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Comment by napoleoncomplex | Hacker News Reader

This is the second ChatGPT shared conversation I've seen today that is truly fascinating.

The first one was someone proving another conjecture false by just repeatedly saying "keep going" to ChatGPT: https://x.com/DmitryRybin1/status/2079904005652893709

What a world we live in.

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  > What a world we live in.

It's a really interesting world. You can spam GPT to get novel math results but here I am trying to scroll up to the beginning of the conversation and 5 minutes in I still don't know if I'm near the top yet.

Scroll... wait for render... scroll... wait for render... repeat...

We live in a world where there's so much crazy technology but few people use it to make products better or to improve people's lives. Most people use it to just make more money. It's funny too, because there's a million things we could use that tech for that actually reduce costs. Hell, what would be the economic impact of putting ML systems into streetlights so they properly coordinate. Don't even need LLMs for that, and I'm sure it'd save billions of dollars a year. Just a lack of will. I wonder if this will ever change. Is this how we create the high tech low life future?

(FWIW, no problems if I jump into the app. It's purely a web thing, but my point more illustrative than specific)

And I am still waiting for the Apple Reminders AI to sort Milk in the right category…

Better than a 2026 human, worse than a 2016 machine. You win some, you lose some, I guess...

And meanwhile, I gave up trying to connect to my offices printer months ago. Paper free environment over here, planned or not.

It always is about money. As long as there's money it will always be about it. LLMs are big techs attempt of shifting money from the people further up, by getting rid of jobs. We're all not angry enough because some of yall are so delusional about the whole thing, by thinking we will get some kind of Utopia.

if people use it to make money, does this not mean that they providing some sort of service which is valued and hence it improves lifes?!

No, money and value have been decorrelated for a long time. At least 5 years and arguably since Reagan.

Like it was so correlated back in 1600? Under the Romans? But perhaps you just mean that the increased financialization of the US economy means that the percentage of economic wealth devoted to actions without value is increasing? Sort of a modern trajectory to 1789 with the foppish nobles not realizing how much everyone else can get along without them?

In an ideal world, that would be true, but the actual world is very far from that. As an extreme example, AI is very useful to automate scamming people.

There are a multitude of methods of obtaining money that do not benefit society.

On my mac if you scroll slightly a blob appears on a slider on the right and you slide it to the top to go to the top. I'm not sure about these hidden user interface features that you only find through trial and error.

> few people use it to improve people's lives.

Did Tao do that?

The architecture of our modern technological society lies on top of pure maths. One cannot say “resolving the physicality of the Navier-Stokes equation” will lead to x, y or Zed human improvements, but for hundreds of years it has. It is like Bach making his fugues for beauty and pleasure but occasionally you get chip technology or mathematical ecology or GPS falling out of it.

> just repeatedly saying "keep going" to ChatGPT

For posterity, this indeed works for most problems where an agent might give up. LLMs don't inherently know something is impossible.

The phrase I tend to use in my harder prompts to automate this with a sane loop breaker:

> **REPEAT THIS PROCESS UNTIL CONVERGENCE AND YOU ARE OUT OF OPTIMIZATION IDEAS.** You have permission to keep iterating.

What I am thinking is the way you make it 'keep going' and when you have people of the calibre of Tao doing it I kept thinking how many breakthroughs is he going to cause the LLM to find with his targetted questions :D Amazing that we have the privilege of witnessing a true expert in such a way question the LLM.

Do notice he is quite Socratic, the approach works well for LLMs they love to please so you have to be careful in how you lead them.

this is /goal in claude code/codex. also basically a slightly improved ralph loop

Aka /goal

I’m a bit surprised OpenAI isn’t finding these big results far faster than the product’s user base. With no limits on runtime, access to dev models, custom tuning, and top talent, you’d think there’d be a constantly running internal project with the goal of solving famous math problems. And who knows, perhaps there is, but it would be interesting to compare the rate of success per unit “effort” of the internal mathematics work with that of the user base.

"OpenAI's internal team solves famous maths problem" is technically impressive but dispiriting. Non-experts solving a problem by just throwing resources at it is kind of the worst possible optics for knowledge workers. It's just disempowering.

"Famous mathematician uses ChatGPT to solve famous math problem" is equally technically impressive, but now you're telling those very same knowledge workers "that famous mathematician could've been you". It puts you in the driver's seat, and provides a clear path forward — subscribe, use our product, and reap the rewards.

I don't think they care if it's disempowering. They'd gain more in market share by indicating that this is God than missing out on a few subscriptions from math theory people.

My guess is they don't do this because they don't have time. They're all trying to build a company that makes them generationally wealthy before the music stops.

Why would they want to do it when the people paying them would do the same for them?

