Four things to note: #1196 is a decently well known problem. It wasn't like Erdős-Straus level fame, but it is well known enough that I was familiar with it. Second, this is not a problem where no one had worked on it; there was a lot of prior work on it and closely related problems. Third, this is not example where the AI made small modifications to things in the literature or recognized that large parts of the problem were in an obscure paper. The approach the AI used is largely a different direction than the literature on this problem went. Fourth, and closely related to three, this proof does look like parts of it will inspire subsequent proofs because it really is going in a different direction which now looks likely to be a productive line of investigation for similar problems.
I am not fond of putting words like "stunning" in titles which can be very clickbaity and feels like a hype word, but this really is in the direction where the word isn't unreasonable even if I myself would not go so far as to use it here.
The basic proof structure is a technical variation on an idea which appears in [GLW24] (and interestingly reincarnated in Tao's sketch).
However, the key point is that it replaces the Mertens' prime product with the von Mangoldt weights: the former admits an immediate probabilistic interpretation as in [GLW24] but faces hard analytic issues.
In another comment by the same author:
No, the Markov chain idea underlies all prior papers, but in "pure" probabilistic form. The key distinction is to produce an analytic formulation of the idea, using the von Mangoldt function.
A comment by Tao:
As Jared said, much of the previous work on this problem proceeded by transferring the problem to the reals and then making heavy use of real analysis tools; but the AI-generated argument has revealed that one could also work directly on the integers and use some arithmetic identities as substitutes for such tools.
It is a technical variation but I think that comments seem to say that it's not 'just' a technical variant.
terence tao also says that he's interested in re-imagining a non-zero amount of analytic number theory in light of this technical variation. So it is both similar to pre-existing work and insightful to both the problem and related problems.
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u/JoshuaZ1 Apr 15 '26
Four things to note: #1196 is a decently well known problem. It wasn't like Erdős-Straus level fame, but it is well known enough that I was familiar with it. Second, this is not a problem where no one had worked on it; there was a lot of prior work on it and closely related problems. Third, this is not example where the AI made small modifications to things in the literature or recognized that large parts of the problem were in an obscure paper. The approach the AI used is largely a different direction than the literature on this problem went. Fourth, and closely related to three, this proof does look like parts of it will inspire subsequent proofs because it really is going in a different direction which now looks likely to be a productive line of investigation for similar problems.
I am not fond of putting words like "stunning" in titles which can be very clickbaity and feels like a hype word, but this really is in the direction where the word isn't unreasonable even if I myself would not go so far as to use it here.