Ever More Relevant,Prime-Number Proofs Chalk Up More Success

Ever More Relevant,Prime-Number Proofs Chalk Up More Success
February 3, 2006; Page B1
The Wall Street Journal
Sharon Begley

Classic math problems can derail a mathematician's career. "You usually can't solve them," says mathematician Daniel Goldston of San Jose State University, so devoting your career to one is likely to leave your list of accomplishments a bit short.

Mathematicians are not letting a little thing like daunting odds deter them, however, and as a result one of the most romantic subjects in math -- prime numbers -- has recently seen what Kannan Soundararajan of the University of Michigan, Ann Arbor, calls "spectacular progress. Only five years in, and already this has been a great millennium for primes."

Ever since Euclid proved that there are infinitely many primes (though they get sparser the farther out on the number line you go), claims about primes have proliferated like rabbits. In many cases, mathematicians know in their hearts that a claim is true but, to their chagrin, can't prove it.

That used to bother only those who inhabit the rarefied world of abstract math, where discussion of a "pseudo complex structure on a manifold X of dimension 2N is a C-module structure on a tangent bundle" counts as small talk. But now that primes are the basis for codes that encrypt financial data as well as national-security transmissions, making sure that mathematicians' hunches about primes are actually true matters in the real world, too.

A prime is a number whose only factors are 1 and itself, like 2, 3, 5, 7 and 18,793, not to mention 2 multiplied by itself 30,402,457 times minus 1 (the largest prime discovered so far, announced on Christmas; testing ever larger numbers for primeness, while doable, is very time consuming). Like the atoms from which the world of matter is constructed, primes are the raw material from which the whole world of numbers can be assembled because every number can be written as the product of primes. For instance, 18 = 2 x 3 x 3. They are also beacons of simplicity and purity -- minimalist art in the otherwise rococo landscape of math.

Yet primes' simplicity belies their mystery. "There are so many basic questions about prime numbers that we can't answer, and many things we think are true but can't prove," says Prof. Goldston.

His quarry is a centuries-old claim called the twin-prime conjecture. It asserts that there are infinitely many primes separated by 2, like 29 and 31. The largest known twin primes are the numbers 16,869,987,339,975 times 2 multiplied by itself 171,960 times plus 1, and that product minus 1.
But just because mathematicians have found a bunch of twin primes doesn't mean there are endless pairs, as claimed. Lots of crackpots have taken a stab at proving the claim, to no avail. Pros have also claimed to have a proof, only to retract it when competitors have spotted an error.

In three papers submitted to math journals for publication, however, Prof. Goldston and two colleagues have come as close as anyone in history.
"We weren't actually working on the twin-prime conjecture, because the feeling was there's no way to attack it," says Prof. Goldston. Instead, he and Cem Yildirim of Bogazici University, Istanbul, thought they might prevail against a weaker claim, namely, that there are infinitely many primes separated by gaps way smaller than the average spacing. They succeeded, proving that if the average gap between primes in some region of the number line is, say, 10 trillion, you can nonetheless find gaps as small as you care to.

Their proof suggests that there are infinitely many consecutive primes that differ by only 16, which is getting close to the 2 claimed by the twin-prime conjecture. "They're tiptoeing up to it," says Prof. Soundararajan.

The breakthrough surprised even its architects. "A year and a half ago I was convinced I'd die without solving" the twin-prime conjecture, says Prof. Goldston. "But now we're so close."

The popular image of mathematicians has them staring into space seeking inspiration, but Prof. Goldston scours old papers to find nuggets that might apply to his quest. Since 1999 he has collaborated with Prof. Yildirim. Their first proof, unveiled in 2003, had a mistake. When János Pintz of the Hungarian Academy of Sciences helped fix it, he became part of the "GPY" team.

That was not the only benefit of the error. The proof contained a sub-proof, like an accurate chapter in a flawed book, that Ben Green of the University of Bristol, England, and Terry Tao of UCLA used to prove a counterintuitive claim: that there exist special number sequences, called arithmetic progressions, containing as many primes as you can imagine.

In such progressions -- such as 5, 11, 17, 23, 29, 35 . . . -- the numbers are the same distance apart; here, 6. The longest such sequence, discovered in 2004, contains 23 primes starting with 56,211,383,760,397 and increasing by 44,546,738,095,860. Prof. Green and Prof. Tao raised that 23: You can find a progression with 1 million primes, one with 10 trillion, or any other number, they proved.

Other prime claims elude proof. The Goldbach conjecture, which dates from the 1700s, says that every even number greater than 2 is the sum of two primes (20 = 7 + 13, for instance). Everyone believes it's true, but no one has proved it. Still, the millennium is young.