Alvaro Lozano-Robledo
817 posts
Alvaro Lozano-Robledo
@mathandcobb
Mathematics professor (arithmetic geometry), author, associate editor at The Ramanujan Journal, Hagoromo chalk ambassador. Views expressed are my own.
Katılım Ocak 2026
97 Takip Edilen1.9K Takipçiler
@mathandcobb I remember when I explained Hausdorff spaces to my dog. He understood it, I think
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If you can't explain it to a puppy, then you don't really understand it.
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Alvaro Lozano-Robledo retweetledi
asked gpt-5.5 Pro to solve Erdos #1196. 5.4 Pro actually proved it, but 5.5 got lazy and gave up on the proof halfway, instead passing off the open problem as a settled Erdos theorem, citing a real but irrelevant paper. the leaked CoT shows the exact moment it decided to bullshit
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Alvaro Lozano-Robledo retweetledi
In collaboration with @mathandcobb we have added an "Erdos' Unit-Distance Conjecture" design! :D
math-gear.myspreadshop.com
Alvaro Lozano-Robledo@mathandcobb
When Nature reached out to use the graph I created (using GPT) to illustrate the new (dis)proof of the unit-distance problem, I reached out to Will Sawin to see if he had other suggestions. So here is a slight modification that bounds the complex norm of the points.
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A very interesting question about OpenAI and DeepMind solving Erdős problems is:
How exactly were the winning insights obtained by the AI models?
Did they simply connect dots between different disjoint papers in the training data, or did something truly novel and out-of-distribution occur?
I tend to assume the first explanation - as it aligns better with current publicly known LLM capabilities and limitations.
But it would be fascinating to analyse the latent spaces, training data, chain-of-thought traces, etc. of the models used, in a scientifically credible way.
A similar question: Can we quantify how well-positioned an Erdős problem was for solving, based on preexisting literature and promising ideas in prior studies? How generously positioned were the problems the AIs solved?
And how did previously solved Erdős problems (by human mathematicians) sit at the time of their solving?
Then we could directly compare the creative mathematical ingenuity of elite human mathematicians versus the AI models.
Important research, in my opinion.
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Alvaro Lozano-Robledo retweetledi
The German Mathematician Gerd Faltings being awarded The Abel Prize 2026 by H.R.H. Crown Prince Haakon today in a wonderful ceremony in Oslo. (Photo: Thomas Brun / NTB Kommunikasjon / The Abel Prize). #AbelPrize2026
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Xiao Ma@MaXiao54704
Everyone’s talking about OAI’s new progress on the Erdős unit distance problem, I used standard GPT-5.5 to reproduce the proof ~ 👇 chatgpt.com/share/6a0e9e04… Apparently Erdős offered a $1,000 bounty for it… which means my 5.5 Pro subscription might actually pay for itself ~
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@mathandcobb @AlexanderKalian Who was it that reproduced it?
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@chi_t_williams You just have to pretend the puppy understands.
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@mathandcobb No to be honest I don’t even fully get those things
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Alvaro Lozano-Robledo retweetledi
Can't agree more. After trying out ChatGPT 5.4/5.5 for two months I feel it's far from replacing human researchers, even though it's enabling me to do things not possible before. The effect on how outsiders perceive academic research is far more worrying.
Konstantin Mishchenko@konstmish
My pure math friends are mostly optimistic about AI doing math. What they are really worried about is not getting grants and positions because people will assume mathematicians are no longer needed.
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@mathandcobb This is true actually, I understand nothing on a deep fundamental level except how to sit, stay, and roll over
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@vy_om_man @PhysInHistory Ah I see the difference now :D
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@PhysInHistory Aliens visited Earth billions of years ago, one of them took a shit in the ocean, and that evolved into bacteria and eventually humanity.
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@PhysInHistory They originated from a hallucinated reference in the bibliography of a paper by aliens.
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@AlexanderKalian I had these messages in mind when I replied:
x.com/i/status/20591…
Guilherme O'Tina@guilhermeotina
the worrying version is that theyre one-shot lucky guesses in latent space rather than reproducible reasoning. deepmind solved 9 erdos problems for ~$200 each in inference. at that price you can brute force the search and throw away the 999 failed attempts. the interesting test is whether the model can reconstruct the same proof from a different random seed
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@AlexanderKalian Sorry you are right... I think I was supposed to reply to someone else in the comments who was discussing whether this was some sort of lucky one shot.
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Alvaro Lozano-Robledo retweetledi
I got GPT 5.5 Pro to solve a few problems on my latest NSF grant proposal... basically fed it my notes and had it complete the job. I also got Claude to finish writing a few papers I'd been procrastinating on during the postdoc.
The process is far from perfect and still requires a lot of human intervention. And formalization in lean is still out of reach for the big models, at least without a ton of effort.
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@mathandcobb This is basically what I'm doing with my research now.
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The point of this example is that an LLM may not be about to prove Theorem X for you, but you may have been able to do enough work on problem X to reduce it to a step that an LLM could actually help with. I think that's an exciting possibility.
I wish I was smart enough to prove it by myself... But hey this would be a great second best option.
Alvaro Lozano-Robledo@mathandcobb
I remember that a few years ago I came up with a very weird formulation of Vandiver's conjecture in terms of certain matrices and I could not make heads or tails of how to work with said matrices, so I gave up. But now I have the sneaky suspicion that an LLM could really figure out the pattern in the matrices that I thought could imply Vandiver... Alas, I completely forget how I came up with the matrices in the first place. But I think I remember though to be able to retrace my steps with the help of one of my graduate students and see if there is something that an LLM can see that I couldn't see back then!
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