Last week, a 26-year-old engineering grad student at the University of Michigan got back from a meeting, sat down at his computer, and discovered that he had earned his own footnote in the history of mathematics. Michael Shafer did a victory dance, then called his wife to let her know that he had found the largest known prime number Prime numbers are those, like 7 and 101, that have no divisors other than 1 and themselves. Although there are an infinite number of primes, it’s impossible to predict precisely where they will show up. The prime number theorem gives a rough estimate of how many primes exist below a given number — the degree of error in this estimate is intimately tied to the Riemann Hypothesis, the most important unsolved problem in number theory (and one worth a million bucks to the mathematician who solves it) — but there’s generally no easy way to tell if a given number is prime. The type of prime that Shafer discovered, mersenne primes, are an exception. Mersenne primes are primes of the form 2p - 1, where p is a prime (although not all primes will form Mersenne primes in this way), and these numbers have special properties that make them easier to find. The five largest known primes are all Mersenne primes, and Shafer’s discovery is a whopper: over six million digits long. Calculating the factors by hand for numbers as big as these is quite difficult. Indeed, Marin Mersenne, the seventeenth-century monk for whom the primes are named, was wrong about two of the eleven numbers he claimed were Mersenne primes. (He also claimed that three Mersenne primes weren’t.) Most mathematicians have a sense of well-written proof being elegant, even beautiful. The Lucas-Lehmer test that shows whether a possible Mersenne prime really is is elegant. Actually running the test, a process that can take the average PC weeks, is not. Given enough time with a calculator, anyone, regardless of their level of insight into the math, could do the same; if spending weeks on pecking at a calculator sounds too intimidating, you could download a program to do it for you. The twenty-fifth and twenty-sixth Mersenne primes were discovered by a couple of mathematically inclined high schoolers, Landon Noll and Ariel Glenn (née Laura Nickel), who had access to a university mainframe. Aristotle’s claim that mathematics exhibits "order, symmetry, and limitation; and these are the greatest forms of the beautiful" isn’t really true for mere computation — the greatest knock on the original proof of the four-color map theorem, a long-standing problem in graph theory, was that it was unelegant, that it relied too much on letting a computer crunch numbers.
But even so, people try it anyway. Noll and Glenn both went on to work with cryptography, a field that makes great use of tremendously large numbers and the difficulty in factoring them, but Mersenne primes aren’t even particularly good for this. They’re there, and they’re huge, and that’s good enough. A similar sort of mania surrounds calculating the digits of pi. Rational numbers are those which can be expressed as whole number fractions, such 2/3 or 6. Although sqrt(2), 1.41421356…, is irrational, it solves the equation x2 - 2 = 0. These numbers, sensibly enough, are known as algebraic; pi is "transcendental". Pi solves no algebraic equations. Its expression, 3.14159625…, , stretches forever and never repeats. There’s no compact way to express it, except geometrically. The great Indian mathematician Srinivasa Ramanujan had a startling, intuitive grasp of number theory, and came up with brilliant methods for calculating digits of pi. Improving on these methods requires genius. Actually implementing them requires a big computer and hard work. But people fall under the spell anyway. Consider the talented mathematicians Gregory and David Chudnovsky and their attempts to beat the world at pi-computation:
The next year, David H. Bailey, at the National Aeronautics and Space Administration, used a Cray 2 supercomputer and a formula discovered by two brothers, Jonathan and Peter Borwein, to scoop twenty-nine million digits of pi. Bailey found nothing unusual. A year after that, in 1987, Yasumasa Kanada and his team got a hundred and thirty-four million digits of pi, using a NEC SX-2 supercomputer. They saw nothing of interest. In 1988, Kanada kept going, past two hundred million digits, and saw further amounts of nothing. Then, in the spring of 1989, the Chudnovsky brothers (who had not previously been known to have any interest in calculating pi) suddenly announced that they had obtained four hundred and eighty million digits of pi — a world record — using supercomputers at two sites in the United States, and had seen nothing. Kanada and his team were a little surprised to learn of unknown competition operating in American cyberspace, and they got on a Hitachi supercomputer and ripped through five hundred and thirty-six million digits, beating the Chudnovskys, setting a new world record, and seeing nothing. The brothers kept calculating and soon cracked a billion digits, but Kanada’s restless boys and their Hitachi then nosed into a little more than a billion digits. The Chudnovskys pressed onward, too, and by the fall of 1989 they had squeaked past Kanada again, having computed pi to one billion one hundred and thirty million one hundred and sixty thousand six hundred and sixty-four decimal places, without finding anything special. It was another world record. At that point, the brothers gave up, out of boredom.
These are exceedingly smart people. Gregory may have proved an important result related to Hilbert’s Tenth Problem when he was 17. The brothers developed (and eventually, with some reluctance, patented) several novel pieces of computer hardware while in the process of building a massive computer in David’s apartment out of seventy thousand dollars worth of spare parts. But the actual work of computation differs very little from that done by "lightning calculator" Johann Dase. Dase was capable of doing hugely complicated problems in his head. He made a living by this talent, and at the request of Carl Gauss, calculated pi to 200 digits over the course of several months, setting what was then the world record. A semi-literate German, Dase had no mathematical education or understanding of math more advanced than arithmatic. He didn’t advance the art of mathematics. He just crunched the numbers. But the numbers, like Sir Edmund Hillary said, are there. The promise of victory — obscure but real triumph — floats just out of reach. Michael Shafer ran a computer program, caught the brass ring, and now holds a world record. To a certain sort of person, big numbers are just cool.
In 1900, German mathematician David Hilbert delivered an address at a math conference in Paris. The list of unsolved problems he delivered is known as Hilbert’s problems, the 23 most important unsolved questions in math at the turn of the last century. Problem ten, the one Gregory Chudnovsky may have taken a run at, involves algebraic equations. Problem eight is the Riemann hypothesis, about (among other things) the distribution of primes. Solving these problems (only a few remain unsolved) earned mathematicians a place in history. But one man earned the only standing ovation in the history of the American Mathematical Society. F. N. Cole spent the entirety of his address, "On the Factorisation of Large Numbers", at the blackboard, proving that one of Mersenne’s guesses was wrong:
267-1 = 193707721 × 761838257287 = 147573952589676412927
"Mathematics," said David Hilbert, "is a game played according to certain simple rules with meaningless marks on paper," but every game has a winner.