I have no exact prompt to share, but he does write:
> Appreciate the insight! If it's at all of interest, this was a one-shot (supposed) solution in about 80 mins, unlike some other problems like 851 that took over 20 continuations totalling perhaps 15-20 hours of reasoning time.
Agree. Additionally, it’s really disheartening that people do this with Erdos problems specifically. They are not major research questions in mathematics, but were intended as little conjectures that people could use as a way into serious number theory with a small cash reward and a little bit of minor fame for being the person who did the work to solve one of them. They are not things where the solution itself provides an amazing amount of insight or moves the frontier of mathematics forward particularly.
So what is happening now is people now are nuking and paving the whole space with AI to prove their model can do maths, and we are all poorer for having this nice thing ruined in this way.
Number theorist Jared Lichtman says this AI proof is from "The Book", the highest compliment one can give. He also says:
> I care deeply about this problem, and I've been thinking about it for the past 7 years. I'd frequently talk to Maynard about it in our meetings, and consulted over the years with several experts (Granville, Pomerance, Sound, Fox...) and others at Oxford and Stanford. This problem was not a question of low-visibility per-se. Rather, it seems like a proof which becomes strikingly compact post-hoc, but the construction is quite special among many similar variations.
> The conjecture is 60 years old and many experts had consulted on the problem, making partial progress. I mentioned this to @thomasfbloom, and he replied: "perhaps the first Book Proof from AI?"
Terence Tao says:
> In any case, I would indeed say that this is a situation in which the AI-generated paper inadvertently highlighted a tighter connection between two areas of mathematics (in this case, the anatomy of integers and the theory of Markov processes) than had previously been made explicit in the literature (though there were hints and precursors scattered therein which one can see in retrospect). That would be a meaningful contribution to the anatomy of integers that goes well beyond the solution of this particular Erdos problem.
- He previously posted on X on the 2025/10/17 the following:
> Hi, as the owner/maintainer of http://erdosproblems.com, this is a dramatic misrepresentation. GPT-5 found references, which solved these problems, that I personally was unaware of. The 'open' status only means I personally am unaware of a paper which solves it. [1]
> GPT-5 has been a very useful tool in searching the literature, and this has been a valuable addition to the website. Its literature searching ability is already useful and impressive enough, no need to describe it as something it's not! [2]
https://xcancel.com/Liam06972452/status/2044051379916882067#...
The proof is here: https://www.overleaf.com/project/69dd1d8437eba662fda82929
Much more interesting than the proof would be to see the exact prompts used by Liam Price to generate the proof.
I have no exact prompt to share, but he does write:
> Appreciate the insight! If it's at all of interest, this was a one-shot (supposed) solution in about 80 mins, unlike some other problems like 851 that took over 20 continuations totalling perhaps 15-20 hours of reasoning time.
Source: https://www.erdosproblems.com/forum/thread/1196#post-5365
One of the people on the Erdös problem website (https://www.erdosproblems.com/forum/thread/1196), Jared Lichtman, is involved in a AI startup:
https://www.math.inc/
That AI startup also partners with Terence Tao:
https://www.math.inc/veritas-fellowships
https://www.math.inc/a-conversation-with-terry-tao
These two AI "enthusiasts" have massive conflicts of interest, which should perhaps be investigated by an ethics commission.
What's the conflict?
Research integrity vs. hyping up AI, obviously. Even in small matters like calling this proof "From The Book".
So where's the proof of this exact problem and why exactly did it end up on the list of unsolved problems if it has been already solved?
Yawn..
Agree. Additionally, it’s really disheartening that people do this with Erdos problems specifically. They are not major research questions in mathematics, but were intended as little conjectures that people could use as a way into serious number theory with a small cash reward and a little bit of minor fame for being the person who did the work to solve one of them. They are not things where the solution itself provides an amazing amount of insight or moves the frontier of mathematics forward particularly.
So what is happening now is people now are nuking and paving the whole space with AI to prove their model can do maths, and we are all poorer for having this nice thing ruined in this way.
Number theorist Jared Lichtman says this AI proof is from "The Book", the highest compliment one can give. He also says:
> I care deeply about this problem, and I've been thinking about it for the past 7 years. I'd frequently talk to Maynard about it in our meetings, and consulted over the years with several experts (Granville, Pomerance, Sound, Fox...) and others at Oxford and Stanford. This problem was not a question of low-visibility per-se. Rather, it seems like a proof which becomes strikingly compact post-hoc, but the construction is quite special among many similar variations.
> The conjecture is 60 years old and many experts had consulted on the problem, making partial progress. I mentioned this to @thomasfbloom, and he replied: "perhaps the first Book Proof from AI?"
Terence Tao says:
> In any case, I would indeed say that this is a situation in which the AI-generated paper inadvertently highlighted a tighter connection between two areas of mathematics (in this case, the anatomy of integers and the theory of Markov processes) than had previously been made explicit in the literature (though there were hints and precursors scattered therein which one can see in retrospect). That would be a meaningful contribution to the anatomy of integers that goes well beyond the solution of this particular Erdos problem.
Well that’s interesting. Probably I’m wrong, but I still feel like something important is slipping away here.
Like what exactly?
Number theorist Jared Lichtman is also involved with an AI startup so he might have a bit of an incentive to frame things this way.
Thank you, that feels like important context!
This guy also says it's a book proof though:
https://en.wikipedia.org/wiki/Thomas_Bloom
Important context too, thanks! More context:
- Thomas Bloom is the current owner of https://www.erdosproblems.com/
- He previously posted on X on the 2025/10/17 the following:
> Hi, as the owner/maintainer of http://erdosproblems.com, this is a dramatic misrepresentation. GPT-5 found references, which solved these problems, that I personally was unaware of. The 'open' status only means I personally am unaware of a paper which solves it. [1]
> GPT-5 has been a very useful tool in searching the literature, and this has been a valuable addition to the website. Its literature searching ability is already useful and impressive enough, no need to describe it as something it's not! [2]
[1]: https://x.com/thomasfbloom/status/1979254235075059732
[2]: https://x.com/thomasfbloom/status/1979254675833549207
I don't have the mathematical chops or knowledge of mathematicians to evaluate any of that.
Source: https://www.math.inc/a-conversation-with-terry-tao
However, I think this is still likely a very significant achievement/milestone.
what are you even yapping about.
Has this problem never been solved before? How do we know it's not just regurgitating a solution that came before it?