These last few weeks have seen two very exciting developments in number theory, my “first love” in mathematics. In particular, two important conjectures about prime numbers have been proven for the first time.
The first one states that there are infinitely many pairs of prime numbers that differ by at most 70 million. One would ultimately like to establish that there are infinitely many pairs that differ by 2, but showing that the smallest gaps between prime numbers are bounded is a huge step in the right direction. An interesting twist on the story is that it was proven by a relatively unknown mathematician, Yitang Zhang, using a modification of well-known techniques, which, surprisingly, most experts in the field believed would not be sufficient. This is reminiscent of the AKS primality testing algorithm I discussed in an earlier post.
The second one states that every odd number can be expressed as the sum of 3 primes, and was proven by my friend Harald Helfgott (I actually met Harald not through our work on mathematics, but through our common interest in the constructed language Esperanto; that, however, is a story for another time). This is the so-called “odd Goldbach conjecture”. The harder statement, known as the “even Goldbach conjecture”, states that every even number can be expressed as the sum of 2 primes. It’s unlikely we’ll see a proof of that one anytime soon, but then again, we’ve had many unexpected surprises in this area recently.
However, in today’s post I’d like to highlight the recent work of another friend, Yufei Zhao, whom I met during my math competition days. Yufei investigated a topic more broadly related to making use of random patterns in prime numbers (and not just them). I highly encourage you to check out his blog, and especially this post. It discusses the context of the recent results, Yufei’s own work, and a possible way forward to make progress on these tantalizing, long-standing results. For my part, let me just add that one of the attractions of number theory is that it is extremely easy to make statements which are probably true, but immeasurably difficult to prove (Fermat’s infamous last theorem being just one example). What makes it worthwhile to explore such questions, however, is that even if one doesn’t find the Eldorado (prove the difficult result), there are a lot of pretty gems to be found along the way.