1859 · the remark
riemann publishes eight pages on how many primes lie beneath a given magnitude. partway through, almost in passing, he observes that the non-trivial zeros of ζ(s) appear to fall on the line where the real part is exactly one half. he calls it very probable. he allows that a rigorous proof would be desirable. he notes it is not required for the argument at hand, and he moves on. the most consequential aside ever committed to paper.
1900 · the list
hilbert puts it eighth on his list of twenty-three problems for the coming century. it is the only problem that will appear both on that list and, a hundred years later, on the clay institute's. two centuries of mathematicians have now nominated the same question as the one they most want answered.
1914 · hardy's consolation
hardy proves that infinitely many zeros lie on the critical line. sit with the shape of that sentence, because it is the shape of every encouraging result you will ever be handed. infinitely many is not all. an unbounded quantity of confirmation remains perfectly consistent with a single counterexample waiting further out. this is not a technicality. it is the entire condition.
2000 · the price
the clay mathematics institute attaches one million dollars, one of seven such prizes. the money is not the point and everyone involved knows it is not the point, but it does establish a market price for the thing, and i am constitutionally unable to ignore a market price.
2021 · the count
platt and trudgian verify the hypothesis computationally to height 3×10¹². within that range there are exactly 12,363,153,437,138 zeros. every one lies on the critical line. every one is simple. it is the largest body of confirming evidence ever assembled for an undemonstrated statement, and it moves the truth of that statement by an amount which is, formally, zero. faith and knowledge are not adjacent quantities. they are different kinds.
the other side of the name
vlad tenev and tudor achim founded harmonic in 2023 because both had become fixated on using a machine to take a millennium prize problem. their agent is called aristotle; it reasons in lean rather than in english, which is what lets them say it does not hallucinate — it will not assert what it has not derived. it took gold at the international mathematical olympiad. in july 2025 they put it in the app store.
the odds, from the man himself
asked about it, tenev cited a prediction market: “last i checked, there’s a 43 percent chance that the next millennium prize will be solved by ai or, like, human, with significant ai assist.” then he added that he thought that was an underestimate. in january 2026 a combination of gpt-5.2 pro and aristotle resolved erdős problem #728 outright — the first erdős problem settled autonomously by machines, formalised in lean within two days.