By Christian Viveros-Fauné
By Miriam Felton-Dansky
By Tom Sellar
By Tom Sellar
By Jessica Dawson
By Tom Sellar
By R. C. Baker
By Tom Sellar
It remains unconquered to this day. Nash's crash-and-burn was more dramatic than mostdistracted and mumbling, he incoherently linked the hypothesis to arcane cosmology. But he is just one in a long line of formidable thinkers to have unsuccessfully tackled this conjecture, formulated by Bernhard Riemann in 1859 in a brief paper titled "On the Number of Prime Numbers Less Than a Given Quantity." In 1900, mathematician David Hilbert, addressing a conference in Paris, outlined what he judged the greatest unsolved problems in mathematics. Of the so-called Hilbert Problems (discounting the handful later deemed too vague for decisive solutions), the only one still standing at the start of the 21st century is the Riemann Hypothesis. In 2000, the Clay Mathematics Institute in Cambridge, Massachusetts, threw down the gauntlet again, announcing seven Millennium Problemsamong them the R.H.each one with a million-dollar prize attached.
Math conundrums have been known to languish unsolved for a seeming eternity. Long considered the holy grail, Fermat's Last Theorem (another Hilbert Problem) had stumped all comers for some 350 years before Andrew Wiles finally cracked it in 1994. There has been a flurry of RH research activity recently, spurred more by Wiles's triumph than the cash prize (mathematicians agree that immortal glory awaits the person who provesor refutesthe hypothesis). "People would never publicly acknowledge that they were working on the Riemann Hypothesis," says Dorian Goldfeld, a professor of mathematics at Columbia. "Now everyone's coming out of the closet." Brian Conrey, director of the American Institute of Mathematics in Palo Alto, California, writes in the March issue of Notices, the journal of the American Mathematical Society, "Now is arguably the most exciting time in its history to be working on R.H."
This month sees the publication of no fewer than three popular-math books focusing on primes and the hypothesis: Marcus du Sautoy's The Music of the Primes (HarperCollins), Karl Sabbagh's The Riemann Hypothesis (Farrar, Straus and Giroux), and John Derbyshire's Prime Obsession (Joseph Henry Press). While Wiles's proof made the front page of the Times, inspiring a bestseller (Simon Singh's Fermat's Enigma) and even an Off-Broadway musical (Fermat's Last Tangosample lyric: "Your proof contains a big fat hole"), it's questionable if the Riemann Hypothesis has similar crossover potential, not least because it still lacks a happy ending. At first glance, Fermat's is straightforward enough to seduce mathophobes; it simply states that there are no whole-number solutions of the equation xn+ yn = zn for n > 2. (The form will be familiar to anyone with even the faintest recollection of the Pythagorean Theorem.) By comparison, even a perfunctory definition of the Riemann Hypothesis necessarily entails an infinite series called the zeta function (1 + 1/2s + 1/3s + 1/4s + . . . ), which, for certain values of s, has a finite answer, and the daunting abstraction of complex numbers, which consist of real and imaginary parts.
From a mathematical standpoint, though, a Riemann proof would be more significant than the Fermat solution. It would shed light on the most basic objects in mathematics: prime numbers. A prime is a number perfectly divisible only by itself and the number 1; while every other number can be expressed as the product of at least two primes, the primes themselves are irreducible (11, 89, and 6,971, for instance, are prime; 15, which can be written as 3 x 5, is not). Pick your science metaphor: The primes are the atoms, the periodic elements, the genetic code of mathematics, and it frustrates mathematicians to no end that something so fundamental should remain so intractably mysterious. Primes seem to be randomly strewn amid the positive integers. Since their sequence conforms to no discernible formula, it's impossible to tell when the next prime will pop up.
Despite their chaotic behavior, as du Sautoy writes, the primes have a "timeless, universal character. Prime numbers would be there whether we had evolved sufficiently to identify them." In Carl Sagan's novel Contact (1985), scientists realize that the messages being transmitted from Vega are prime-number pulsesand thus signs of intelligent life. In discussing the primes, mathematicians often use the vocabulary of first love. "They're objects of great beauty, no question," says AIM's Conrey, explaining how as a teenager he fell for the Twin Prime Conjecture, which states that there are an infinite number of pairs of primes separated by 2, like 3 and 5, 17 and 19 (it's still unproved).