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# From 70 million to two

## It's not that mathematicians have been searching for numbers that differ by exactly two. But what they want to know is if there an infinity of them?

A major advance in mathematics, and the gist of it made a 14 year-old I know chuckle. Well, here’s a parallel. Say you’re searching for objects that are exactly two inches long—some school project perhaps. Along comes a friend who says: “Pssst! I know where you’ll find many objects, all less than 70 million inches long!"

You’d chuckle too. You’d probably think he was batty. So then what would you think if the rest of your school, even the rest of the world, sees this as a revelation and applauds your friend as a pioneer? Probably chuckle some more.

Yet in April, something just like this happened in that goldmine of fascinating goings-on, mathematics. Mathematicians have long studied certain pairs of numbers that differ by exactly two. It’s not that they have been searching for them—they are all over the place. But precisely for that reason, mathematicians want to know, is there an infinity of them? Or do they stop appearing, somewhere in the upper reaches of the number line? This is a dilemma that mathematicians have never been able to resolve.

Along comes a totally unknown mathematician—so unknown and unheralded that he once worked at a sandwich shop to make ends meet—who says: “Pssst! I’ve just proved that there’s an infinity of these numbers that differ by 70 million or less!"

Worth a 14-year-old’s chuckle? And yet mathematicians the world over are in a whirl. They have grappled so long with the question of whether there is an infinity of such pairs that it even has a name, the Twin Prime Conjecture. For it is primes we are talking about; specifically, primes that differ by two, like 17 and 19, 29 and 31, 41 and 43, 101 and 103 (called, no prizes for guessing, twin primes). The conjecture is that there is an infinity of such pairs. But nobody has managed to prove it.

In fact, this is really a special case of a broader conjecture: that there is an infinite number of primes that differ by any given even number, not just 2. (Why even, and not odd? Puzzle for you to tease out—though ignore the special-case prime of 2). Nobody has proved that one either.

Along comes Yitang Zhang, a professor at the University of New Hampshire. Remember, we still don’t know whether there are an infinite number of pairs of primes that have a gap between them of two, or any larger even number. But in April, Zhang proved that there is an upper limit to these gaps: 70 million. That is, you can wander as far as you like into the upper reaches of the number line, and you will never run out of prime pairs that differ by 70 million or less.

All right, I don’t blame you for being underwhelmed. But people interested in these things have described Zhang’s proof as “astounding", “thrilling", “a huge step forward" and “one of the great results in the history of number theory." From being an obscure member of the faculty at a small university, Zhang has suddenly been catapulted into a dizzying series of lectures about his work at top-flight universities, and praised for his “clarity" in delivering them.

The reason for all this attention is really Zhang’s discovery of a limit. Because an upper bound on the gaps between prime pairs is, yes, a huge step forward from not knowing if there is such a bound at all. And to mathematicians, there is intrinsically little to choose between 70 million and two. Both are just numbers. The very existence of the 70 million limit holds out hope that it can one day be pushed down to two. It will be hard work, and there are already people who think it may not happen. But new proofs like Zhang’s also bring new thinking in their wake. That’s their great value, and where the real hope of proving the Twin Prime Conjecture lies.

And there is an interesting insight in all this. Primes and prime pairs are scattered randomly among the numbers, though they get slowly rarer as you get to those fabled upper reaches. However, this does not mean they are scattered evenly: you might have a bunch of them, followed by a long dearth of them. This is really the essence of randomness: that when you have random distributions, patterns naturally form. Patterns, that is, including twin primes.

In other words, the truly counter-intuitive idea would be if the twin prime conjecture was not true. Because then something would be working to prevent primes from forming the two-apart pattern; almost by definition, that militates against randomness. The mathematician Jordan Ellenberg says that with Zhang’s result, “we might…be on our way to developing a richer theory of randomness." That’s an exciting prospect.

Zhang himself said this about how he found the way to his proof: “The important thing is to keep thinking."

There’s good advice. Certainly worth rather more than a chuckle.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. A Matter of Numbers will explore the joy of mathematics, with occasional forays into other sciences. To read Dilip D’Souza’s previous columns, go to www.livemint.com/dilipdsouza