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Math Trek

A field where breakthroughs are hard to come by produces two big advances on a single day

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Problems in number theory often have a certain exasperating charm: They are extraordinarily simple to state, but so difficult to prove that centuries of effort haven’t sufficed to crack them. So it’s pretty remarkable that on one day this May, mathematicians announced results on two of these mathematical conundrums. Both proofs address one of the most fundamental questions in all of mathematics, the relationship between multiplication and addition.

On May 13, a virtually unknown lecturer at the University of New Hampshire, Yitang Zhang, shocked experts when he announced in a talk at Harvard a proof that takes steps toward solving one of the oldest problems in all of mathematics: the twin prime conjecture.

“Zhang’s result came completely out of the blue,” says Andrew Granville of the University of Montreal. “It’s huge.”

Prime numbers — those divisible only by 1 and themselves — are like the fundamental particles of mathematics, the indivisible building blocks out of which all other numbers are formed. Mathematicians have long noticed that primes often occur in pairs that differ by 2, like 5 and 7 or 137 and 139. They suspect that there are infinitely many pairs of primes that differ by 2, as well as infinitely many that differ by any even number.

Since primes get rarer as they get larger, it’s easy to imagine the opposite — that the gaps between them also grow.

Zhang has shown that that’s not true. He has shown that there’s at least one number N such that there are infinitely many pairs of primes that differ by N. Zhang can’t tell you what N is — but he does know it’s smaller than 70 million.

True, 70 million is a lot more than 2. But it’s a start.

“All of the experts, including me, had thought about how to develop the idea,” says Granville. “We thought we thought through the idea that Zhang ultimately used, but we decided there was no way. Zhang fortunately didn’t know the experts, so he didn’t get put off. He found the way that we all missed, not by a little bit but by a lot.”

Perhaps Zhang was able to find a different way because after earning his doctorate in 1991, he chose an unorthodox path. He apparently didn’t seek an academic job and worked for a while in a sandwich shop before finding work as a lecturer. Still, he kept doing mathematics. His thesis advisor, T.T. Moh of Purdue University, describes him in extraordinary terms: “When I looked into his eyes, I found a disturbing soul, a burning bush, an explorer.”

The same day that Zhang gave his talk, Harald Helfgott of the École Normale Supérieure in Paris posted a paper online with a proof closing in on another major open problem in number theory. The Goldbach conjecture says that any even number greater than 2 is the sum of two primes. The number 8, for example, is 3+5. Helfgott has proven something a bit easier, a proposition known as the odd Goldbach conjecture: that any odd number greater than 5 is the sum of three primes. The number 7, for example, is 2+2+3.

The Goldbach conjecture implies the odd Goldbach conjecture: subtract 3 from any odd number greater than 5, express the result as a sum of two primes, add back the 3, and you’ve expressed your original number as a sum of three primes. Unfortunately, the trick doesn’t work the other way.

Helfgott’s finding didn’t shock mathematicians the way Zhang’s did because he was following a well-understood approach. In 1919, mathematicians G.H. Hardy and John Littlewood produced a function that counts how many different ways one can represent a number as the sum of three primes. As long as the result of that function is always greater than 0, the odd Goldbach conjecture is true. Because this function is created using the same methods that decompose a radio signal into its basic frequencies, Helfgott describes this as “listening to the primes.”

Generally, big numbers can be represented in many different ways as the sum of three primes. In 1937, Ivan Vinogradov used Hardy and Littlewood’s function to prove the odd Goldbach conjecture for big numbers. Really, really big, that is: at least 106,846,168 or so.

That number is boggling, far beyond the number of atoms in the universe and the magnitude computers can handle. So the challenge has been to bring that limit down to something manageable and then to allow computers to do the rest, which Helfgott has finally managed to do.

“Helfgott’s treatment must have 60 or 70 clever new ideas in it, and each of them produces a small improvement, and eventually he got it down to something he could deal with in practice,” says Roger Heath-Brown of the University of Oxford. “It’s a great achievement.”

Unfortunately, proving either Goldbach or the twin primes in full is likely to require an approach fundamentally different from the ones Helfgott and Zhang used.  Mathematicians hope to hone down Zhang’s 70 million number — the most wildly optimistic guess is that they might reduce it to 16 — but very significant new ideas will be needed to get all the way to 2. And Helfgott’s method is known not to extend beyond sums of three primes.

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