In September, OpenAI announced an apparent solution to one of the seven “Millennium Prize problems” in mathematics: Navier-Stokes, a longstanding question about a set of equations describing the behavior of fluids such as water and air. It was an exciting development, but one that also caused turbulence and discombobulation in the mathematics community.
On Tuesday, when the firm released a trove of hundreds of results, no new Millennium proofs were among them. But select findings seemed to represent significant progress on some, if not all, of the five remaining open questions that stand as the most prominent in the field.
What, exactly, are these very specialized open questions? They are difficult for non-mathematicians (and even most mathematicians) to understand. But here is an attempt at a summary, and a bit of insight into how OpenAI’s new results may — or may not — help pave the way toward resolution.
What is the Riemann hypothesis?
Formulated in an 1859 paper by German mathematician Bernhard Riemann, the problem is about prime numbers (a whole number greater than 1 that can only be divided evenly by itself and 1, such as 2, 3, 5, 7, 11, 13 and so on) and how they are distributed along the number line stretching to infinity.
Moving further along the number line, primes become rarer: Up to 100, some 25 percent of the numbers are prime; up to one million, only about 7.8 percent are prime. Riemann made a working hypothesis that provides a precise formula for estimating, or counting, how many primes exist below any given number.