OpenAI makes breakthrough on 80-year-old maths problem
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https://www.theguardian.com/technology/2026/may/21/openai-paul-erdos-maths-problem-breakthrough
One of my classmates from undergrad, as well as one of my TAs, were on the team at OpenAI that did this (I am not currently in touch with either of them).
There's still a lot of uncertainty as to how far this is going to go, but it's clear that LLMs are a transformative technology and I feel that this is something that the free software movement needs to take more seriously. From what I've seen, Richard Stallman seems oblivious to how good the latest models have gotten.
Even if AI doesn't replace humans completely, it seems inevitable that it will be a requirement to remain competitive in any kind of knowledge work. Fields such as pure mathematics that in the past only required pencil and paper will now be dependent on proprietary models from OpenAI, Anthropic, etc. all run on NVIDIA GPUs sitting in data centers. And we may see a future in which the most successful people will be whoever can afford to pay for the most advanced models.
Here is a related slashdot article:
Three reasons why this is not an actual breakthrough, from the article itself:
1. The problem was not solved, merely a different model of patterning that had not been previously used was shown to work better
2. The advance that was made were done with significant human input and manipulation
3. The advance that was made was due to the AI following paths that no human would have bothered with because they look unworkable. That's one of the actual values of AI - it can quickly do actions that humans wouldn't bother with because they don't want to waste months or years on unprofitable work.
So, it did not solve the problem, it made its advance with significant direct human input, and it did it by doing the grunt work on lines of inquiry that had already previously been considered but set aside because they looked like a waste of time. The same outcome could have been achieved by forcing 100 grad students to follow lines of inquiry that had previously been abandoned as worthless.
This isn't greater intelligence. This is a greater willingness to do complex tasks that humans had already refused to do themselves.
I am a translator!
Thanks for the nice explanations. I am not sure about the 100 grad students: they could have refused to do the work or pretended to have done it while minimizing their efforts and having provided something incomplete. But that changes nothing to your point :)
1. The problem was not solved, merely a different model of patterning that had not been previously used was shown to work better
The solved problem is exactly the one Erdős asked 80 years ago, i.e., https://www.erdosproblems.com/90
As written behind that link, the "model at OpenAI constructed (for infinitely many n) a set P of n points in R² such that the number of unit distance pairs in P is at least n^(1+c), where c > 0 is an absolute constant". Maybe what confused you is that, after being shown that proof, Will Sawin from the Princeton showed that c = 0.014 works.
2. The advance that was made were done with significant human input and manipulation
That is not al all what https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf (the actual source, the mathematical article) says in its "Statement on AI Use" paragraph:
This problem was solved in a completely automated fashion. Our internal model was given an AI-written statement of the problem, and its output was sent to an AI grading pipeline, which indicated high confidence that the solution was correct. It was only after this point that internal human researchers and mathematicians began to examine the solution carefully. After preliminary AI-assisted verification and rewriting, a draft was sent to external mathematicians, including several number theory experts, who confirmed the proof’s correctness (and have already simplified and strengthened the argument).
The "strengthened" part certainly refers to the computation of a value for c, but that could not been achieved without the proof that such a constant exist.
Right after the paragraph I have just quoted comes the "Prompt", which is exactly https://www.erdosproblems.com/90 stated in a more formal way. The "Final Response from Internal Model" disproves Erdős' conjecture on the unit distance problem.
>"disproves Erdős' conjecture"
It disproves his conjecture, but it is not a final solution to the problem, according to the Guardian article:
>"the broader problem remains unsolved because the AI did not come up with a new answer for how fast the pairs of dots rise, but merely showed that the limit Erdős proposed was too low"
It only had its "breakthrough" by following seemingly unprofitable paths - again, forcing 100 (or 1,000 or 1,000,000) grad students to do the same would likely end up in the same result:
>"Bloom wrote that the AI system had attained its results by “persevering down paths that a human may have dismissed as not worth their time to explore”."
It disproves his conjecture, but it is not a final solution to the problem
The asked problem was to prove or disprove Paul Erdős' conjecture. Now that it is disproved, that we know that the maximum number of pairs of points of R² at a unit distance "grows strictly faster than what was thought for decades", one can ask: "how much faster?". That is a new problem. Will Sawin "only" refined the proof that it "grows strictly faster than what was thought for decades" to find that, with n points, more than n^1.014 pairs can be at a unit distance, for infinitely many values of n. Maybe the exponent will be further increased.
OK, so it disproved the conjecture, which was the problem on the table. Good enough. And I read a comment on the OpenAI page[1] by a mathematician saying that this was a novel way to apply number theoretic constructions to an open problem in discrete geometry, and this may lead to other algebraic number theorists tackling additional open problems in discrete geometry. I don't know what either number theoretic constructions or discrete geometry are, although I'm sure you do quite well, but it's impressive sounding stuff. So I'll grant that. Good on you, ChatGPT.
[1] https://openai.com/index/model-disproves-discrete-geometry-conjecture/
[I am answering you in two parts because the Trisquel forum is today limiting my text input for some reason.]
I also disagree that there was not significant human input and manipulation. The paper by OpenAI discusses an "internal model" of their AI which was given the prompt - we would need to know more about this internal model, but the OpenAI paper curiously does not give details on this "internal model". The Guardian article notes that "OpenAI ... is preparing to float on the US stock market", and that OpenAI "has been tripped up before by its attempts to solve Erdős’s problems, having hailed a supposed breakthrough last year that was in fact based on already existing literature absorbed by the model".
Clearly a lot of work went into this particular model that is not being discussed in the OpenAI paper, that somehow came up with a workable way to disprove Erdős' earlier conjecture through massive grunt work, at a time precisely prior to OpenAI's public stock offering, which gives an indication of how important this work was from a marketing perspective.
Clearly a lot of work went into this particular model
In the model. Sure. Not in disproving Paul Erdős' conjecture:
This problem was solved in a completely automated fashion.
https://cdn.openai.com/pdf/74c24085-19b0-4534-9c90-465b8e29ad73/unit-distance-proof.pdf
But what's in the "internal model"? I'll bet that more people than me will be asking that question in the coming days, weeks and months. It does seem like an awfully opportune time for OpenAI to get an impressive sounding bit of press, right before taking the company public and potentially turning a bunch of executives into instant mega-billionaires.
"Replace humans completely"? Is the software going to install and execute itself, create its own goals, update its own code, kill off humans and populate the planet with descendant-LLMs? It's so weird when people talk about this technology as if it's all-powerful.
Competitiveness in the field of pure mathematics is largely about getting university jobs and tenure. A lot of it has no commercial value, so I don't see why LLM companies would pay much attention to it.
It's already the case that the most successful people in the world can afford the most. That won't be new.
Why should people in the free software movement take this "more seriously"?
>"Fields such as pure mathematics that in the past only required pencil and paper will now be dependent on proprietary models from OpenAI, Anthropic, etc. all run on NVIDIA GPUs sitting in data centers."
I'm pretty sure that mathematicians have been using supercomputers for decades to tackle some of the hardest problems, so it's no surprise at all that the most powerful AI models would be used as tools in this field. Magic Banana can probably tell me if I'm wrong on the use of supercomputers, but from what I'm reading through a couple of quick online searches, supercomputers have been very much in use to crack tough mathematical problems for a long time now.