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OpenAI's Astra solves ten math problems
Daniel Litt of the University of Toronto, one of the most prominent skeptics of AI's potential for frontier math research, conceded a bet about what AI could do. He wrote that the bet has not strictly resolved, since no Annals-quality number theory paper has appeared yet, but that he was wrong about the capabilities needed to produce one and it is a matter of time. He also wrote that the models don't yet seem able to do certain kinds of high quality research, and that he expects this to be a matter of not too much time.
Kirwin Hampshire posted that he is suffering a profound spiritual crisis after LLMs produced a number of counterexamples to significant long-standing conjectures over roughly the past week. Math PhD student Tasmin Chu called on mathematicians to avoid working with AI companies and to preserve the way math research is currently done.
Jacob Tsimerman, who just won the Fields Medal, promptly took a job at OpenAI.
Why does one counterexample sink a conjecture?
A conjecture usually claims something holds for every case in an infinite family, so a single failing case is enough to refute it. Leonhard Euler conjectured in 1769 that at least n nth powers are needed to sum to another nth power. In 1966, L. J. Lander and T. R. Parkin found four fifth powers that sum to a fifth power, using a computer search. Their equation was 27 to the fifth power plus 84, 110, and 133 each to the fifth power, equal to 144 to the fifth power.
Why is the Annals of Mathematics so prestigious?
The Annals of Mathematics was founded in 1884 and is now run jointly by Princeton University and the Institute for Advanced Study. It publishes only a small number of papers each year, and its editors look for results that change the direction of a field. Referees often take a long time to check the proofs in submitted papers. Andrew Wiles's proof of Fermat's Last Theorem appeared there in 1995.
Has a computer settled a famous problem before?
In 1976, Kenneth Appel and Wolfgang Haken proved the four color theorem, which says any map can be colored with four colors so that no neighboring regions match. They reduced the problem to 1,936 configurations and used more than a thousand hours of computer time to check them. Many mathematicians at first distrusted a proof that no human could read in full. In 2005, Georges Gonthier verified a version of it with the Coq proof assistant.
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