# Proving the Twin Prime Conjecture — sort of

Most people understand the concept of prime numbers — whole numbers that are divisible only by one and themselves. Most people also understand that there is an infinite number of prime numbers. There is a new development in that esoteric branch of mathematics that deals with prime numbers. This is a bit technical, but it is so cool!

First a bit of background. Primes occur frequently among smaller numbers, but as the numbers grow, primes become less frequent. Furthermore, normally the gap between primes also grows — except for what are known as twin primesTwin primes are pairs of prime numbers that differ in value by 2. Here are three examples: 3 and 5, 17 and 19, and 2,003,663,613 × 2195,000 − 1 and 2,003,663,613 × 2195,000 + 1.

Greek mathematician Euclid of Alexandria is thought to have originally formulated the twin prime conjecture in 300 BCE: There is an infinite number of twin primes. Mathematicians have been trying to prove this conjecture ever since — that is until now, sort of.

On May 13, 2013, at Harvard University, Yitang Zhang (of the University of New Hampshire) presented a paper proving there are infinitely many pairs of primes (as opposed to twin primes) that are less than 70 million units apart. 70 million seems like a very large number, but what Zhang has demonstrated to the satisfaction of anyone who can actually understand his logic, is that there is a limit to the size of the gaps between pairs of primes. The gaps between consecutive prime pairs don’t keep growing forever. In effect, Zhang crammed the gap between pairs of primes down from infinity to just 70 million.

Dan Goldston, an analytic number theorist at San Jose State University in California, along with two colleagues showed in 2005 that there is an infinite number of prime pairs that differ by no more than 16. There was a catch, however. They were assuming a conjecture that no one knows how to prove. Zhang assumed no such conjectures. Goldston does not think the value can be reduced all the way to 2 to prove the twin prime conjecture, but he says the very fact that there is a number at all is a huge breakthrough. “I was doubtful I would ever live to see this result,” he quipped.

The jump from 70 million to 2 is nothing compared with the jump from infinity to 70 million — and Zhang just did that!

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