Mining For The 'Prime' Jewels Of Numbers

Largest Mersenne Prime

hide captionThese are the first few digits of the nearly 13 million digit Mersenne prime, the longest found — so far.

Alyson Hurt/NPR

Set To Type...

NPR asked the typographers at Hoefler & Frere-Jones to estimate how long this Mersenne prime would be if typed out — all 12,987,189 digits. The answer, it turns out, depends not only on the font but also the kerning, or the space between characters.

Set to 12 point Mercury Text: 17 miles, 202 feet, 1.5 inches. And to 12 point Gotham Book? That'd be 20 miles, 3,131 feet, 1.75 inches.

From The Archives

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."

See The Largest Known Prime, All 13 Million Digits

We all think we remember prime numbers from grade school, but just in case your memory is a little hazy, here's a quick refresher.

A prime is a number that can only be divided by two other whole numbers: itself and 1.

So, 1? Not a prime — it can only be divided by 1. You need two divisors to qualify for prime status.

Marked in orange are some of the easy primes — because they don't take much brain power to figure out:

Table of Prime Numbers

But then, there are the special primes. These numbers come from a variety of simple equations, but present mind-bogglingly large numbers. Finding primes was sort of an elite hobby for some of the early math greats, like Pierre de Fermat, Carl Gauss and Sophie Germain.

But the most celebrated of all primes is the Mersenne — what mathematician Chris Caldwell calls "the jewels of number theory."

Rare, Huge Numbers

Mersenne primes, named after 17th-century mathematician Marin Mersenne, are numbers derived from the simple equation:

Mersenne Prime Equation

The number n is a prime, and the result is prime. And what makes the Mersenne primes so interesting is how rare they are. And their gargantuan size.

Currently, a Mersenne prime clocking in at nearly 13 million digits holds the record of being the largest prime number ever found. You can take a look at just a snippet of the number here. But if you want to see all 13 million digits, you can download this whopping 17-megabyte text file. (That's almost 3,500 single-spaced pages in a Word document, at 12-point Times New Roman font.)

And here's a glimpse at how quickly Mersennes ramp up in size:

Sequence of Mersenne Primes

Notice that as the value of the number n — the exponents 2, 3, 5, 7, 13, 17 and 19 in the example above — increases, the value of the Mersenne prime skyrockets. If we plot the value of the Mersenne prime vs. the value of the exponent, here's what the pattern looks like:

Mersenne Prime Chart

And here's an easier way to think of that Mersenne prime at the top right of the chart: It's more than a trillion trillion trillion. You get that number with this equation:

Mersenne prime, n=127

So just using an exponent of 127 gives you that huge number. But the largest Mersenne prime's exponent is 43,112,609.

To find such a large number is beyond the brain power of anyone. Microsoft Excel can't even calculate any Mersenne prime with an exponent larger than 607. If you try, it gives you a nice #NUM! error — that's Excelese for FAIL.

You need massive computers to do the number crunching. So a team called the Great Internet Mersenne Prime Search, or GIMPS, is farming the computing power out to the Internet. Of the 46 Mersenne primes found so far, GIMPS has discovered 12.

Users can lend their desktop computing power out to a program using a type of software called distributed computing. This type of large-scale marshaling of computing power is not new to the Internet. Other groups, like the Search For Extraterrestrial Intelligence, or SETI@home, use personal computers for data crunching, as does the Folding@home program, based at Stanford University, whose goal is to learn more about how the proteins found in living organisms fold into complex shapes and interact with other biological molecules.

So why bother searching for these primes? Any math geek will tell you that it's just cool and they're in it for the chase. But the effort has been reciprocal, too. As they search for these ever-larger primes, they have also been able to contribute important theorems and principles to the field of mathematics.

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