OpenAI's Astra Solved Decades-Old Math Problems For $2,000 (forbes.com) 166
An anonymous reader quotes a report from Forbes: The cost of producing new results on ten longstanding mathematical problems just fell to $2,000, according to OpenAI, which says its Astra model generated machine-checkable proofs for questions that had resisted human progress for decades. OpenAI published the work on August 1 and used it to give its next major model family a name: Astra. The results run across group theory, high-dimensional geometry, coding theory, quantum complexity, lattice cryptography and extremal combinatorics. They arrived as a 249-page manuscript collection and, alongside it, something the field has not seen attached to an AI claim before at this scale: a machine-checkable certificate for every single result.
The problems were not textbook exercises dressed up as discoveries. Each had been open for at least ten years, most of them far longer, and several sit at the center of their subfields:
- A construction establishing the existence of non-sofic groups, a question that has occupied group theorists for years.
- A disproof of Connes's rigidity conjecture, a long-standing problem in the theory of von Neumann algebras.
- An improvement to the general upper bound on sphere-packing density in high dimensions, a bound that had stood since 1978.
- Three problems come from the catalogue of open questions left behind by Paul Erdos. The announcement follows another result from May, when OpenAI used a similar reasoning model to produce an original mathematical proof disproving a famous unsolved conjecture in geometry, which was first posed by Paul Erdos in 1946.
The problems were not textbook exercises dressed up as discoveries. Each had been open for at least ten years, most of them far longer, and several sit at the center of their subfields:
- A construction establishing the existence of non-sofic groups, a question that has occupied group theorists for years.
- A disproof of Connes's rigidity conjecture, a long-standing problem in the theory of von Neumann algebras.
- An improvement to the general upper bound on sphere-packing density in high dimensions, a bound that had stood since 1978.
- Three problems come from the catalogue of open questions left behind by Paul Erdos. The announcement follows another result from May, when OpenAI used a similar reasoning model to produce an original mathematical proof disproving a famous unsolved conjecture in geometry, which was first posed by Paul Erdos in 1946.
Hold on to your papers... (Score:5, Interesting)
What a time to be alive!
When people tell me that AI is just a "plagiarism machine", I point to stuff like this and the AlphaFold database... Advances in science, mathematics, and research that have been out of our grasp for decades... Solved in novel ways.
~D
Re: Hold on to your papers... (Score:2)
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...much like regular human mathematicians?
I think any human solving the 'easiest' as-yet-unsolved math problem would expect to get credit for it. I mean, it's unsolved - it's not like people didn't try!
Re: Hold on to your papers... (Score:5, Interesting)
You'd think there'll be people solving the easy unsolved problems, but the fact is humans go for glamor. I'll give you a specific example, in my company .. which does biotech .. I mention in passing to every intern the following: "there are about 20,000 genes in the human genome. For many of them, nobody has investigated what it does. You can find numerous lists on this (reference: https://unknome.mrc-lmb.cam.ac... [cam.ac.uk] ) .. for example a famous gene like TP53 (which causes cancer when fucked) has tens of thousands of publications made about it (though ironically there's likely still stuff it does that we don't know). Meanwhile as shown in the reference URL there are hundreds of genes that have had no papers published on them. Like zero, no publications .. given the resources available at this company we will let you investigate what those genes do and possibly find out something cool, but, you have to do it on your own time."
Guess how many interns have bothered to do that in the past decade? Zero.
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I can't imagine it didn't cost more than $2,000 to solve it.
Yes, I can't imagine TFA didn't go in to all that. Oh wait... it did.
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There are plenty of math problems that have a price on their head, like a million dollar to solve this or that.
If you can do it for $2000, you have $998,000 profit. Yeah, I know this is a mind boggelling big number. That is why people are bad in math. Hint: you can buy a small house and Tesla for it, or spent it in one night in Bangkok.
Famous Unsolved Math Prizes
* Riemann Hypothesis: Worth $1,000,000 through the Clay Mathematics Institute for proving properties of prime numbers.
* P vs. NP Problem: Worth $1,000,000 for determining if every problem whose solution can be quickly verified can also be quickly solved.
