TLDR (story, not math) - Knuth poses a problem, his friend uses Claude to conduct 30 some explorations, with careful human guidance, and Claude eventually writes a Python program that can find a solution for all odd values. Knuth then writes a proof of the approach and is very pleased by Claude's contribution. Even values remain an open question (Claude couldn't make much progress on them)

looks like he is trying to make a point that the actual (formal) proof for 2Z + 1 (odd numbers) is still human - by himself that is. Not sure who came up with the core modular arithmetic idea of with s = 0 k increasing by 2 mod m.

> with careful human guidance,

I think this is pretty clearly an overstatement of what was done. As Knuth says,

"Filip told me that the explorations reported above, though ultimately successful, weren’t really smooth. He had to do some restarts when Claude stopped on random errors; then some of the previous search results were lost. After every two or three test programs were run, he had to remind Claude again and again that it was supposed to document its progress carefully. "

That doesn't look like careful human guidance, especially not the kind that would actually guide the AI toward the solution at all, let alone implicitly give it the solution — that looks like a manager occasionally checking in to prod it to keep working.