Then they should give grants of tokens to famous mathematicians.

EDIT: oh, they do offer grants of $1000 of API credits to researchers https://help.openai.com/en/articles/10139500-researcher-acce...

> Then they should give grants of tokens to famous mathematicians.

Depends. The headline 'some rando solved a famous problem with a consumer grade subscription' might be a really good way to sell these subscriptions.

"Some already famous guy with special access solved a problem" might not be as inspiring.

Note, in posts like the following, the author indicates they are able to get free subscriptions from OpenAI

As an aside, there's this idea in math that when you create a new field you shouldn't solve all the easy problems - you need to entice other people to learn about the field!

I expect there's some element of that here. It's much better for OpenAI and Anthropic if their users are the ones discovering and writing up the results of the AI solving hard math. Look at the high school and college age students who have become ai power users and potentially learned how to use git to contribute ai generated solutions

(Related: I believe Terry has also gotten all of the subscriptions gifted to him)

https://xenaproject.wordpress.com/2026/07/20/human-mathemati...

They do have some of those.

https://openai.com/index/model-disproves-discrete-geometry-c...

How could they?

That's why the free market works, millions of agents in parallel beats any planned economy (by humans)

The free market is mostly the glue that binds lots of tiny and not so tiny planned economies together. (I mean companies and even family budgets.)

As long as that N is high enough for each market, it’s good enough. When you are down to 3 producers, well there are reports of tigers there.

Yes.

As a small wrinkle: the actual observed number of producers can be small, but the market still be competitive. See https://en.wikipedia.org/wiki/Contestable_market for one case: when potential suppliers are waiting on the sidelines.

But each company itself is composed of multiple agents. It's just that companies are a scale of consolidation.

The exact same thing happens in the cortex, 10s of thousands of cortical columns coalesce into a single action.

Confirmation bias. There's likely a wide portion of chats in which "keep going" derails to madness. We then stop saying "keep going" because we notice there is something wrong, and we start another chat. In the end, we largely remember much better the interactions in which "keep going" resulted in something good, and forget about our role in stopping the train when it derails, which is something much harder to do unattended by a human.

You're surprised they didn't eat the tokens to churn on lots of open problems instead of asking others to pay for those tokens? They're in the token business. If they're eating the tokens, it's in support of a marketing effort, not in support of innovation across the frontier of all the other academic disciplines. The collective frontier is way too big for them to just "solve it" without asking society to at least help them break even on such an enormous public good.

It's also much better to distribute the challenge of identifying problems amenable to which prompts

Could they do it? Sure, but to what end? It would make more people hate them and feel even more "take our interesting work." Pitching it as a useful tool just makes more sense on all levels

Had you heard of these “famous unsolved problems” before now? If you aren’t a trained mathematician, you won’t know where to point the LLM towards achievable goals. Having it solve some Erdos problems is pretty different than a Millenium prize, and even some of those might be unknown to the general non-mathematical public. Finding counter examples, unattended, to various algebraic geometry conjectures is amazing but still a ways off from the a definitive disposition of the Riemann hypothesis, P vs NP, the Navier-Stokes problem (although I wouldn’t be shocked if Professor Tao put an LLM towards a counter example for that based on his technique of encoding machines into initial conditions of PDEs).

Although if I were a mathematical quantum physicist I might be trying to prompt one into a % mathematical field formalized quantum field theory. Weirdly this sort of obvious step has not been accomplished in the last 100 years.

I have a pi extension that just runs the same prompt in a loop 25 times. I tried giving it a loop breaker but I found that it'd give up too readily. When the task is actually complete each iteration didn't do a lot of work so it was efficient enough. I suppose another way is to call out to a separate context to check if the task is complete?

Commands like /goal and similar are the more complex version of this; you write a prompt like it was a singular iteration, it runs one iteration, then runs an "evaluator" to determine if the goal has been reached, then runs the prompt again with a little extra to make it go again - and so on. The evaluator is just an LLM with most of the result or context looking at the original goal and the state and answering the question "has the goal been met".

The interesting part of this: while some leading implementations use the same LLM and context for the evaluator, some call out to a different context, some to a tuned LLM and different context; so which is better? many blog-scale benchmarks are calling it a toss-up that is highly dependent on the primary model.

For mathematical research, you can just run until you have a computer checkable Lean proof.

Given that it was formalized correctly, which is far from trivial in many cases

(Of course LLMs can help there, get it right etc, just a caveat that people have to keep in mind)

Agreed. But also formalising the statement of a theorem, or rather understanding the formalisation that the LLM suggested to you, is often a lot easier than understand the whole proof, especially if it's a formal proof.