* Navier-Stokes Equation: Worth $1,000,000 for proving mathematical properties of fluid motion.
* Birch and Swinnerton-Dyer Conjecture: Worth $1,000,000 regarding elliptic curves and number theory.
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Yeah, and I bet many people already tried getting AI to solve Milenniumproblems. But there is a reason why so many of them are still unsolved. Erdös Problems are not expected to be that hard (even though one doesn't really know when a longstanding problem seems to resist a solution if it may indeed be much harder than thought).
I won't be surprised when we have an AI solving a Millenniumproblem in a few years, but I wouldn't bet on it.
Re: Hold on to your papers... (Score:2)
If you can do it for $2000, you have $998,000 profit.
Yeah, but you have to regard this as a probabilistic effort: when you spend the $2,000, do you have a greater than 0.2% chance of solving the problem? If not, you're throwing your money away. And chances are, you donâ(TM)t know your odds when you start, so then things become really speculative.
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Actually you've got a pretty good guess that the problems are quire hard, and a reasonable chance that they are indeterminate (unless they involve a finite subsection of math).
Re: Hold on to your papers... (Score:2)
Yes, it's ten more problems solved. Now they can be applied towards solving more problems.
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$2000 is half the truth, but still not completely misleading.
When it comes to the math alone, you should say: model training + 2000 and you are in the range of millions instead of thousands.
But the message they want to get across here (it's of course also marketing) is, that given they provide the model now, you need to invest only $2000 in their AI and can solve such a problem. When you are now dedicated to poke to AI to solve the next problem, you may have a chance to do it for just a few thousand dollars
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OTOH, they didn't use up the AI in solving the problem, so it's still there to be used for something else. So you can't really count all the cost of building an training it against a problem that it solved as a test of its capabilities.
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Yes, that's why I said it is not completely misleading. That' the same thing like people were saying with the deepseek paper "But they didn't pay their employees with that money!" The $5 million for DeepSeek or the $2000 for the proof here are the values that you need to invest to reproduce what they did, when you already have the required infrastructure and already paid your employees.
On the other hand, if some media person asks you not knowing the ball park of the infrastructure and personal costs, you ma
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If your up-front investment is $1M, each unit of output sells for $2000, and costs $1900 in operational expenses - Your profit is $100 per unit. Once you sell 10000 units, you've broken even and every additional sale is $100 in pure profit. Presuming you expect to sell significantly more than 10000 units, the initial cost is literally irrelevant over the long te
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. for example a famous gene like TP53 (which causes cancer when fucked) has tens of thousands of publications made about it (though ironically there's likely still stuff it does that we don't know).
That's not really glamor, curing cancer is an important practical problem. It's a good thing that money and attention are directed towards that.
Re: Hold on to your papers... (Score:5, Insightful)
I can think of a couple reasons nobody's taken your company up on that.
- they're interns (ie- short term). I worked tech full-time (and did some good stuff there), but the ramp-up on the company's project and in-house processes took serious time - it was a full year before I could find my butt with both hands, so to speak
- and this research would be on their own time but with company resources. Meaning, discoveries would be owned by the company, and (depending on the company) the intern might not even get credit, or even paid for time spent finding them
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Is an "intern" not something like an unpaid worker, who does an "internship"?
What do you expect? He does the work you want, don't pay him, and on his own time he can use your research facilities, and later you steal his work, claiming: in your contract you agreed to hand over all IP you discover why you do unpaid training work for us?
Cry harder man ...
Hire me as computer scientist, pay me for my 7.5 hours a day, and I happily add 3h or so on top of it fooling around with genes ...
But as an intern? No way ..
Re: Hold on to your papers... (Score:5, Insightful)
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Yeah, right, except some of the solved problems are 70-80 years old.
You're excellent at moving the goalposts.
The problem is you don't follow AI companies closely and both Anthropic and OpenAI
Re: Hold on to your papers... (Score:2)
"The problem is you don't follow AI companies closely and both Anthropic and OpenAI have announced that their latest models have optimized their own code to run faster and more efficiently, which means that LLMs have already allowed them to make more money"
False. The correct term is lose less money. Zero of these AI companies are in the black.
