Some dismiss Erdős problems as trivialities - this couldn't be further from the truth! While many are amusing novelties, some of them are the most central problems in number theory and combinatorics.
A blog post with, in my view, the 10 most important:
erdosproblems.com/forum/thread...
A new blog post by @acerfur.bsky.social describing his experience as a pioneer of using AI tools to solve Erdős problems:
www.erdosproblems.com/forum/thread...
I think that having zero research should count as a 'defect in your research'.
AI is capable now of generating new interesting mathematics.
But it's much easier for it to generate plausible-sounding nonsense.
I am concerned that the latter, copied and promoted by users with no understanding of the mathematics, is going to drown out the former.
I expect new human insights to happen first - the AIs can be a powerful force multiplier, so once the initial big new idea/insight is had AI can quickly fill in the details and find other applications. (At least for some problems, certainly not most/all.)
My prediction is that by the end of the month there'll be between 4 and 8 new Erdos problems with solutions mostly or entirely AI generated.
But then we'll have seen all the easy wins available, and progress will slow until a significant jump in model capability or new human insights.
If you don't have an Erdos number and would like one, I'm open to an exciting collaboration on the intersection of Erdos-style mathematics and Old English literature and the benefits of free borders. (The first important question being whether this intersection is empty.)
This may lead to mathematicians becoming (even more) closed off than now about their half-baked ideas and insights - why share an observation on MathOverflow which solves 50% of a problem when a week later someone will ask GPT, which will fill in the other 50%, and give it 100% of the credit.
This leads to people viewing proofs as more 'purely AI-generated' than they are. Often AI is capable of correcting this when prompted, and can dig up references/sources, but people sometimes don't dig deeper when they should.
One of the big challenges now in using AI for mathematics is the credit/attribution problem. AI has a tendency to use observations/techniques without giving credit as to where it 'learnt' about them (mainly because it's forgotten itself).
Presumably with the same construction again?
Yes, Cloudflare...not heard about La Liga before!
I haven't checked the original paper again, but I suspect it's just another way they thought of to ask about a!b! dividing n! " up to small primes", and they didn't necessarily give much thought to the precise question. But surely infinitely many n, not all n.
In fact via a very similar argument to the previous one? (And there may be another to follow, since there at least three very similar problems. )
I'm confused by what you say, since this very thread is evidence where it has written novel proofs?
Hmm, it's working fine for me? It is behind a CDN already.
Any scenario involving Tom the Clown.
If you want something to read for the next couple of years, I highly recommend The Wandering Inn - 16 million words and still in progress!
wanderinginn.com
(It even has a mathematician canine character, with cool shades, though you have to wait about 10+ million words for them to show up.)
There are certainly teams trying to make things like this (e.g. AlphaProof), so we will see...
(The number of 'rules' and 'moves' is orders of magnitude greater than that of Go though, so not sure how much harder this will be.)
But we haven't (yet!) seen any genuinely new ideas/techniques come from AI in maths. But I'd expect those to come from an AlphaGo style AI rather than an LLM based one.
They're still doing 'new research' since a lot of research is applying standard techniques to new problems. (There are really, at the end of the day, very few people thinking about these problems compared to how many problems there are.)
I agree - part of what made AlphaGo so interesting is that it was doing (successful) things that no human had ever tried or thought of trying. We haven't seen any evidence of that from public AIs, since they're trained on all the tricks and techniques that humans have become familiar with.
Yeah, it's definitely an interesting time. I keep being surprised by what they can do. At the moment the most it's done is not very deep, judging by human standards, but it's definitely the sort of output that I'd expect from a human graduate student, rather than a machine.
A famous quote by Renyi (often falsely attributed to Erdős) is "A mathematician is a machine for turning coffee into theorems."
I recently learnt that in German this is actually a great pun: the word 'satz' for 'theorems' can also be translated as 'coffee grounds'.
(Your thread is, I mean. The website itself is a summary of all states of all problems for experts and non-experts alike.)
Hi Zach! Owner/maintainer of erdosproblems.com here. Just wanted to say how exciting is for me personally to see one of my favourite comic artists/authors posting about Erdos problems.
This is a great summary of the state of play for non-experts.
Do they say anything about why?
To be fair, some of the time she actually is a genius(ish) (kind of) mastermind only pretending to be a huge airhead.
No? Here a_n are specifically the Taylor coefficients of the entire function f, and we want f(n)=F(a_n) for all n (where F is some fixed function).
But why are the a_n fixed points of F? The condition was just that F(a_n)=f(n). (You seem to be assuming that F(a_n)=a_n for all n?)