arbitrary massive odd number, how are you going to determine whether or not it’s prime?

One thing you could try is to divide it by every number smaller than it and see if any of them divide it cleanly. But then enormous numbers come with an enormous number of cases to check, and that quickly becomes infeasible even for computers to manage this level of exhaustive search. So instead, researchers design advanced algorithms and employ every clever little optimization trick they can to either speed up the process of checking whether a number is prime or to narrow the search somehow.

IRA FLATOW: Well, let’s talk about that. I understand that an engineer named Luke Durant figured out this giant number. How did he go about it?

JACK MURTAGH: That’s right. So he was actually part of a broader initiative, and the key idea is– since these enormous numbers gets very time consuming and cumbersome– is to focus on certain types of prime numbers. So these numbers are called Mersenne primes. I can explain what those are, if you’d like.

IRA FLATOW: Sure.

JACK MURTAGH: OK. So a Mersenne prime– they are primes that take a very special form. So you get one by taking the number two, multiplying it by itself a bunch of times, so 2, 4, 8, 16. You keep doubling. You do that some number of times, and then you subtract 1. So 2 times 2 times 2 times 2 times– all minus 1. And when you do this, sometimes, you get a prime number out depending on how many 2s you multiplied. So you don’t always get a prime.

You don’t even often get a prime. But sometimes, you do. And why this matters for prime hunters is that we know methods for checking whether numbers of this form are prime that are way faster than general methods for checking whether arbitrary numbers are prime. So the bigger the number, the longer it takes to check. But since we have a super fast method for Mersenne primes, we can afford to go big with them.

IRA FLATOW: There’s a whole community of people looking for these Mersenne numbers online.

JACK MURTAGH: Exactly. Yeah. The “Great Internet Mersenne Prime Search,” it’s called, and this is an organization that began in 1996. And what’s really cool about them is that they took a crowdsourced approach. So anybody can go to mersenne.org and download free software to run on their computer at home and join in the hunt and maybe be the next discoverer.

Luke Durant was one of these people, downloaded the software, but he did break a tradition in the community. So since the program began, they found 18 new prime numbers, and 17 of those were found on personal computers. So these are volunteers who maybe have a computer running in the corner of their office hoping to make a discovery. And Luke broke that trend by just throwing an absolute barrage of computational muscle at this problem.

IRA FLATOW: He strung a lot of computers together?

JACK MURTAGH: Yes, exactly. So first of all, he used GPUs instead of CPUs. So GPUs are the specialized computer chips that are powering much of the current AI boom. Luke formerly worked at NVIDIA, which designs these chips, so he’s very familiar with them.

IRA FLATOW: ‘Nuff said. And did he spend a lot of money on this?

JACK MURTAGH: Yeah. So when asked this question, he said, “I think under $2 million.”

IRA FLATOW: Oh, wow. That’s serious stuff just to find a prime number. Is there a prize for this effort?

JACK MURTAGH: Yeah. So funnily, you do win a monetary prize for finding a Mersenne prime, and that prize is $3,000, so this didn’t exactly pay for itself.

IRA FLATOW: Not good rate of return on that one. Is there going to be an end to this search or does it just continue forever?

JACK MURTAGH: Well, we know there are infinitely many prime numbers, and I don’t think people are going to stop searching for them. Curiously, we don’t know if there are infinitely many Mersenne primes, so that search could dry up. It is conjectured that there are infinitely many, but I think that this crowdsource project is not slowing down. It is going to take more and more computational resources as we go, unless there are algorithmic breakthroughs.

IRA FLATOW: Right. Can any of our listeners join this Mersenne search?

JACK MURTAGH: Yeah, that’s the cool thing. Anyone can do this. Go to mersenne.org, download the software, and you might strike gold. You don’t need $2 million to do this.

IRA FLATOW: Jack, as a math nerd yourself, do you have a favorite prime? Now, mine is 3. Turns out 3 is my lucky number. Do you have something like that?

JACK MURTAGH: I don’t think I do. I sort of like all primes. Maybe I’ll pick 2, because 2 is special in that it’s the only even prime number.

IRA FLATOW: Oh, I never thought about that. There’s only one even prime number.

JACK MURTAGH: It’s the only one.

IRA FLATOW: Wow. You just blew my head away on that. All these years thinking about primes. You’re also a puzzle maker, right?

JACK MURTAGH: That’s right.

IRA FLATOW: Do prime numbers ever make it into your puzzles?

JACK MURTAGH: I have not used prime numbers in a puzzle before. One of the main types of puzzles I make are crossword puzzles. And so for crosswords, people who make crosswords are always collecting little phrases in the language that maybe could be reinterpreted in some kind of wordplay type way.

And so I’ve thought of the phrase prime real estate as a possible revealer in a puzzle that has to do with prime numbers. And locations. Fifth Avenue, for example. 5 is a prime number, and maybe Fifth Avenue is where prime– haven’t fully worked out the kinks, but that’s the kind of idea.

IRA FLATOW: You could have prime rib.

JACK MURTAGH: Prime rib, yeah. Why not?

IRA FLATOW: Well, Jack, good luck to you and everybody else searching for that prime number.

JACK MURTAGH: Thank you very much.

IRA FLATOW: Jack Murtagh, math writer and columnist for Scientific American, based in Jersey City.

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