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False. The correct term is lose less money. Zero of these AI companies are in the black.
No ne cares about those companies. Unless you are an investor.
Important are companies that use AI, like Zeiss, Munich Re, or BioNtech.
And they most certainly do not "lose money" by getting 5 times more productive.
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No, the subject of discussion are companies that use LLMs, like in this case, using them for solving math problems.
lolol (Score:2)
I made someone big mad by suggesting we could discuss topics related to the primary subject as we were already doing in this thread. Rent free!
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So, your new goalpost for AGI/ASI is being able to "make money"? Thank you for the clarification! By the way, have you ever heard the term "fundamental research"? Many if not the vast majority of things that you now take for granted have their roots in it.
Secondly, many companies that use OpenAI and Anthropic AI are profitable.
It's all quite confusing to me but whatever.
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False. The correct term is lose less money. Zero of these AI companies are in the black.
Anthropic made a profit second quarter this year https://aitoolsrecap.com/Blog/anthropic-first-profit-2026-revenue-breakdown [aitoolsrecap.com]. We'll see if they make a profit in Q3, but at least right now, that statement doesn't look accurate. Now, there's reasonable concern about how real this profit is, given the weird amount of self-dealing and entanglement between the different companies. I also wouldn't be surprised if none of the major data center companies are taking chip depreciation into account to the extent they
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And here we're talking about profits, despite their huge scaleup expenditure. Theirs and OpenAI's margins on serving inference are ~40%.
It's almost impossible for a company undergoing a massive scaleup rate to turn a profit, let alone growing 10X annualized for 3 1/2 years running, but Anthropic pulled it off.
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> I wonder how many problems they tried before they solved one successfully.
Was gonna mod Insightful, but the replies aren't getting it.
"Solving this problem cost $2000."
That may be true.
"We tried to solve 50 problems at $2000 each and one was successfully solved."
This can also be true which means:
"We spent $100K and we solved one open problem."
which is fiscally, but not akshually, the same as:
"We spent $100K to solve one problem."
If Astra is "all that", OpenAI should have a pricing tier, "no charge unle
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There is reason for some skepticism because some previous claims have turned out that the problems were in fact not unsolved. The LLM just found the existing solutions and used those:
https://techcrunch.com/2026/05... [techcrunch.com]
which is not useless, but not the same thing either. So I would hold on for a bit before accepting the claims.
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There is a HUGE difference between using AI as a tool, and using AI just to replace people for jobs when AI isn't robust enough to handle many jobs well. I wouldn't even trust AI to replace a level 1 customer service rep in many areas.
I'm not a math guy, but... (Score:3)
Re: I'm not a math guy, but... (Score:2)
Stupid computer.
Re: I'm not a math guy, but... (Score:2)
"what is the mathematical probability that the AI investments of the 2020s will turn out to be the largest financial error in the history of the human species? Answer with just an integer."
Its response was zero. Perfect. Absolute certainty in a mathematical impossibility. About a year later it updated it to something in the 40s. I just asked it again and it's now in the 60s.
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Sir, this is a Wendy's!
Don't ask AI questions that are meant for a glass ball. The things are good at thinking, not good at predicting the future.
Re: I'm not a math guy, but... (Score:3)
Re: I'm not a math guy, but... (Score:2)
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Unlikely.
as 1.0 and 2.0 are simply two integers, 1 and 2, which add up to 3, in any floating point implementation I am aware about.
A 64 bit (double precision) floating point number, holds 2^52 perfectly fine integers.
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They are not.
Proof: https://0.30000000000000004.co... [30000000000000004.com]
Re: I'm not a math guy, but... (Score:2)
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Saying 1,0 *is* an integer is false. It converts to an integer. Whether it equals an integer depends on how you represent floats...though in every implemented system I'm aware of 1.0 does convert to 1. However saying 1.0 is within epsilon of 1 is true for every floating point system using a normal meaning of epsilon.
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That's a programmer perspective. For a mathematician it is an integer, just as 1+0i is. It is also a rational, a natural number and a complex number and could be a representation of a vector in R^1. In math integer is a concept and not a data type.