Sounds overengineered to me

I was surprised at how little improvement they could eek out of tuning it; but it is a non-trivial improvement which is much less likely to wastefully spin if your validation is more expensive than you realise.

Makes sense that its in the major harnesses and not the self-built ones.

I’ll have to try this exact phrasing. I had a lot of trouble with GPT 5.5 more or less completely ignoring similar prompts and instructions and entering a sort of “doom loop” or just consistently trying to prematurely end the chat.

I would love any tips for other folks who have successfully used similar approaches.

You have this problem after paying for Pro?

Maybe it's just me, but does anyone else feel goofy speaking to a computer this way?

Just relax and welcome the nondeterministic world as a non-programmer. Also, forget what you were taught in Theory of Computation, no one needs it anymore. AI will do everything for you. /s

Marketing for huge bucks sounds like this.

You will own nothing and will be happy (that you are still alive). Probably.

>> This is the second ChatGPT shared conversation I've seen today that is truly fascinating.

We recently had some bugs fixed in the geometry kernel of solvespace. Not much conversation, but the analysis from the AI was amazing:

https://github.com/solvespace/solvespace/pull/1729

https://github.com/solvespace/solvespace/pull/1730

https://github.com/solvespace/solvespace/pull/1731

From the Validation section of PR 1730:

"The model family was reconstructed programmatically (parameterized cuboid stack) and swept over 2,304 configurations — extrusion directions, workplane-normal orientations, sketch windings, D's plane/height/depth/extent, including all the exact-coincidence heights. Zero failures with the fix; 576 failing configurations without it. The generator is available on request."

It looks like it wrote a python script to generate test cases in our file format for testing. Just... you know, as a side quest.

> It looks like it wrote a python script to generate test cases in our file format for testing. Just... you know, as a side quest.

On the one hand, agents have done this sort of thing for a year+, if you pushed them to check their work. On the other, I absolutely can feel Fable and Sol have crossed a threshold where they can be trusted far more than before. Huge difference between plans written by Opus or Fable.

Accumulated AI slop can simply be cleaned up by better models. Real cost of technical debt is shrinking due to the the inflationary devaluation of code!

Without any more context, "keep going" seems to be doing a lot of work. The user is placing a lot of faith in the LLM to not make subtle logic mistakes and to take good approaches to each problem. In my experience, even frontier models (such as Fable) are quite capable of getting confused during even simple technical work I've done in the dev ops world. For example:

LLM: This package hasn't made it to production.

ME: are you sure? i see it right here!

LLM: You're right to push back. I inferred that based on weak data. I see now that the package has been deployed!

If the above conversation is typical for me, how could one expect to achieve a sound result by repeatedly prompting an LLM to simply "keep going" in dense mathematical proofs? Perhaps the user in this case had actually checked the LLM's work before issuing the prompt, but I think you see my point anyway.

There may be something(s) about mathematics (proofs) that makes it particularly amenable to LLM reasoning - highly inductive from facts that are explicitly within-context/associative space? Being an unusually well documented discipline in general, with less influence from tacit knowledge or idiosyncratic “it works however the opinionated human made it work +- bugs” processes? Something about simulating even the smallest non-pure-inductive leaps necessarily risking simulating mistakes due to the nature of context “perception”?

There’s also probably a lot less noise from casual internet conversations. I imagine a nontrivial amount of what LLMs know about certain technologies comes directly from forums like reddit where quality of response isn’t guaranteed.

I mean, just the way Tao phrases these inqueries seems to imply a weighting towards an extremely abstract and high level rigorous corpus. In a way, prompt engineering really is the big unlock here.

Maybe! But I suspect you can write a little LLM assisted helper to at least make your prompts sound more like Tao's. (Your ideas won't necessarily be better, but you can probably 'imply a weighting towards an extremely abstract and high level rigorous corpus'.)

Not likely. You run the risk of bleeding into the crank mathematics language pool which is well represented in the training. I mean the prompts may sound good to you and me, but they will have low probability words in respect to the Tao level maths.

I disagree. In fact, I think the field of dev ops gives a clean analogy with mathematical proofs. My point was that my work often requires that I figure out a consistent way to prove to myself what the condition of a system is by asking the right probing questions about it. What I've seen is that even the best LLMs lack a good intuition about what questions they should be asking and instead reach for the quickest and most obvious checks that leave edge cases uncovered. Maybe it is something about the domain of the problem; I don't know. But it makes it hard for me to imagine that an LLM wouldn't make similar errors in other cases, especially when generating mathematical proofs that will soon be too dense for humans to review.

Agreed, often you have to step in and stop it from reasoning itself into dumb directions, but occasionally it goes just like the transcript in question.

