AI companies are unable to solve complex number theory problems like bounding gaps between primes despite significant compute investment.

Aug 10, 2005: Goldston, Pintz and Yildirim develop a method that can obtain a bound of 16 on gaps between primes assuming the (still unproven) Elliott-Halberstam conjecture, and a "near miss" towards any finite bound at all without that conjecture. They also observe that any improvement over the Bombieri-Vinogradov equidistribution theorem for primes would produce *some* finite bound for gaps between primes. Late August, 2005: Multiple AI companies race to release unconditional versions of the GPY theorem, with various bounds. Sep 2005: After dozens of such releases, and millions of dollars of compute collectively spent, the bound stabilizes in the low triple digits. Progress stalls, and the AI companies move their attention on to the next benchmark. 2005-: The problem is now considered "saturated", requiring enormous amounts of compute to make further progress. Mathematicians now also move their attention to other problems in the field (cf., Thurston's account in Section 6 of https://arxiv.org/abs/math/9404236 of the field of foliation theory being "evacuated" after he focused on solving all the major problems in that area; it took over a decade for the field to be reinvigorated by external developments). With almost noone willing or able to digest the results, no professionally written preprint, let alone a published paper, is generated from the race to improve the bound. The AI-generated arguments are never taught in classrooms, or make it to a number theory textbook. (6/8)

Aug 10, 2005: Goldston, Pintz and Yıldırım (GPY) develop a method that can obtain a bound of 16 on gaps between primes assuming the (still unproven) Elliott-Halberstam conjecture, and a "near miss" towards any finite bound at all without that conjecture. They also observe that any improvement over the Bombieri-Vinogradov equidistribution theorem for primes would produce *some* finite bound for gaps between primes. The GPY method forms the foundation for all subsequent work in the area. Feb 27, 2006: Motohashi and Pintz observe that one only needs to improve the Bombieri-Vinogradov equidistribution theorem for *smooth* moduli. However, even this restricted improvement remains open. May 14, 2013: Zhang, who at the time was working as an adjunct lecturer, rediscovers the Motohashi-Pintz observation, and cleverly adapts older (and somewhat out of fashion) work of Bombieri, Friedlander, and Iwaniec to obtain the required equidistribution theorem. This leads to the specific finite bound of 70 million. Late May, 2013: Multiple announcements appear on blogs and on the arXiv making various incremental improvements to the 70 million bound. A competitive race to rush to announce these bounds begins to develop. [In retrospect, this race was an early harbinger of what competitive benchmarking might do to a problem such as this, with time pressure incentivizing authors to cut corners with respect to exposition and verification.] (2/8)