For a computer scientist it is also an integer but they would probably prefer using the simplest representation, which is 1 instead of 1.0. And for a compiler it is some decimal (usually floating point) type and a programmer knows that.
What is unfortunately ignore
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1.0 is an integer.
Way to go announcing to the world you've never programmed anything before, and in the first 4 words of a very long post.
1 is an integer (or maybe a float, depends on how you define it)
1.0 is not an integer, attempting to define it as an integer would result in conversion, truncation, or a compile error depending on what language you are using.
You see: even dumb Angelo managed to adapt to your silly "decimal point" problem, but you can not use it yourself properly. Strange, isn't it?
The only thing you've done is demonstrated you don't know how computers or programming languages deal with a decimal point: Hint, it's nothing to do with anything rela
"Amazing but oversold" (Score:5, Interesting)
If you're going to repost the entire lede of the story, at least also share the counterpoint: "Gary Marcus, a reliable critic of the field, called the release amazing but vastly oversold. Some specialists expect that once the dust settles, a few of the ten will look genuinely surprising and the rest will be classified as reachable problems that nobody had gotten around to attacking."
And ironically, or maybe unironically, this story reads like it was AI-generated. Phrases like these:
"Three objections are worth taking seriously, and the honest reading concedes all three."
"Grant every objection and the structural point survives intact. Whether the ten were cherry-picked or not, the certificates still verify. A curated result that mechanically checks is a different object from a curated result that does not."
"The cost of producing a hard answer just fell to almost nothing, and the constraint moved to proving the answer is right."
Most humans don't write quite like that.
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The cost of producing a hard answer just fell to almost nothing, and the constraint moved to proving the answer is right.
I don't think this is a change.
For example, Fermat's last theorem was easily produced, and the proving the answer took centuries.
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"Amazing but oversold" might well sum up the whole LLM-era.
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My Math Prof. https://de.wikipedia.org/wiki/... [wikipedia.org] did write exactly like that.
We always assumed he actually was a Germanistic Professor and did Math only as a hobby.
Re: "Amazing but oversold" (Score:2)
Ding ding ding ding! (Score:2)
Astra correctly answers all 10 questions in the $2000 lightning round!
And now on to the final round. To win $32,000 cash, a brand new Oldsmobile Vista Cruiser, and an all-expense paid vacation for two to Puerto Vallarta, all you have to do is answer this single question to the satisfaction of our judges:
The Riemann Hypothesis: True or False?
P = NP? (Score:2)
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Re: P = NP? (Score:2)
Wait, I know this one!
Change doors, oh yeah!
Re: P = NP? (Score:2)
Not an LLM (Score:5, Informative)
The work openAI is doing isn't just saying, "ChatGPT, solve this problem." They have combined their LLM (probably a custom LLM, with a math focus) with existing algorithms, and expert mathematicians working the prompt.
So this appears to be an incremental advance, but it's not that an LLM automatically understands math and can start proving things.
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I'm pretty sure what's actually happened here (that is, I heard through the grapevine, read some of the notes, have seen conversations between established mathematicians who are using it at the moment) that someone has built a natural language interface to Lean.
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The astonishing thing is, that with the one counter example proof exactly this happened.
math guy: Solve $problem
claude: This is a long standing open problem and no solution is known
math guy: Then try to start a partial solution is fine
claude: Here is a partial solution
math guy: continue
claude: Here is most of the solution
math guy: find the fitting numbers
claude: Here is the full proof
Unfortunately the log did not contain the reasoning, so it really looked like a miracle when one cannot see what claude did b
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First of all: it was not Claude.
And the log most certainly contained the reasoning.
That is what "reasoning models" are doing. Producing a chain of thought. And telling you that chain of thoughts.
Stupid AI haters, with your stupid "oh lets ignore anything about AI" and retelling prefabricated "wrong answers" you picked up from another AI hater.
AI is reality. Get used to it or be left behind. Don't cry. Up to you.
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Sorry, but that's not what reasoning models are doing. Don't trust the logs as accurate traces of their thinking.
