In September, OpenAI announced an apparent solution to one of the seven Millennium Prize problems in mathematics: Navier-Stokes, a longstanding question about equations describing fluid behavior. While the firm later released hundreds of results, no new Millennium proofs were included, though some findings showed progress on the five remaining open questions.
What are these specialized questions? They are difficult even for most mathematicians. One is the Riemann hypothesis, formulated in an 1859 paper by German mathematician Bernhard Riemann. It concerns prime numbers and their distribution along the number line. As numbers grow larger, primes become rarer. Riemann proposed a hypothesis providing a precise formula for estimating how many primes exist below any given number. He died at 39 before returning to the problem. Peter Sarnak, a mathematician at Princeton University and the Institute for Advanced Study, called it a "great problem" and a proof "would be monumental mathematically."
A result related to the quasi-Riemann hypothesis was verified by Lean, a computer proof assistant, and considered likely correct. However, it falls short of solving the original Millennium problem. Alex Kontorovich, chair of the mathematics department at Rutgers, described it as a "fundamental breakthrough" but noted that genuinely new ideas would be needed to prove the full hypothesis.
Other remaining problems include the Birch and Swinnerton-Dyer conjecture, formulated in the early 1960s by British mathematicians Bryan John Birch and Sir Peter Swinnerton-Dyer using an early computer, the EDSAC-2.
Source: New York Times Top Stories · Summarized by HeadlinesBriefing