---
source: newsletter
source_url: https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/
ingested: 2026-05-12
sha256: 4eaeb90102fca4ad12633a08a1d4b06536490525bc27a7a647060c97029303b1
---
Title: A recent experience with ChatGPT 5.5 Pro
URL Source: https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/
Published Time: 2026-05-08T15:40:04+00:00
Markdown Content:
# A recent experience with ChatGPT 5.5 Pro | Gowers's Weblog
# [Gowers's Weblog](https://gowers.wordpress.com/)
Mathematics related discussions
* * *
« [Group and semigroup puzzles and a possible Polymath project](https://gowers.wordpress.com/2026/03/20/group-and-semigroup-puzzles-and-a-possible-polymath-project/)
## A recent experience with ChatGPT 5.5 Pro
We are all having to keep revising upwards our assessments of the mathematical capabilities of large language models. I have just made a fairly large revision as a result of ChatGPT 5.5 Pro, to which I am fortunate to have been given access, producing a piece of PhD-level research in an hour or so, with no serious mathematical input from me.
The background is that, as has been widely reported, LLMs are now capable of solving research-level problems, and have managed to solve several of the Erdős problems listed on [Thomas Bloom’s wonderful website](https://www.erdosproblems.com/). Initially it was possible to laugh this off: many of the “solutions” consisted in the LLM noticing that the problem had an answer sitting there in the literature already, or could be very easily deduced from known results. But little by little the laughter has become quieter. The message I am getting from what other mathematicians more involved in this enterprise have been saying is that LLMs have got to the point where if a problem has an easy argument that for one reason or another human mathematicians have missed (that reason sometimes, but not always, being that the problem has not received all that much attention), then there is a good chance that the LLMs will spot it. Conversely, for problems where one’s initial reaction is to be impressed that an LLM has come up with a clever argument, it often turns out on closer inspection that there are precedents for those arguments, so it is still just about possible to comfort oneself that LLMs are merely putting together existing knowledge rather than having truly original ideas. How much of a comfort that is I will not discuss here, other than to note that quite a lot of perfectly good human mathematics consists in putting together existing knowledge and proof techniques.
I decided to try something a little bit different. At least in combinatorics, there are quite a lot of papers that investigate some relatively new combinatorial parameter that leads naturally to several questions. Because of the sheer number of questions one can ask, the authors of such papers will not necessarily have the time to spend a week or two thinking about each one, so there is a decent probability that at least some of them will not be all that hard. This makes such papers very valuable as sources of problems for mathematicians who are doing research for the first time and who will be hugely encouraged by solving a problem that was officially open. Or rather, it used to make them valuable in that way, but it looks as though the bar has just been raised. It is no longer enough that somebody asks a problem: it needs to be hard enough for an LLM not to be able to solve it.
In any case, a little over a week ago I decided to see how ChatGPT 5.5 Pro would fare with a selection of problems asked by Mel Nathanson in a paper entitled [Diversity, Equity and Inclusion for Problems in Additive Number Theory](https://arxiv.org/abs/2603.15556). Nathanson has a remarkable record of being interested in problems and theorems that have later become extremely fashionable, which has led him to write a series of extremely well timed and therefore highly influential textbooks. In this paper, he argues for the interest of several other problems, some of which I will now briefly describe.
