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Year in Review: Progress made toward twin prime proof

Surprising advance sparks flurry of work on mathematical conjecture

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Some pan for gold, some pan for prime numbers. Yitang “Tom” Zhang may have found both.

Zhang, a mathematician at the University of New Hampshire, stunned the mathematical world in May when he reported a major step toward solving one of the oldest outstanding problems in number theory, called the twin prime conjecture (SN: 10/19/13, p. 38).

For more than 100 years, mathematicians have known that prime numbers — such as 7 or 17, with no divisors except for themselves and 1 — get sparser and sparser as the numbers get larger. They are like flecks of gold in a stream that is gradually running out of gold.

Nevertheless, mathematicians believe you can always find two prime “gold nuggets” in the same pan no matter how far downstream you go. If the separation between the primes is 2, they are called twin primes. Examples are 17 and 19, or 1,607 and 1,609. The twin prime conjecture says such pairs never run out. It’s called a “conjecture” because no one has been able to prove it.

Zhang proved that you can catch as many pairs of primes as you want if you use a pan whose width is at least 70 million. (Of course, they may or may not be twin primes.) Zhang did so in spite of being a relative unknown in the field of number theory. “No one had a clue that he was working on the problem,” says Andrew Granville of the University of Montreal. Nevertheless, Granville says, Zhang’s paper, to be published in Annals of Mathematics, is “beautifully written, just stunning, masterful work.”

Mathematicians have been frantically competing since May to improve on Zhang’s result. So far the narrowest pan known to work is 4,680 numbers wide. But James Maynard of the University of Montreal announced in November that he can bring the number down to 600. Further advances are almost certainly in the offing, but it is considered unlikely that the size will soon be brought down to 2, which would prove the twin prime conjecture.

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