On Monday, Harald Helfgott of the École Normale Supériure in Paris posted a proof of one of the oldest open problems in number theory to the preprint repository arxiv. The ternary Goldbach conjecture, like so many questions in number theory, is easy to state but hard to prove. Every odd number greater than 5 can be written as the sum of three prime numbers. (Prime numbers have no factors other than themselves and the number 1.) For example, 7=2+2+3 and 91=7+41+43.
The ternary Goldbach conjecture is sometimes called the weak Goldbach conjecture. The strong Goldbach conjecture states that every even number greater than 2 can be written as the sum of two primes. Both conjectures were formulated in correspondence between Christian Goldbach and Leonhard Euler in 1742, hence the name. Logically enough, if you prove the strong Goldbach conjecture, you get the weak one for free: if you have an odd number greater than 5, subtract 3 from it. Now you have an even number greater than 2. So if you know that every even number greater than 2 is the sum of two primes, you can add 3 (a prime) to it to get your odd number, decomposed into the sum of three primes.
Sadly, it doesn't work the other way. If you have an odd number written as the sum of 3 primes and subtract one of the odd primes, you're left with an even number written as the sum of two primes, but there's no guarantee that all even numbers will show up this way. But the ternary Goldbach conjecture does establish that every even number can be written as the sum of at most 4 primes: just subtract any odd prime number (for example, 3, or 257885161-1 ) from the even number you want to split up, and you're left with another odd number, which we now know can be written as the sum of three primes. This improves Olivier Ramaré's 1995 theorem that every even number is the sum of at most 6 primes.
Helfgott's result is a big deal, but it didn't come as a lightning bolt from the heavens. His work is part of a long line of papers using a technique called the Hardy-Littlewood-Vinogradov circle method. (Catchy, huh?) The very general idea of the circle method is that we turn a question about a set of numbers, in this case the primes, into a question about integrals over circles using techniques originally coming from analysis in the complex plane. It seems kind of miraculous that it's even possible to convert questions about integers, which are spaced out discretely on the number line, into questions about continuous functions. "Questions of distribution of primes, or integers, can be expressed naturally in terms of the properties of continuous functions defined in terms of them," Helfgott wrote in an email. A more concrete explanation of the circle method is beyond me, but if you want to dig into it and its limitations a bit more, you can check out this post by Terence Tao. It's not for the equation-averse.
In the 1930s, Soviet mathematician Ivan Vinogradov established that the ternary Goldbach conjecture holds for all but finitely many odd numbers, so if someone could "just" check the odd numbers below a certain huge number C, everything would be good. There was just the pesky problem that Vinogradov's bound was on the order of 106846168, an impossibly huge number for today's computational resources, much less those available to Vinogradov. Over the next 70 or so years, the upper bound was whittled down to around 101346 as of 2002, but it was still far too large to handle.
Helfgott started working on Goldbach's conjecture in 2006 as a postdoc in Montréal. "I had been trying to see what some different ways of proving Vinogradov's theorem would be," he wrote in an email. "I realized that one could prove it without the circle method," he wrote, but, "it didn't seem possible to give reasonable bounds C by the alternative proofs." But papers and conversations with other researchers gave him hints of how to improve the bounds coming from the circle method.
Helfgott eventually managed to wrestle the upper bound down to 1030, a much more manageable size, and with David Platt of the University of Bristol, he verified the conjecture for all numbers below that bound by computer. But the heavy computational resources were dedicated to proving the Generalized Riemann Hypothesis (GRH) for a large but finite number of cases. The GRH is one of the most important unsolved problems in mathematics. If solved, it would help us understand the distribution of prime numbers much better than we do. In fact, if the GRH were proved, the ternary Goldbach conjecture would be a corollary. But for the time being, computer-assisted checks of GRH for certain numbers are the best we can do.
Of course, making substantial progress on a problem that some of the most brilliant mathematicians of the past century have worked on was not an easy task. "There were several blind alleys-at one point I had to throw away a 50-page manuscript," Helfgott wrote. "It was difficult to tell down the middle whether the plan would truly succeed. After all, if I had brought C down to 10100, that would still have been larger than the number of subatomic particles in the universe multiplied by the number of seconds since the Big Bang-there would have been no hope of checking things that far!" Helfgott wrote that keeping track of the bounds explicitly was one of the most difficult parts of the work. "One annoying thing about the problem was that it turned out not to be the kind of thing I could work on in my head while at the movies or at a concert (not that I should)," he wrote. "I did get some good ideas in the shower, though."
Helfgott's paper has not been peer reviewed yet, but number theorists seem to be optimistic that the theorem will hold up to scrutiny. Unfortunately, it doesn't provide much illumination on the strong Goldbach conjecture. Terence Tao, who proved last year that every odd number can be written as the sum of at most five primes, wrote on Google Plus that "the circle method is very unlikely to be able to settle the even Goldbach conjecture by itself." Helfgott wrote that the problem is essentially that the strong Goldbach conjecture would require asymptotic estimates—more refined information about the values of certain quantities—at key points, rather than the coarse upper bounds available through current methods.
I asked Helfgott how he had celebrated his accomplishment. "Well, I gave a talk on this yesterday, and then I was taken out for dinner by the locals, as usually happens when one visits a place to give a talk. My parents are coming to visit now, so it will be a very good time to take a short break." Helfgott is understandably relieved to have this major project finished and get back to his normal routine, which includes non-mathematical study as well. "I sometimes faced the difficult dilemma of whether to work into the night or prepare for a Russian test instead," he wrote. "Hopefully I will be catching up on languages now that this is done." Helfgott is fluent in English, French, Spanish, German and Esperanto, and according to his blog "badly needs to practice" Polish, Quechua (an indigenous language in his home country Peru), and Russian.*
The title of this post is an allusion to Bach's Goldberg Variations. I can only hope that Vi Hart or another talented person is writing a song about the Goldbach conjecture to the tune of the theme from the Goldberg Variations. In the meantime, here's a Wired article from 2012 about a cool visualization of the notes.
*This sentence was edited after publishing. Helfgott emailed me to correct the list of languages he speaks. He also sent me a more recent photo of himself, which I have added to the top of the post.