A Divine Language by Alec Wilkinson

Author:Alec Wilkinson
Language: eng
Format: epub
Publisher: Farrar, Straus and Giroux

11.

“Bounded Gaps Between Primes” is a backdoor attack on the twin-prime conjecture, which was proposed in the nineteenth century and says that, no matter how far you travel on the number line, even as the gap widens between primes, you will always encounter a pair of primes that are separated by two, such as 5 and 7. The twin-prime conjecture is still unsolved. Zhang established that there is a distance within which, on an infinite number of occasions, there will always be two primes.

“You have to imagine this coming from nothing,” Eric Grinberg said. “We simply didn’t know. It is like thinking that the universe is infinite, unbounded, and finding it has an end somewhere.” Picture it as a ruler that might be applied to the number line. Zhang chose a ruler of a length of seventy million, because a number that large made it easier to prove his conjecture. (If he had been able to prove the twin-prime conjecture, the number for the ruler would have been two.) This ruler can be moved along the line of numbers and enclose two primes an infinite number of times. Something that holds for infinitely many numbers does not necessarily hold for all. For example, an infinite number of numbers are even, but an infinite number of numbers are not even, because they are odd. Similarly, this ruler can also be moved along the line of numbers an infinite number of times and not enclose two primes.

From Zhang’s result, a deduction can be made, which is that there is a number smaller than seventy million which precisely defines a gap separating an infinite number of pairs of primes. You deduce this, Amie told me, by means of the pigeonhole principle. You have an infinite number of pigeons, which are pairs of primes, and you have seventy million holes. There is a hole for primes separated by two, by three, and so on. Each pigeon goes in a hole. Eventually, one hole will have an infinite number of pigeons. It isn’t possible to know which one. There may even be many, there may be seventy million, but at least one hole will have an infinite number of pigeons.

Having discovered that there is a gap, Zhang wasn’t interested in finding the smallest number defining the gap. This was work that he regarded as a mere technical problem, a type of manual labor—“ambulance chasing” is what a prominent mathematician called it. Nevertheless, within a week of Zhang’s announcement mathematicians around the world began competing to find the lowest number. One of the observers of their activity was Terence Tao, who had the idea for a cooperative project in which mathematicians would work to lower the number rather than “fighting to snatch the lead,” he told me.

The project, called Polymath8, started in March of 2013 and continued for about a year. Incrementally, relying also on work by a young British mathematician named James Maynard, the participants reduced the bound to 246. “There are several problems with going lower,” Tao said.

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