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W e are living in a mathematical golden age! Back in January 98 they found a new Mersenne prime number, the largest prime number yet. Roland Clarkson gets the discovery credits, along with George Woltman and Scott Kurowski, who designed the software and network he used. Clarkson is a 19 year old college student who found it — the 37th known Mersenne Prime, 23021377-1, a number with over 900,000 digits — using his home computer.
Wow!
A bit of explanation just in case you're rusty: a prime number, you remember, is a number that is only divisible by itself and 1 (counting whole integers, that is) — so 7 is prime, but 6 isn't, because it's also 2x3. Anyway, prime numbers have weird qualities that have always fascinated mathematicians.
The ancient Greeks figured out that there was no predictable formula for finding primes, seeing as they appear completely irregularly — strange in itself, since numbers usually obey patterns of some sort. But some early math whizzes stumbled on a formula:
2P-1
where P is a prime number.
But don't burst your pocket protector yet. Unfortunately, although that formula fruitfully yields numbers, smarter math whizzes figured out that many of those numbers aren't prime. Oh well. It's still a good way to generate a number, and then test it for whether it's prime. A monk named Mersenne got his name attached to it, basically for the same reason we call America America — Amerigo Vespucci wasn't the first in the new world, but he wrote really well and had a cool sounding name. Same thing for Mersenne. (Well, to be fair, he was a good number-cruncher, apparently.) So, we call any prime number that turns out to be 2P-1 a Mersenne prime.
Cool, yes?
Wellllll, in 1876 Mr. Lucas (who also gave us the Lucas sequence: 1, 3, 4, 7, 11, 18, 29....) proved that
2127-1 = 170, 141, 183, 460, 469, 231, 731, 687, 303, 715, 884, 105, 727
which is not only prime, but the largest prime anyone had ever come up with.
The amazing thing is that it stayed the largest known prime until 1951. And it will probably always hold the record as the largest prime found by hand, with no machine used at all.
In 1951 Ferrier found the next one with a desk calculator, and in the same year someone used a newfangled computer to find an even larger one, with 79 digits.
The next year, 1952, was a great year for Mersenne primes. Raphael Robinson found five of them, using a computer program that was the first he'd ever written. The program ran flawlessly on the first try; obviously this was before Microsoft came along.
The acceleration of Mersenne-prime-finding is a cool thing to see. Look at it by year:
1951
1952 (5 new primes)
1957
1961
1963 (3 new primes)
1971
1978
1979 (2 new primes)
1982: 25,962 digits
1983: 39,751 digits
1985: 65,050 digits
1989: 33,265 digits
1992: 227,832 digits
1994: 258,716 digits
1996: 378,632 digits
1996: 420,921 digits
1997: 895,932 digits
1998: 909,526 digits
1999: 2,098,960 digits*
see the update below
At this rate, Chris Caldwell estimates that we'll find a 1 million digit megaprime sometime in 1999. And — a billion digit prime by 2009! As I said, we're living in a golden age.
A friend made a snide comment about how important this all is (the implication was that it isn't). I'm still allowing that person (paradoxically, but not surprisingly, a Christian) to be my friend, even though they can't see the beauty of this. Numbers and their relationships are part of God's universe, and it's just as much of a thrill to discover new aspects of it as it is to see a new flower.
But then again some folks can't see flowers either.
The primes are the building blocks of the positive integers, we're told: every positive integer is a product of prime numbers in one and only one way. The weird thing is that even though there is an infinite number of primes, only around 6000 have been found. 6000! And that's all the primes, not just the Mersenne ones. So that just shows you how little we know.
The interesting thing to me is that the process of mounting a huge project often brings byproducts that are really 'useful' in everyday life — we think of the knowledge and consumer products that came from NASA in the 60s, or that Sir Georg Solti's recording of Wagner's Ring Cycle, the first complete stereo recording of it ever attempted, turned out to be a great touchstone for the butt-kicking-factor of your stereo equipment (besides beating Elvis Presley in the Billboard top 20).
One of the neat byproducts of the search for the next Mersenne Prime (and something that has a lot of Web-people excited) is that we're beginning to use computers in a different way. There's this thing called the Great Internet Mersenne Prime Search — that's GIMPS — started by George Woltman and Scott Kurowski a couple of years ago. The deal is that instead of having some giant government/university supercomputer do the searching (at something like a quarter-million bucks per day), you can break it up into bits that smaller computers could handle. The net makes it possible to gather enough interested people (right now about 4000), and there you have it: a decentralized, very democratic, science project.* You just take a certain range of numbers, and your computer works on it to see if that range has the next Mersenne number in it. Cool! And that's how the largest prime number in history so far was found by a 19-year-old college student.
So, let us celebrate a new way of doing things. Let us celebrate the big-mindedness of science whizzes who realize the best way of solving their problems is to get the rest of us to help. And let us dare to stand on the shoulders of friendly giants, and stare together in wonder at the secrets of the universe.
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U P D A T E
april 25, 2000
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Well, it looks like Chris Caldwell's 1998 prediction that we'd see a million-digit megaprime last year came true in megaspades. On June 1, 1999, Nayan Hajratwala found the 38th known Mersenne prime,
26972593-1
on his home computer, in three weeks of computing time. It weighs in at 2,098,960 digits. Although it's the 38th known Mersenne prime, it may not be the 38th in order of size: they haven't been able to check out all the smaller exponents yet.
More uses for Mersenne primes: one of the project's programmers, Richard Crandall, also developed Apple Computer's encryption system, which uses Mersenne primes to encrypt and decrypt messages. Also, the GIMPS software is apparently great for testing your computer for bugs. And, naturally, teachers of all sorts are turning their kids on to math with the GIMPS search.
By the way, since I wrote the original article, the project has grown to around 15,000 people. Together, they form a virtual supercomputer whose sustained throughput, according to entropia.com, is around 1145 billion floating point operations per second (gigaflops), or 95.1 CPU years (Pentium 90Mhz) computing time per day. They report that "for the testing of Mersenne numbers, this is equivalent to 20 of Cray's most powerful T932 supercomputers, at peak power."
That's why ya gotta love the internet.
And speaking of information and cooperation, it turns out that naming this project after Mersenne has further resonance. Check out this passage, courtesy of britannica online:
One of Mersenne's most important contributions was his long service as a communication link between philosophers and scientists throughout Europe. Because there were then no scientific journals, men might work a lifetime on the same project, never knowing of one another's existence.
Mersenne met regularly and corresponded at length with eminent figures, including Descartes, Girard Desargues, Pierre de Fermat, Blaise Pascal, and Galileo, and it was said that "To inform Mersenne of a discovery, meant to publish it throughout the whole of Europe."
