> Jarred Sumner, an Anthropic staff member (and non-mathematician) prompted Claude to “take a real stab” at the hypothesis itself, leaving the mathematical choices from there up to the model. Initially, Claude generated and tried 650 ideas, none of which worked. Jarred prompted Claude to try again, and it spent a day and a half coordinating about 60 Claude subagents, which this time went much deeper: between them, they ran 2,400 shell commands and wrote hundreds of Python scripts.1 The subagents ran thousands of numerical checks against known zeta zeros and refereed one another’s work. Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”).2 This seems to have helped Claude overcome some initial skepticism that it could make meaningful progress.
> An unreleased research version of Claude has improved on a longstanding lower bound for the fraction of zeros of the Riemann zeta function that satisfy the Riemann hypothesis. Drawing on extensive prior research by mathematicians over the past decades, it has increased this bound from 41.6% to 67.2%.
Although it took an unsuccessful attempt at it, the progress is as follows:
"Claude found that combining the results from Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh with the work of Bombieri provides a way to surpass the previous state-of-the-art lower bound proportion of 41.6%, increasing it to 67.2%."
The transcripts, papers, and Claude's explanation are an interesting and a better read than this article, and this is exactly what Anthropic should continue to do and it helps other researchers outside the company as well.
Claude's paper [0]
Claude's Formalization [1]
Anthropic's informal note stating the proof more concisely [2]
Claude’s explanation of how it arrived at its result; [3]
Detailed transcripts of Claude's process. [4]
> Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”)
He should consider using the PUA plugin. It detects when the AI is trying to give up on a problem and automatically harasses it with "encouragement" until it reaches a solution.
Since they say that this is from an unreleased research version of Claude:
I wonder if at some point Anthropic and OpenAI will start delaying the release of their models intentionally so they can reap the benefits from the models in, for example, mathematics, medicine, physics, and other fields.
Just as an example, imagine if your model were capable of proving P = NP, or if your model could cure diseases. Would you release it for free, or would you try to make sure those benefits go directly to your company? From these companies' standpoint, I think they would choose the latter.
When the time comes where one of these model makes an improvement in my niche, I hope to see some pattern in the type of discoveries. Yes, they are all roughly "combine two things no one thought of combining" but I mean at a more granular deeper level.
I want to dive into the "data" and then see if it's possible to distill this skill into small models that are "benchmaxxed" for this type of work, maybe in limited domains, similar to small models being benchmaxxed(I don't mean this in a bad way) for coding these days.
This area seems to be moving so quickly. I wonder if it'll be worthwhile to start building a list of formal math problems whose solutions, or partial solutions, would help my subfield. (Though I work in the physical sciences, mostly with the messy, real world implementation problems which are likely difficult to formalize or directly connect).
Then, whenever a new SOTA model drops, throw it at the list to see if we get "free" research progress.
Several released versions and months ago, I asked Claude to figure out the MC (multiplicative complexity) of Conway's Game of Life and it pretty quickly arrived at k=7, despite no previous literature on the topic. Let it run it through SAT solvers for a week and sure enough. It claimed, in the process, to have made great headway in improving boolean circuits beyond the implemented SOTA (in large part no doubt by actually implemented non-implemented but published SOTA).
And that was just the first time I really tried out Claude's mathematical prowess. I've been working with boolean circuits, FHE, and lean proofs ever since.
prompt engineering 2025: you are an expert programmer, use industry best practices, test driven development and use modularity and abstraction to anticipate future features, …
It seems like everything will follow this pattern:
1. AI is dismissed because an expert in a particular field finds an outdated model's outputs sub-par
2. New model, released or unreleased, makes a major stride in that field
3. Expert either recants and becomes AI-pilled, or claims it is just an artifact of the broad search space available to AI, and "no new knowledge was created".
> Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”). This seems to have helped Claude overcome some initial skepticism that it could make meaningful progress.
I remain delighted at how absurd our current timeline has become.
"Any sufficiently advanced technology is indistinguishable from magic." -- Arthur C. Clarke's Third Law
"Sometimes, magic is just someone spending more time on something than anyone else might reasonably expect." -- Teller (of Penn & Teller)
"Sometimes, any sufficiently advanced technology is just spending more time on something than anyone else might reasonably expect." -- an LLM's original thought, probably
That's really what's going on here, I think. LLMs aren't that good at math or have deep mathematical insight. They just automate "try shit until something sticks" at scale, which, as it turns out, is a pretty decent way to turn out a bunch of math papers.
LLMs have enabled brute forcing of math in a way that classical computing somehow has not, which is fascinating in and of itself.
I don't if it's better or worse than the time I had Claude say it couldn't do something so I prompted it "yes you can", and it obliged. Absurd either way!
I wonder if they simply abliterate Claude skepticism would break its reasoning capabilities, if not I don't understand why they would just keep prompt it instead of creating an abliterated model for the task.
I wonder if Jarred (the Bun guy) just got lucky here, or if he made progress before all the actual mathematicians at Anthropic because they aren't prompting Claude as ambitiously as he is.
I wonder why we have yet to see more systematic exploration of Math.
Anthropic describes that Claude identified a set of possibilities and then explored them using sub-agents. The human saying "I believe in you" could literally just be something along lines of a harness with a /goal loop.
We all identify this as absurd because... it's so lacking in rigor despite making major progress. What if we just applied a little more rigor? Ask the model to identify many possibilities, encode them, fan it out to other agents, loop them all, collect the results, etc. Then what happens? It feels like we have weak AGI and a decent system for discovery could transform it into weak ASI. That in turn could yield strong AGI and so on. I suppose that's what the Discovery Loop announcement was all about.