If ![Image 1: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002) is a set of integers, then its _sumset_![Image 2: A+A](https://s0.wp.com/latex.php?latex=A%2BA&bg=ffffff&fg=333333&s=0&c=20201002) is defined to be ![Image 3: \{a+b:a,b\in A\}](https://s0.wp.com/latex.php?latex=%5C%7Ba%2Bb%3Aa%2Cb%5Cin+A%5C%7D&bg=ffffff&fg=333333&s=0&c=20201002). For a positive integer ![Image 4: h](https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=333333&s=0&c=20201002), the ![Image 5: h](https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=333333&s=0&c=20201002)–_fold sumset_, denoted ![Image 6: hA](https://s0.wp.com/latex.php?latex=hA&bg=ffffff&fg=333333&s=0&c=20201002), is defined to be ![Image 7: \{a_1+\dots+a_h: a_1,\dots,a_h\in A\}](https://s0.wp.com/latex.php?latex=%5C%7Ba_1%2B%5Cdots%2Ba_h%3A+a_1%2C%5Cdots%2Ca_h%5Cin+A%5C%7D&bg=ffffff&fg=333333&s=0&c=20201002). Nathanson is interested in the possible sizes of ![Image 8: hA](https://s0.wp.com/latex.php?latex=hA&bg=ffffff&fg=333333&s=0&c=20201002) given the size of ![Image 9: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002). To that end one can define a set ![Image 10: \mathcal R(h,k)](https://s0.wp.com/latex.php?latex=%5Cmathcal+R%28h%2Ck%29&bg=ffffff&fg=333333&s=0&c=20201002) to be the set of all ![Image 11: t](https://s0.wp.com/latex.php?latex=t&bg=ffffff&fg=333333&s=0&c=20201002) such that there exists a set ![Image 12: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002) with ![Image 13: |A|=k](https://s0.wp.com/latex.php?latex=%7CA%7C%3Dk&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 14: |hA|=t](https://s0.wp.com/latex.php?latex=%7ChA%7C%3Dt&bg=ffffff&fg=333333&s=0&c=20201002).
An obvious first question to ask is simply “What is ![Image 15: \mathcal R(h,k)](https://s0.wp.com/latex.php?latex=%5Cmathcal+R%28h%2Ck%29&bg=ffffff&fg=333333&s=0&c=20201002)?” When ![Image 16: h=2](https://s0.wp.com/latex.php?latex=h%3D2&bg=ffffff&fg=333333&s=0&c=20201002), the answer is the set of all integers between ![Image 17: 2k-1](https://s0.wp.com/latex.php?latex=2k-1&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 18: \binom{k+1}2](https://s0.wp.com/latex.php?latex=%5Cbinom%7Bk%2B1%7D2&bg=ffffff&fg=333333&s=0&c=20201002). It is an easy exercise to show that if ![Image 19: |A|=k](https://s0.wp.com/latex.php?latex=%7CA%7C%3Dk&bg=ffffff&fg=333333&s=0&c=20201002), then ![Image 20: 2k-1\leq|A+A|\leq\binom{k+1}2](https://s0.wp.com/latex.php?latex=2k-1%5Cleq%7CA%2BA%7C%5Cleq%5Cbinom%7Bk%2B1%7D2&bg=ffffff&fg=333333&s=0&c=20201002), so this result is saying that all sizes in between can be realized. However, it is not true in general that ![Image 21: hA](https://s0.wp.com/latex.php?latex=hA&bg=ffffff&fg=333333&s=0&c=20201002) can take every size between its minimum and maximum possibilities, and we do not currently have a complete description of ![Image 22: \mathcal R(h,k)](https://s0.wp.com/latex.php?latex=%5Cmathcal+R%28h%2Ck%29&bg=ffffff&fg=333333&s=0&c=20201002).
