Disclaimer: I only scanned the article quickly; I might be re-stating something already in the article.

We have just got some very strong evidence about the way in which LLM-based systems solve mathematical problems and this evidence supports what many have already suspected including myself.

Here's what I'm talking about. On 10 August Anthropic released an article [1] claiming that:

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%.

The same article describes the methodology followed by Anthropic's employee, Jarred Sumner, who prompted Claude, as follows:

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.

Jarred got Claude to throw stuff at the wall repeatedly (650 initial "ideas" plus unspecified more by "60 Claude subagents" ... running "2400 shell commands" and "hundreds of Python scripts") and then kept whatever happened to stick. In this case, by happy accident, what stuck was an improved bound of the zeroes of the zeta function etc.

This is how every single mathematical result reported by an AI company has ever been generated. They throw stuff at the wall and take whatever happens to stick.

This approach works. Not only it works, it is, in principle, a universal problem solver. "Millions of monkeys on typewriters" will eventually produce a proof of the Riemann hypothesis; or a disproof of it.

The key point being "eventually". Is this a way to do mathematics research? Can that replace mathematicians?

In AI, this method is well-known as the "generate-and-test" method. It is ancient, basal to AI if I may be so bold. It first appeared to my knowledge in the Logic Theorist, the proof-finding program that Simon and Newell presented in the 1956 Dartmouth convention that named "Artificial Intelligence", to such luminaries of AI and CS as John McCarthy (the real "godfather of AI" who named the field), Marvin Minsky, Claude Shannon and others.

We've had the ability to brute-force all of mathematics "eventually", given "enough" compute for nearing a century now. Why haven't we solved all of mathematics? Are LLMs really so special that they can out-brute force search every previous brute force searcher?

Well, you tell me, HN. I say: no.

___________

[1] https://www.anthropic.com/research/riemann-zeta

▲

ozgung an hour ago | parent | next [-]

This is not "brute-force" though. It's an iterative search algorithm. You learn things at each iteration. You also don't search blindly. You use "something" (heuristics, experience, intuition) to come up with "ideas" at each iteration. You don't try 650 random programs. You try 650 different ideas each learning from the results of previous trials.

Yes this is the "Universal Problem Solving Algorithm". It's actually the same algorithm used by Evolution.

Also this algorithm is vastly different than "monkeys with typewriters". Monkeys don't learn or evolve their writing. There is no memory, no constraints, no learning-curve. At each iteration they freshly sample from a Uniform Distribution. Expected time for a solution is infinitely long.

"The Universal Algorithm" on the other hand is incredibly fast. Humans (designers, researchers) also use the same algorithm but they are much slower to iterate than computers. Instead of trying 650 different ideas at a single run, we have 100s of researchers each try few different ideas independently.

	▲	famouswaffles 7 minutes ago \| parent [-]
		Thank you. I've never seen 'brute force' and 'type writing monkeys' abused so much than these LLM discussions.

▲

yorwba 3 hours ago | parent | prev [-]

> We've had the ability to brute-force all of mathematics "eventually"

In a way, yes. You can easily write a program that recursively enumerates all provable theorems in some order. But if you want a proof of a specific theorem, how do you find it in the list? You need to encode the theorem in a formal syntax first, and since mathematics is built on towers of definitions referencing other definitions, that alone is a significant amount of work before you can even write down what you want to prove.

If you want brute force alone, specialized solvers are likely a better choice than LLMs, but what LLMs add to the table is the ability to work with mathematics as it has already been written down. And even though they're bad at brute-forcing, they're still better at it than humans.

An example of a good division of labor is the SAT Attack on Tarski's High School Algebra Problem https://arxiv.org/abs/2608.08421 where they construct a formula with O(n⁴) variables and O(n⁶) clauses and use a SAT solver to show that it is unsatisfiable for n ≤ 11 but satisfiable for n = 12. Then they use an LLM to help them write a Lean proof that the SAT solver input is equivalent to the human-readable description of what they wanted to prove.

	▲	YeGoblynQueenne 3 hours ago \| parent [-]
		>> In a way, yes. You can easily write a program that recursively enumerates all provable theorems in some order. But if you want a proof of a specific theorem, how do you find it in the list? You need to encode the theorem in a formal syntax first, and since mathematics is built on towers of definitions referencing other definitions, that alone is a significant amount of work before you can even write down what you want to prove. My current framing of this is that the advantage of LLMs lies in their ability to generate the text of a proof without having derived each of its steps in order, like a theorem prover (automated or not) would have to. There's nothing forcing an LLM to derive conclusions from premises (or indeed making it at all capable to do that). They don't have to understand what the proof they generate means, or to be able to tell whether it's true. In fact, they can't do either. But that's fine as long as it's possible to check the proof with an external verifier. So most LLM-based proofs use the LLM as the generator and an external verifier as the tester, either a solver like Lean or a mathematician. That's the best of both worlds as far as generate-and-test goes. A powerful generator tied to a powerful tester. EDIT: yeah, like this: >> An example of a good division of labor is the SAT Attack on Tarski's High School Algebra Problem https://arxiv.org/abs/2608.08421 where they construct a formula with O(n⁴) variables and O(n⁶) clauses and use a SAT solver to show that it is unsatisfiable for n ≤ 11 but satisfiable for n = 12. Then they use an LLM to help them write a Lean proof that the SAT solver input is equivalent to the human-readable description of what they wanted to prove. I'm not disergarding the fact that LLMs don't generate text completely at random. They generate likely text. I suspect that can make it more likely to generate the text of some correct proofs. But I have no idea how likely that "more likely" is or what proofs are those.