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Mining For The 'Prime' Jewels Of Numbers

These are the first few digits of the nearly 13 million digit Mersenne prime, the longest found — so far.
Alyson Hurt/NPR
These are the first few digits of the nearly 13 million digit Mersenne prime, the longest found — so far.

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.

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."

Copyright 2023 NPR. To see more, visit https://www.npr.org.

Joe Palca is a science correspondent for NPR. Since joining NPR in 1992, Palca has covered a range of science topics — everything from biomedical research to astronomy. He is currently focused on the eponymous series, "Joe's Big Idea." Stories in the series explore the minds and motivations of scientists and inventors. Palca is also the founder of NPR Scicommers – A science communication collective.