Another natural question one can ask, and this is where ChatGPT came in, is how large a diameter you need if you want a set ![Image 23: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002) with ![Image 24: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 25: hA](https://s0.wp.com/latex.php?latex=hA&bg=ffffff&fg=333333&s=0&c=20201002) having prescribed sizes. (Of course, the size of ![Image 26: hA](https://s0.wp.com/latex.php?latex=hA&bg=ffffff&fg=333333&s=0&c=20201002) must belong to ![Image 27: \mathcal R(h,k)](https://s0.wp.com/latex.php?latex=%5Cmathcal+R%28h%2Ck%29&bg=ffffff&fg=333333&s=0&c=20201002).) Nathanson showed that for every ![Image 28: t\in[2k-1,\binom{k+1}2]](https://s0.wp.com/latex.php?latex=t%5Cin%5B2k-1%2C%5Cbinom%7Bk%2B1%7D2%5D&bg=ffffff&fg=333333&s=0&c=20201002) there is a subset ![Image 29: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002) of ![Image 30: \{0,1,2,\dots,2^k-1\}](https://s0.wp.com/latex.php?latex=%5C%7B0%2C1%2C2%2C%5Cdots%2C2%5Ek-1%5C%7D&bg=ffffff&fg=333333&s=0&c=20201002) with ![Image 31: |A|=k](https://s0.wp.com/latex.php?latex=%7CA%7C%3Dk&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 32: |A+A|=t](https://s0.wp.com/latex.php?latex=%7CA%2BA%7C%3Dt&bg=ffffff&fg=333333&s=0&c=20201002), and asked whether the bound ![Image 33: 2^k-1](https://s0.wp.com/latex.php?latex=2%5Ek-1&bg=ffffff&fg=333333&s=0&c=20201002) could be improved. ChatGPT 5.5 Pro thought for 17 minutes and 5 seconds before providing a construction that yielded a quadratic upper bound, which is clearly best possible. It wrote up its argument in a slightly rambling LLM-ish style, so I asked if it could write the argument up as a LaTeX file in the style of a typical mathematical preprint. After two minutes and 23 seconds it gave me that, after which I spent some time convincing myself that the argument was correct.
The basic idea behind both Nathanson’s argument and ChatGPT’s was that in order to obtain a set of a given size with a sumset of a given size, it is useful to build it out of a Sidon set, which means a set with sumset of maximal size (that is not quite the usual definition but it is the simplest to use in this discussion), and an arithmetic progression. Also, for a bit of fine tuning one can take an additional point near the arithmetic progression. Then if one plays around with the various parameters, one finds that one can obtain sets of all the sizes one wants. Nathanson doesn’t express his argument this way (it is Theorem 5 of [this paper](https://arxiv.org/pdf/2411.02365)), instead giving an inductive argument, but I think, without having checked too carefully, that if one unravels his argument, one finds that effectively that is what he ends up with, and the Sidon set in question consists of powers of 2. ChatGPT obtained its improvement by simply using a more efficient Sidon set — it is well known that one can find Sidon sets of quadratic diameter. (One might ask why Nathanson didn’t do that in the first place: I think it is because the obvious idea of using a more efficient Sidon set becomes obvious only after one has redescribed his inductive construction. Is that what ChatGPT did? It is very hard to say.)
Next, I asked ChatGPT to see whether it could do the same for a closely related question, where instead of looking at the size of the sumset, one looks at the size of the _restricted_ sumset, which is defined to be ![Image 34: \{a+b:a,b\in A, a\ne b\}](https://s0.wp.com/latex.php?latex=%5C%7Ba%2Bb%3Aa%2Cb%5Cin+A%2C+a%5Cne+b%5C%7D&bg=ffffff&fg=333333&s=0&c=20201002). Unsurprisingly, it was able to do that with no trouble at all. I got it to write both results up in a single note, to avoid a certain amount of duplication. If you are curious, you can see the note [here](https://drive.google.com/file/d/11r-ggU__GMmHIrgEHQVULUIR1VxKSwmi/view?usp=drive_link).
I then asked what it could do for general ![Image 35: h](https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=333333&s=0&c=20201002). I was much less optimistic that it would manage to do anything interesting, because the proof for ![Image 36: h=2](https://s0.wp.com/latex.php?latex=h%3D2&bg=ffffff&fg=333333&s=0&c=20201002) makes fundamental use of the fact (due to Erdős and Szemerédi) that we know exactly which sizes we need to create. If we don’t know what the set ![Image 37: \mathcal R(h,k)](https://s0.wp.com/latex.php?latex=%5Cmathcal+R%28h%2Ck%29&bg=ffffff&fg=333333&s=0&c=20201002) is, then it seems that we are forced to start with a hypothetical set ![Image 38: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002) with ![Image 39: |A|=k](https://s0.wp.com/latex.php?latex=%7CA%7C%3Dk&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 40: |hA|=t](https://s0.wp.com/latex.php?latex=%7ChA%7C%3Dt&bg=ffffff&fg=333333&s=0&c=20201002) and build out of it a set of small diameter with the same property. As it happens, I still don’t know how to get round that difficulty (I’m mentioning that just to demonstrate that my mathematical input was zero, and I didn’t even do anything clever with the prompts), but Nathanson mentioned in his paper a remarkable paper of Isaac Rajagopal, a student at MIT, who must have got round the difficulty somehow, because he had managed to prove an exponential dependence of ![Image 41: \mathcal R(h,k)](https://s0.wp.com/latex.php?latex=%5Cmathcal+R%28h%2Ck%29&bg=ffffff&fg=333333&s=0&c=20201002) on ![Image 42: k](https://s0.wp.com/latex.php?latex=k&bg=ffffff&fg=333333&s=0&c=20201002) for each fixed ![Image 43: h](https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=333333&s=0&c=20201002).