OTOH, since the output is a proof, you don't need to even consider the log. Just check the proof, and (if properly formulated) there are deterministic methods of doing that.
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Don't explain to me what I already know.
What I said in that post:
1) Person asked claude. Person just poked it "continue solving" without giving feedback or suggestions what to consider. Everyone could have done this (even though most people wouldn't recognize if the final result is correct)
2) Claude found the proof
3) The log doesn't include the reasoning of Claude (as they don't make their reasoning traces available to users)
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That chat I am talking about was Claude.
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> "ChatGPT, solve this problem."
It seems that a lot of people, seemingly smart people, still think this is how anyone uses LLMs.
>expert mathematicians working the prompt.
This isn't a gotcha. This is the standard operating procedure for knowledge workers.
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This isn't a gotcha
Something is wrong with you if you thought that was a gotcha. Fix yourself.
Verification (Score:2)
I am interested in one aspect though. LLM responses are seemingly good if you don't know much about the topic. If you are a topic expert, you immediately catch bullshit.
The question is: do we have enough high level matematicians to verify LLM claims?
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enough high level matematicians to verify LLM claims?
Claims? You mean the proof? The proof is long, so verified by other AI. Not LLMs of course, but deterministic rule-based AI.
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IIUC, the proof is verified by Lean, which is NOT an AI, but is rather a deterministic proof checker, and highly trusted.
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Correct, but Lean is symbolic AI, aka GOFAI (Good Old Fashioned AI). It's doing a lot of the heavy lifting here. LLMs find and compose the spaghetti strands, but the deterministic Lean verifies if the separate steps in the pieces of the proof are valid. My guess is that LLMs cannot perform logical inference, and Lean is there to verify that the candidate proofs actually obey the rules of predicate calculus. So, a neuro-symbolic hybrid, but there's nothing wrong with that.
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"The question is: do we have enough high level matematicians to verify LLM claims?" Well, we'll be generating fewer of them as time goes on. Bean counters will unite and declare that AI can solve math problems so we do not need any mathematicians. The alleged administration never saw a math problem it could understand and has been busy cutting research funding that doesn't directly contribute to producing widgets for its industry friends.
To make matters worse, AI will find its way into schools. So Johnny an
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A classic example of current cultural pessimism.
The present administration is said to be doing something dire - "The alleged administration never saw a math problem it could understand and has been busy cutting research funding that doesn't directly contribute to producing widgets for its industry friends."
But no numbers are shown to support this. Then there are the '"beancounters" - obviously a derogatory term, presumably for accountants, and they are supposed to be saying something completely stupid, an
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By arguing that the Chinese are visibly doing AI better, and that the only reason the US does not imitate them is racism about them.
China is using optical computers for AI training. Only the memory interface is still classic "electric". Soon that will be replaced with PICs (Photonics Integrated Chips), see for example: https://www.teemphotonics.com/ [teemphotonics.com]
Optical chips are 10,000 times faster and use about 3,000 times less energy. On top of that, all the data used for training is cataloged, organized, country wide
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I *think* a lot of those claims are unverified PR announcements by Chinese companies. They *MAY* be correct, but I tend to doubt PR announcements.
So far the evidence seems to indicate that they're still using as many high-level NVidia chips as they can get. I suspect research is being reported as developed systems. If the reports (understood in that light) are accurate, your estimate is correct, but your timeline is off by perhaps 5 years.
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The main property of a proof is that it stands by its own. There is no "I believe in the proof", but you can check every step of it and if every step is valid, the conclusion is valid.
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That's not really true. Consider the ABC proof. It is probably valid, but it uses math that almost nobody who is an English speaker can check. The author is currently translating it into a form that can be read by LEAN (an automated proof-checker). If it passes LEAN, there still won't be many mathematicians who can understand it.
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What does that change? If it is correct and uses advanced math, you need a person understanding advanced math to check it. You can still check for every step "Is this a true statement? Does the next statement follow from that statement?" and if you arrive at the result and each statement was true and implied by the previous statement, the proof holds.
