Mining For The 'Prime' Jewels Of Numbers
Largest Mersenne Prime - These are the first few digits of the nearly 13 million digit Mersenne prime, the longest found — so far. (Alyson Hurt/NPR)
When they're not being used to send e-mails or play solitaire, about 50,000 personal computers around the world are engaged in a search -- for the world's largest prime number.
Prime numbers -- like 2, 3, 5 and 7 -- are numbers that are divisible only by the number 1 and by themselves.
The largest prime found to date is nearly 13 million digits long. To get an idea of just how big this prime number is, if you write 10 digits per inch -- all 12,978,189 of them -- the number would extend for 20.45 miles. The number is currently the world's largest prime. But there's always a larger one to find.
(Alyson Hurt/NPR)
These gargantuan primes fall under a special category known as Mersenne primes, named for a 17th century French theologian who made some predictions about them. (His early predictions ultimately turned out to be wrong.)
Computational Torture
The current reigning champ of Mersenne primes was discovered last summer as part of a program called the Great Internet Mersenne Prime Search, or GIMPS.
Apparently, Mersenne primes are easier to find than other primes, and George Woltman, a software designer in Florida and the man behind GIMPS, has written a free, downloadable program to search for them. But you need a lot of computing power.
"It takes about two or three weeks to test a single number," he says. "And everybody is plugging away, trying to find yet another prime number."
Chris Caldwell, a mathematician at the University of Tennessee, Martin, says that the main obstacle in proving that these numbers are prime is just doing the arithmetic with numbers that are millions of digits long. Caldwell says there is a formula for testing whether a large number is a Mersenne prime, but it's computationally intense.
"Not only do you have to multiply a 13 million-digit number by a 13 million-digit number, but you have to do that about 13 million times," Caldwell says. "And that just takes a tremendous amount of computation."
GIMPS has been plugging away at this for 13 years now, and has found 12 Mersennes so far.
'The Jewels Of Number Theory'
So why are people anxious to find another, larger Mersenne prime? The response from many in the math community is generally the same: because.
"Mersennes, in a way, are kind of like a large diamond," Caldwell explains. When he went to Washington, he says, he took his kids to see the Hope Diamond. That's the 45-carat diamond that sits in a special case in the National Museum of Natural History, usually with crowds around it.
"Nobody there looking at the Hope Diamond ever asks, 'Why did they bother to dig it up?' or 'What is it good for?' -- even though it really isn't good for much other than to just hang there and people to look at," Caldwell says. "And in many ways the Mersennes play that same role -- that they really are the jewels of number theory."
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RENEE MONTAGNE, host:
Right now, when they're not being used to send emails or play solitaire, about 50,000 personal computers around the world are engaged in a search. They're looking for the world's largest prime number. The largest prime found to date is nearly 13 million digits long, but there's always a larger one to find.
NPR's Joe Palca reports.
JOE PALCA: First, a quick reminder: Primes are numbers that are divisible by only one and themselves. So, 2 is prime, 3, 5, 7 and so on. The largest prime is actually 12,978,189 digits. You want an idea of how big that is?
Well, let's say you wanted to write that out on a piece of paper, and you write 10 digits per inch.
(Soundbite of calculator printing)
PALCA: So, that's 12,978,189 divided by 10 digits per inch divided by 12 inches per foot divided by 5,280 feet per mile. And the answer is…
(Soundbite of calculator printing)
PALCA: …20.48 miles. Now, that's a long number.
It was discovered last summer as part of something called GIMPS.
Mr. GEORGE WOLTMAN (Software Engineer): GIMPS is the great Internet Mersenne prime search.
PALCA: That's George Woltman, the man behind the great Internet mersenne prime search. He's a software engineer in Florida. Mersennes are a special kind of prime. They're named for a 17th century French theologian who made some predictions about them that actually turned out to be wrong. If you want to see the mathematical formula that describes Mersennes primes, it's on our Web site.
Turns out, Mersenne primes are the easiest to find, and Woltman has written a free, downloadable program to search for them.
Mr. WOLTMAN: It takes about two or three weeks to test a single number, and everybody's plugging away trying to find yet another prime number.
PALCA: They've been doing it for 13 years and found 12 so far.
Dr. CHRIS CALDWELL (Math, University of Tennessee at Martin): The main obstacle in proving these numbers prime is just doing the arithmetic with numbers that size.
PALCA: Chris Caldwell is a mathematician at the University of Tennessee at Martin. Caldwell says there's a formula to test if a large number is a Mersenne prime, but it's computationally intense.
Dr. CALDWELL: Not only do you have to multiply a 13 million-digit number times a 13 million-digit number, you have to do that about 13 million times. And then that just takes a tremendous amount of computation.
PALCA: So, what's the big deal here? Why are some people so anxious to find the next largest Mersenne prime? I've asked several people, and the response is generally the same: because.
(Soundbite of laughter)
Dr. CALDWELL: Well, actually, that's really not a bad answer.
PALCA: I wasn't about to let him off the hook that easily, so he expanded a bit.
Dr. CALDWELL: Mersennes, in a way, are kind of like a large diamond. When I go to Washington - I took my kids to see the Hope Diamond.
PALCA: The 45-carat diamond sits in a special case in the National Museum of Natural History, usually with crowds around it.
Dr. CALDWELL: Nobody there looking at the Hope Diamond ever asks, you know, why did somebody bother to dig it up? Or, what is it good for? You know, even though it really isn't good for much other than to just hang there and people to look at. And in many ways, the Mersennes play that same role - that they really are the jewels of number theory.
PALCA: No one can say for sure when the next Mersenne prime will be discovered. But I told Chris Caldwell I'd made a little bet with George Woltman, the man who wrote the program that searches for Mersenne primes. Woltman bet the next one wouldn't be found until the second quarter of 2012.
So, if it's the first quarter or sooner, I win, and if it's the second quarter or later, he wins.
Dr. CALDWELL: Let me think.
PALCA: You thinking?
Dr. CALDWELL: Yeah, I'm thinking. I think I'm going to go with George.
PALCA: Big mistake, because I intend to win this bet. If all the people listening to this story, and their friends and relations, download Woltman's program, that means the search for the next largest prime will be faster. Don't worry, Woltman assures me the program doesn't interfere with anything. It runs when a computer isn't doing something more important.
There's no financial glory here, no money riding on this bet, just the joy of victory for me and the chance for someone to enter the Mersenne prime hall of fame. It could be you.
You can find a link to the searching software at our Web site, NPR.org.
Joe Palca, NPR News, Washington. Transcript provided by NPR, Copyright National Public Radio.