I’ll leave the previous paragraph there, but Isaac has subsequently explained to me that that isn’t really the difficulty. His argument gives a complete description of ![Image 44: \mathcal R(h,k)](https://s0.wp.com/latex.php?latex=%5Cmathcal+R%28h%2Ck%29&bg=ffffff&fg=333333&s=0&c=20201002) when ![Image 45: k](https://s0.wp.com/latex.php?latex=k&bg=ffffff&fg=333333&s=0&c=20201002) is sufficiently large, and if one wants to prove a polynomial dependence for fixed ![Image 46: h](https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=333333&s=0&c=20201002), then assuming that ![Image 47: k](https://s0.wp.com/latex.php?latex=k&bg=ffffff&fg=333333&s=0&c=20201002) is sufficiently large is clearly permitted. The real difficulty is that constructing the sets with given sumset sizes was significantly more complicated, and necessarily so because the degree of the polynomial grows with ![Image 48: h](https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=333333&s=0&c=20201002), and one therefore needs more and more parameters to define the sets.
In any case, the task faced by ChatGPT was not to solve the problem from scratch, but to see whether it was possible to tighten up Isaac Rajagopal’s argument. Here’s what happened.
1.   After 16 minutes and 41 seconds, it came back with an argument that claimed to have improved the upper bound from exponential in ![Image 49: k](https://s0.wp.com/latex.php?latex=k&bg=ffffff&fg=333333&s=0&c=20201002) to exponential in ![Image 50: k^\alpha](https://s0.wp.com/latex.php?latex=k%5E%5Calpha&bg=ffffff&fg=333333&s=0&c=20201002) for any ![Image 51: \alpha>1/2](https://s0.wp.com/latex.php?latex=%5Calpha%3E1%2F2&bg=ffffff&fg=333333&s=0&c=20201002). 
2.   I asked it to write that in preprint form too, which took it a further 47 minutes and 39 seconds.
3.   That preprint would have been hard for me to read, as that would have meant carefully reading Rajagopal’s paper first, but I sent it to Nathanson, who forwarded it to Rajagopal, who said he thought it looked correct.
4.   Both ChatGPT and Rajagopal speculated a little on what might need to be done to push things further and get a polynomial bound, so I got greedy and asked ChatGPT to give that a go.
5.   After 13 minutes and 33 seconds it told me it felt optimistic about the existence of such an argument but there were a couple of technical statements that needed checking. 
6.   I asked it to check them.
7.   After 9 minutes and 12 seconds it got back to me with the check having been done, so I asked for this too to be written in preprint form. 
8.   After 31 minutes and 40 seconds the “preprint” was ready. [Here it is.](https://drive.google.com/file/d/1IkJBcWYz_3J_QGsESBmMa-jrEHAJDcJB/view?usp=sharing)
9.   Isaac Rajagopal looked at it and declared it to be almost certainly correct. It was clear that he meant this not just at a line-by-line level but at the level of ideas.
