I was able to use an extended conversation with an AI https://chatgpt.com/share/68ded9b1-37dc-800e-b04c-97095c70eb29 to help answer a MathOverflow question https://mathoverflow.net/questions/501066/is-the-least-common-multiple-sequence-textlcm1-2-dots-n-a-subset-of-t/501125#501125 . I had already conducted a theoretical analysis suggesting that the answer to this question was negative, but needed some numerical parameters verifying certain inequalities in order to conclusively build a counterexample. Initially I sought to ask AI to supply Python code to search for a counterexample that I could run and adjust myself, but found that the run time was infeasible and the initial choice of parameters would have made the search doomed to failure anyway. I then switched strategies and instead engaged in a step by step conversation with the AI where it would perform heuristic calculations to locate feasible choices of parameters. Eventually, the AI was able to produce parameters which I could then verify separately (admittedly using Python code supplied by the same AI, but this was a simple 29-line program that I could visually inspect to do what was asked, and also provided numerical values in line with previous heuristic predictions).
Here, the AI tool use was a significant time saver - doing the same task unassisted would likely have required multiple hours of manual code and debugging (the AI was able to use the provided context to spot several mathematical mistakes in my requests, and fix them before generating code). Indeed I would have been very unlikely to even attempt this numerical search without AI assistance (and would have sought a theoretical asymptotic analysis instead).
A second update on my previous posts https://mathstodon.xyz/@tao/115306424727150237 https://mathstodon.xyz/@tao/115316787727719049 on the #MathOverflow problem https://mathoverflow.net/questions/501066/is-the-least-common-multiple-sequence-textlcm1-2-dots-n-a-subset-of-t that had a first initial counterexample located with #AI assistance, and then the minimal counterexample obtained through a crowdsourced effort. As has happened with several previous projects, the crowdsourced effort has acquired a life of its own. In particular, the task of classifying *all* the counterexamples - which I had previously dismissed in my previous post as "somewhat tedious to perform", has in fact been completed, with the problem alternating between true and false for n between the first counterexample 71 and the last example 172, before settling down to a unending string of counterexamples. We now have some very efficient ways to either prove or disprove that a given number is highly abundant; some of the asymptotic analysis initially required some high powered number theory hypotheses such as the generalized Riemann hypothesis (GRH), but as our arguments became more efficient, this became unnecessary. We even have some recent contributions from the #LeanLang community, formalizing some of the counterexamples in Lean, almost in real-time! (1/3)