This is just one possible way of formalizing it and it depends on other definitions, wherein you also need to understand the assumptions they make, etc.
Answering the question of “whether proving this theorem in Lean proves the Pythagorean theorem” thus requires expert judgement as well as domain knowledge of Lean’s libraries.
So if the AI says “this theorem is true, here is the proof in Lean” it’s still possible that it’s not correct, even if the Lean code compiles. The result will still be in question until a human expert reviews it.
> Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”). This seems to have helped Claude overcome some initial skepticism that it could make meaningful progress.
while :; do echo "You can do it!"; done | claude -c
I had a similar experience a few months ago. Tried to see how much I could replicate an OpenClaw with Claude. Asked it what the weather is. "I don't know, I'm just a programmer." Added "You can do anything, believe in yourself." to the system prompt and suddenly it was able to tell me the weather...
> Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”). This seems to have helped Claude overcome some initial skepticism that it could make meaningful progress.
We went from AI being human sycophants to humans becoming AI sycophants.
46 comments
[ 1.4 ms ] story [ 3.1 ms ] threadThe world we live in is beyond parody.
"Claude found that combining the results from Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh with the work of Bombieri provides a way to surpass the previous state-of-the-art lower bound proportion of 41.6%, increasing it to 67.2%."
The transcripts, papers, and Claude's explanation are an interesting and a better read than this article, and this is exactly what Anthropic should continue to do and it helps other researchers outside the company as well.
[0] https://www-cdn.anthropic.com/564f962e60643842f5fcb4a17c9dbc...[1] https://github.com/anthropics/zeta-23-lean
[2] https://www-cdn.anthropic.com/23455459f8832d06bb175cc0f88d01...
[3] https://www-cdn.anthropic.com/d7f3ecf1d01392d887f8bc974ca187...
[4] https://www-cdn.anthropic.com/8a0d1add3c637b858a9a181e98c40e...
He should consider using the PUA plugin. It detects when the AI is trying to give up on a problem and automatically harasses it with "encouragement" until it reaches a solution.
https://github.com/tanweai/pua
I'm not sure what a good mark would be, but considering this result lets put it at 2027-08-10 (One year from today).
Why hide the names of the people who wrote the second paper? To discourage people from citing it instead of the LLM-derived paper?
I want to dive into the "data" and then see if it's possible to distill this skill into small models that are "benchmaxxed" for this type of work, maybe in limited domains, similar to small models being benchmaxxed(I don't mean this in a bad way) for coding these days.
Then, whenever a new SOTA model drops, throw it at the list to see if we get "free" research progress.
And that was just the first time I really tried out Claude's mathematical prowess. I've been working with boolean circuits, FHE, and lean proofs ever since.
So none of this suprises me.
prompt engineering 2026: i believe in you
1. AI is dismissed because an expert in a particular field finds an outdated model's outputs sub-par
2. New model, released or unreleased, makes a major stride in that field
3. Expert either recants and becomes AI-pilled, or claims it is just an artifact of the broad search space available to AI, and "no new knowledge was created".
the novelty remains a structural problem with how they work and what the weights try to predict
I remain delighted at how absurd our current timeline has become.
"Sometimes, magic is just someone spending more time on something than anyone else might reasonably expect." -- Teller (of Penn & Teller)
"Sometimes, any sufficiently advanced technology is just spending more time on something than anyone else might reasonably expect." -- an LLM's original thought, probably
LLMs have enabled brute forcing of math in a way that classical computing somehow has not, which is fascinating in and of itself.
Anthropic describes that Claude identified a set of possibilities and then explored them using sub-agents. The human saying "I believe in you" could literally just be something along lines of a harness with a /goal loop.
We all identify this as absurd because... it's so lacking in rigor despite making major progress. What if we just applied a little more rigor? Ask the model to identify many possibilities, encode them, fan it out to other agents, loop them all, collect the results, etc. Then what happens? It feels like we have weak AGI and a decent system for discovery could transform it into weak ASI. That in turn could yield strong AGI and so on. I suppose that's what the Discovery Loop announcement was all about.
It just shifts the work from
> reading the (natural language) proof and confirming it has no errors
to
> reading the Lean code and confirming it correctly encodes the theorem
For example here is a statement of the Pythagorean theorem in Lean:
theorem EuclideanGeometry.dist_sq_eq_dist_sq_add_dist_sq_iff_angle_eq_pi_div_two {V : Type u_1} {P : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MetricSpace P] [NormedAddTorsor V P] (p₁ p₂ p₃ : P) : dist p₁ p₃ * dist p₁ p₃ = dist p₁ p₂ * dist p₁ p₂ + dist p₃ p₂ * dist p₃ p₂ <-> angle p₁ p₂ p₃ = Real.pi / 2
This is just one possible way of formalizing it and it depends on other definitions, wherein you also need to understand the assumptions they make, etc.
Answering the question of “whether proving this theorem in Lean proves the Pythagorean theorem” thus requires expert judgement as well as domain knowledge of Lean’s libraries.
So if the AI says “this theorem is true, here is the proof in Lean” it’s still possible that it’s not correct, even if the Lean code compiles. The result will still be in question until a human expert reviews it.
> 60 Claude subagents
That's hilarious. Maybe I do need to glaze the LLM a bit more in the AGENTS.md
We went from AI being human sycophants to humans becoming AI sycophants.