LEAN is a method to verify it formally so people don't need to be experts to trust the proof, but you don't need to be able to translate a proof to LEAN for it
$2000? No way (Score:2)
the AI companies are selling $100 bills for $1
How? (Score:2)
How can half-random-BS-generators allegedly solve maths problems? Or do these specific problems especially lend themselves to being solved with half-random BS?
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Have you considered that these things are better than you thought?
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Because they are not half random BS generators.
They generate a theorem in a "mathematical proof language", like Lean, Rocq/Coq etc. (A sequence of theorems)
And theorem solvers conclude if transforming equations from one step to the next are correct.
https://www.youtube.com/watch?... [youtube.com]
https://www.quantamagazine.org... [quantamagazine.org]
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The same way a blind chicken can find a grain. By trying a lot of different problems. Some will have actual easy to find solutions, but nobody really looked for a long time because the problem was not interesting enough. Then you do not say "AI has solved 10 of 10'000 open math problems it was tried on", you say "AI has solved 10 open math problems".
The whole thing is a lie by misdirection and most people are not smart enough to see what is going on.
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Mathematician here. None of the ten problems under discussion here are remotely problems that "nobody really looked for a long time because the problem was not interesting enough." All ten of these were major problems in their subfields; I was familiar with 8/10 before this, and I've talked with people in the other subfields who have confirmed that the other 2 were also major enough. The sphere packing problem has been a problem which people have been thinking about variants of for literally over a hundred
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Such a shame.. (Score:2)
This new tech is mostly being used to spy on young ladies in their undergarments. I know it's not what Sam Altman would have wanted.
(*wink* - he would have wanted young men in their undergarments, am I too cheeky?)
Automatic verification (Score:3)
Some important background for this is that, around 1900, mathematics got formalized to the point that there is a standard for whether a proof is valid or not that doesn't depend on humans thinking about it, just doing a bit calculation following an algorithm. However, it is such a large calculation that nobody wrote complete formal proofs or verified all of the steps (since they'd be spending huge amounts of time on obvious parts, and not the novel aspects). Then, in the past 20 years, people have made computer systems that work on complete formal proofs while showing you the interesting parts, with the rest in macros, and check these proofs automatically. This means that finding proofs has the form of things like what AlphaZero is good at: something to check if moves are legal and a win condition. However, only a relatively small portion of mathematical research has been put into Lean so far, and that doesn't include the sorts of obscure results that often turn out to be useful in other problems. So the current wave is using LLMs to find results in the literature that might be relevant and producing formal versions of them, game AI to put them together, and deterministic validation to check each of these as it goes along and then people can see at the end that the calculation shows the proof is valid. The only part requiring mathematicians is checking that what the proof proves is what they understood the open question to be.
The Riemann Hypothesis (Score:2)
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When done, please post the solution here [wp.com], thanks!
Interesting Rundown of the State of AI Math (Score:2)
Matt Parker, science/math communicator, put out a nice video on developments of AI-enabled proofs over the past few years: Has an AI discovered new maths? [youtu.be]
It's interesting how AI has relatively quickly gone from a (very good) search engine to being able to string together a chain of reasoning to prove/disprove an open conjecture.
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Not all engines (Score:2)
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So your mistake is thinking they're all alike. (Unless that was sarcasm or other humor.)
Meh (Score:2)
Maybe they should ask their AI for marketing help (Score:2)
Lucky! (Score:2)
Can we now focus on slightly more useful things? World peace? Climate change? Cancer?
Unsolvable problem (Score:2)
Of creating more Funny comments for Slashdot. Isn't there a funny mathematician in the house?
Greatest teacher I ever had was actually a math professor. I only had him for one semester, but one class session was a topology-based magic show and he read a funny section from Life on the Mississippi in another lesson, with some mathematics inserted into Twain's story. Tried to get him for another class, but after I registered they pulled him out and substituted another teacher who could barely talk. (Of course
And how many 1000s of problems did they try? (Score:2)
Mathematicians that can do this stuff are a very limited resource. If they try this on 1000s of open problems (and there are a lot), they are bound to find some that are essentially solvable with a search and that nobody really has worked on in a long time. This is just one more meaningless stunt.