Big Advance on Simple-Sounding Math Problem Was a Century in the Making

Hector Pasten, a mathematician at the Pontifical Catholic University of Chile, solved a long-standing problem in number theory while procrastinating on writing a final exam. He proved that numbers in the sequence n² + 1 must always have at least one fairly large prime factor. Using elliptic and Shimura curves, Pasten showed that the largest prime factor grows at least as fast as (log(log n))², significantly improving previous estimates. This breakthrough may contribute to progress on related number sequences and certain cases of the abc conjecture, potentially advancing research in number theory.