A longstanding question in mathematics that has puzzled countless bright minds throughout the ages is the twin prime conjecture. The conjecture, that some believe was first stated by ancient Greek mathematician Euclid of Alexandria, says that that there is an infinite number of prime “twin pairs” – pairs of prime numbers that differ in value by 2. These are 3 and 5 or 11 and 13 or just as well 2,003,663,613 × 2195,000 − 1 and 2,003,663,613 × 2195,000 + 1. Whether there’s an infinity of such pairs hasn’t been proven thus far, however.
A Chinese mathematician recently made an important contribution to solving this mathematical problem, after he proved that there are infinitely many pairs of primes that are less than 70 million units apart. Now, this might not seem like much considering that the goal is to prove that there are an infinite amount of prime pairs spaced only by two units, but in mathematics you need to dwell into the abstract more to experience the milestone.
“That’s only [a factor of] 35 million away” from the target, quips Dan Goldston, an analytic number theorist at San Jose State University in California who was not involved in the work. “Every step down is a step towards the ultimate answer.”
What’s remarkable about the discovery, is that Yitang Zhang of the University of New Hampshire in Durham, who came up with the new results, relied on standard mathematical techniques. 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.
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Hopefully, like Zhang claims, the paper’s mathematical machinery will allow for the value of 70 million to be pushed downwards. It’s unlikely however that this value will be reduced all the way back to 2 to prove the twin prime conjecture any time soon. Still, Goldston 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 says.
Zhang’s paper is set for publication in the Annals of Mathematics in a few weeks time. [source]