A partial solution to a centuries-old problem known as the twin prime conjecture now affirms the idea that an infinite number of prime numbers have companions — and that a maximum distance between these pairs does in fact exist.
Prime numbers are those non-composite numbers that can only be divided by one or itself. On average, the gap that separates these numbers gets larger as their values increase. But a neat quirk about primes is that every once in awhile they also come in pairs, so-called twin primes. These numbers differ from another prime by two. Examples include 3 and 5, 17 and 19, 41 and 43, and even 2,003,663,613 × 2195,000 − 1 and 2,003,663,613 × 2195,000 +1.
Ever since the time of Euclid, however, mathematicians have wondered if these twin primes keep on appearing for infinity. They have no doubt that primes themselves appear for infinity, but because mathematicians lack a useful formula to predict their occurrence, they have struggled to prove the twin prime conjecture — the idea that there are infinitely many primes p such that p+2 is also prime (i.e. the two number gap).
But now, as the Mathematician Zhang Yitang from University of New Hampshire in Durham has shown, there is a kind of weak version of the twin prime conjecture. He didn’t prove that a distance of 2 exists for an infinite number of primes, but he did prove that there are infinitely many prime gaps shorter than 70 million.
A gap of two is obviously far removed from a gap of 70 million. But considering that the previous estimate was infinity, Zhang’s assertion is incredible. As Maggie McKee noted inNature News, “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.”
Zhang presented his research yesterday (May 13) to an audience at Harvard University, so his work will still have to withstand the scrutiny of peer review. But according to McKee, a referee with the Annals of Mathematics is recommending that his paper be accepted for consideration.