College Math Teaching

May 16, 2013

Big Breakthrough in Number Theory: progress toward the twin primes conjecture.

Filed under: advanced mathematics, number theory — Tags: , , — collegemathteaching @ 7:26 pm

It is a long standing conjecture in number theory that there exists an infinite number of twin primes: twin primes are prime integers that differ by 2.

Example: 3 and 5, 11 and 13, 17 and 19 are examples of twin prime pairs.

Very large twin primes have been found: (2,003,663,613 \times 2^{2195,000}) - 1 and (2,003,663,613 \times 2^{2195,000}) + 1 .
But, up to now: We don’t know if this pairing “stops” at some point (is there a largest pair?)

In fact, up to recently, we had no statement of the following form: given a finite integer M there exists an infinite number or pairs of primes p, q such that p - q \le M (assuming that p is the greater of the pair).

Well, now we do. The Annals of Mathematics (the top ranked mathematics journal in the world) has accepted a paper that shows the infinite pairs statement is true, if M = 70,000,000 :

The twin prime conjecture says that there is an infinite number of such twin pairs. Some attribute the conjecture to the Greek mathematician Euclid of Alexandria, which would make it one of the oldest open problems in mathematics.

The problem has eluded all attempts to find a solution so far. A major milestone was reached in 2005 when Goldston and two colleagues showed that there is an infinite number of prime pairs that differ by no more than 16. But there was a catch. “They were assuming a conjecture that no one knows how to prove,” says Dorian Goldfeld, a number theorist at Columbia University in New York.

The new result, from Yitang Zhang of the University of New Hampshire in Durham, finds that there are infinitely many pairs of primes that are less than 70 million units apart without relying on unproven conjectures. Although 70 million seems like a very large number, the existence of any finite bound, no matter how large, means that that the gaps between consecutive numbers don’t keep growing forever. The jump from 2 to 70 million is nothing compared with the jump from 70 million to infinity. “If this is right, I’m absolutely astounded,” says Goldfeld.

Zhang presented his research on 13 May to an audience of a few dozen at Harvard University in Cambridge, Massachusetts, and the fact that the work seems to use standard mathematical techniques led some to question whether Zhang could really have succeeded where others failed.

But a referee report from the Annals of Mathematics, to which Zhang submitted his paper, suggests he has. “The main results are of the first rank,” states the report, a copy of which Zhang provided to Nature. “The author has succeeded to prove a landmark theorem in the distribution of prime numbers. … We are very happy to strongly recommend acceptance of the paper for publication in the Annals.”

Hey, 70 million is a LOT less than “infinity”. 🙂

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