Blog HQ is closing up shop for the summer. But don’t despair we’ll be back mid-August!
But before we go, we have to point you to two new cool results in number theory.
The famous Twin Primes Conjecture says that even though on average the gaps between successive prime numbers becomes larger and larger (indeed, you can find gaps of arbitrarily large size), there are also infinitely many pairs of prime numbers which are only two apart (like 3 and 5, or 11 and 13). Nobody knows how to prove this, but Yitang Zhang just proved that there are infinitely many pairs of prime numbers which are less than 70,000,000 apart! This is a major breakthrough! Of corse, seventy million is a long way from two, but since there was no bound before Dr. Zhang’s theorem, it’s a huge leap forward towards proving the Twin Prime conjecture.
To read more about Dr. Zhang and his result, read the excellent article on the Simon’s Foundation website. There is a more detailed explanation of the proof in this blog post about a recent talk by Dr. Zhang.
You might guess such a result would be proven by a famous Field’s medalist but, remarkably, Dr. Zhang is relatively unknown among mathematicians. As the famous number theorist Andrew Granville put it:
“Basically, no one knows him,” said Andrew Granville, a number theorist at the Université de Montréal. “Now, suddenly, he has proved one of the great results in the history of number theory.”
– From the Simons Foundation article.
The other big result (which would have been headline news if it wasn’t for Dr. Zhang!) is Harald Helfgott‘s recent proof of the “weak” Goldbach conjecture that every odd number greater than five is the sum of three primes. We talked about the “strong” and “weak” Goldbach conjectures here. You can go there to read more about the conjecture, but suffice it to say that it is right up there with the Twin Prime Conjecture as a long standing open problem in number theory. We were happy to discover that Dr. Helfgott is a fellow blogger!