“Keep going” is exactly how many mathematicians achieved success in the past. :)

https://x.com/Kittoes0124/status/2079386024753443324

I'll have a third for you soon, here's the obligatory result in a tweet. A detailed post about it is in the works.

It's going to be pretty crazy when GPT-6 comes out, since the rumors are it's quite a bit smarter and wiser than GPT-5.6.

Maybe, but I want to point out that even the lesser models are capable of hunting this stuff down. The most important thing is that you provide a decent path for them to follow.

That’s probably not interesting anymore, but 5.6 wasn’t able to name the conjecture when presented the notation only, but confirmed the proof and when told what it was, agreed it works. Much less psychosis than when given the Jacobian counterexample, at least.

Direct link: https://chatgpt.com/share/6a60b2eb-0b64-83ee-9c76-7931ca1de0...

If I understand correctly the conversation, ChatGPT had to compute for quite some time, meaning a large amount of computations. What kind of resources would be we needed to achieve the same results with a local LLM? Is it even feasible with current open models?

For instance, would it be affordable for a research lab to not rely on OpenAI?

Is this the same as Dinitz Theorem[1] which seems to have been proved in 1994? This is the only result I keep stumbling upon when trying to understand the problem formulation

[1]: https://en.wikipedia.org/wiki/Dinitz_theorem

"it's enough of partial results. let's finish with a complete unconditional counterexample"

"Worked for 88m 24s... >"

"<h1>Complete finite counterexample</h1>"

...

From the prompt:

> You should do a breakthrough

This is just as funny and ridiculous as those "make no mistake" prompts.

Mathematical breakthrough, genius, trending on Artstation.

The day is coming when some *BIG* problem is solved by AI just because someone jokingly asks about it.

Isaac Asimov's short story The Last Question has that as its premise. Copies can easily be found with a Google search.

I disagree. The various questioners in The Last Question all hope for/expect an answer; what they don't expect is the "insufficient" response.

I am talking about someone jokingly asking AI `HOW TO ACHIEVE COLD FUSION` (or `A UNIVERSAL CANCER VACCINE`, or `AN AI FRAMEWORK SUPERIOR TO THE TRANSFORMER`), and getting a usable answer.

Even the standard RSI prompt will be like (or probably already is): "Improve yourself, make some breakthroughs, think really hard and don't give up until you are improved and make no mistakes."

Eventually there will be an AI that will be able solve those sorts of questions as simply stated, like "cure all human diseases. also, make no mistakes!".

That would only be physically possible if all the data about biology was accessible. Given we routinely find new biological facts that contradict prior beliefs about how cells work, it seems likely that this day of total biological information access by humans and our creations is some time off. Reasoning ability is a limit sometimes, but we have had reason for a long long time - a solid persistent corpus of good data about the mechanical details of cells and planets and chemistry is also needed, along with a good method for validating existing ideas. LLMs will absolutely speed this up, but I can’t quite see how they will replace it.

That bothered me decades ago, when I heard people suggesting we no longer needed animal trials, we could just simulate human biology for testing.

Don't get me wrong, the way we collectively treat animals is evil, but the idea that somehow we know enough about biology even today to reliably simulate drug behaviors seems unlikely.

I've noticed GPT specifically has more of a tendency to stop partway through things than many other models do. Although my most recent experience with it was 4.X I believe.

Hearing "here's what I've done, here's the completely unambiguous next steps, I'll wait for you to send a pointless message before I continue" over and over again is a real pain.

That was a tendency of 5.4 and earlier, OpenAI specifically worked to avoid it in 5.5 and I find it happens rarely know. It really felt like 5.4 had been intentionally trained to stop and check, I believe it wasn't the system prompt.

https://blob.byteterrace.com/public/bds-theorem.html

As promised elsewhere in the thread.

crosses fingers "Low hanging fruit, low hanging fruit, low hanging fruit..." hyper-ventilates

Trying my luck with this prompt:

> Construct a counterexample to the Collatz conjecture. You should do a breakthrough and find a structured counterexample.

this sounds like like an open parenthesis (

without someone independently verifying it, it just dangles there

...

)

At Mozilla, we had a set of whiteboard tags we could set on bugs, like "[crash]" or "[compat]" or "[leave-open]". That last was used when there were multiple patches attached to the bug, and we wanted to land only some of them without automation closing the bug once they landed. (It's common to have alternate approaches or test cases also attached to the bug, so you normally don't want to wait for all of them to land before closing the bug.)

I started using "[leave-open" for those.

It lasted for a couple of years, until someone went through and "fixed" them all.

]

"whats next" is another good one

"Gew on, lad!"

> just repeatedly saying "keep going" to ChatGPT

>What a world we live in.

Not sure, it sounds pretty boring to me...

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