Mathematicians Make a Major Discovery About Prime Numbers

Erica Klarreich in Wired:

In May 2013, the mathematician Yitang Zhang launched what has proven to be a banner year and a half for the study of prime numbers, those numbers that aren’t divisible by any smaller number except 1. Zhang, of the University of New Hampshire, showed for the first time that even though primes get increasingly rare as you go further out along the number line, you will never stop finding pairs of primes that are a bounded distance apart — within 70 million, he proved. Dozens of mathematicians then put their heads together to improve on Zhang’s 70 million bound, bringing it down to 246 — within striking range of the celebrated twin primes conjecture, which posits that there are infinitely many pairs of primes that differ by only 2.

Now, mathematicians have made the first substantial progress in 76 years on the reverse question: How far apart can consecutive primes be? The average spacing between primes approaches infinity as you travel up the number line, but in any finite list of numbers, the biggest prime gap could be much larger than the average. No one has been able to establish how large these gaps can be.

“It’s a very obvious question, one of the first you might ever ask about primes,” said Andrew Granville, a number theorist at the University of Montreal. “But the answer has been more or less stuck for almost 80 years.”

This past August, two different groups of mathematicians released papers proving a long-standing conjecture by the mathematician Paul Erdős about how large prime gaps can get. The two teams have joined forces to strengthen their result on the spacing of primes still further, and expect to release a new paper later this month.

More here.