Isaac made some very interesting remarks about the nature of what the additional ideas were that ChatGPT contributed. Since, as I have already said, my mathematical input was zero, I invited him to write a guest section to this post. Just before we get to that, I want to raise a question (that will undoubtedly have been raised by others as well), which is simple: what should we do with this kind of content? Had the result been produced by a human mathematician, it would definitely have been publishable, so I think it would be wrong to describe it as AI slop. On the other hand, it seems pointless even to think about putting it in a journal, since it can be made freely available, and nobody needs “credit” for it (except that Isaac deserves plenty of credit for creating the framework on which ChatGPT could build). I understand that arXiv has a policy against accepting AI-written content, which makes good sense to me. So maybe there should be a different repository where AI-produced results can live. But various decisions would need to be made about how it was organized. I myself think that one would probably want to have some kind of moderation process, so that results would be included only if a human mathematician was prepared to certify that they were correct — or, better still, that they had been formalized by a proof assistant — and perhaps also that they answered a question that had been asked in a human-written paper. On the other hand, I wouldn’t want a moderation process that created vast amounts of work (unless the work was itself done by AI, but there are obvious dangers in going down that route). Anyway, until these questions are answered, this result is available from the link above, and perhaps, now that LLMs are so good at literature search, that will be enough to make it findable by anyone who wants to know whether Nathanson’s problem has been solved.
## Isaac’s evaluation of what ChatGPT achieved
With just a few prompts, ChatGPT was able to improve the upper bound on ![Image 52: N(h,k)](https://s0.wp.com/latex.php?latex=N%28h%2Ck%29&bg=ffffff&fg=333333&s=0&c=20201002) (which I will define very soon) from exponential in ![Image 53: k](https://s0.wp.com/latex.php?latex=k&bg=ffffff&fg=333333&s=0&c=20201002) to polynomial in ![Image 54: k](https://s0.wp.com/latex.php?latex=k&bg=ffffff&fg=333333&s=0&c=20201002). While its first improvement of the bound, from exponential in ![Image 55: k](https://s0.wp.com/latex.php?latex=k&bg=ffffff&fg=333333&s=0&c=20201002) to exponential in ![Image 56: k^{\frac{1}{2} + \varepsilon}](https://s0.wp.com/latex.php?latex=k%5E%7B%5Cfrac%7B1%7D%7B2%7D+%2B+%5Cvarepsilon%7D&bg=ffffff&fg=333333&s=0&c=20201002), was a routine modification of my work, the improvement to polynomial in ![Image 57: k](https://s0.wp.com/latex.php?latex=k&bg=ffffff&fg=333333&s=0&c=20201002) is quite impressive. To do this, ChatGPT came up with an idea which is original and clever. It is the sort of idea I would be very proud to come up with after a week or two of pondering, and it took ChatGPT less than an hour to find and prove, using similar methods to those in my own proof. My goal is to explain that idea, in a manner that will be digestible to my friends who are computer science majors as well as my math major friends.
The problem of bounding ![Image 58: N(h,k)](https://s0.wp.com/latex.php?latex=N%28h%2Ck%29&bg=ffffff&fg=333333&s=0&c=20201002) is closely related to a problem I worked on at the Duluth REU (Research Experience for Undergrads) program, of determining ![Image 59: \mathcal{R}(h,k)](https://s0.wp.com/latex.php?latex=%5Cmathcal%7BR%7D%28h%2Ck%29&bg=ffffff&fg=333333&s=0&c=20201002). In particular, ![Image 60: \mathcal{R}(h,k)](https://s0.wp.com/latex.php?latex=%5Cmathcal%7BR%7D%28h%2Ck%29&bg=ffffff&fg=333333&s=0&c=20201002) is the set of possible ![Image 61: h](https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=333333&s=0&c=20201002)-fold sumset sizes ![Image 62: |hA|](https://s0.wp.com/latex.php?latex=%7ChA%7C&bg=ffffff&fg=333333&s=0&c=20201002), where ![Image 63: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002) can be chosen to be any set of ![Image 64: k](https://s0.wp.com/latex.php?latex=k&bg=ffffff&fg=333333&s=0&c=20201002) integers. ![Image 65: N(h,k)](https://s0.wp.com/latex.php?latex=N%28h%2Ck%29&bg=ffffff&fg=333333&s=0&c=20201002) is the minimal ![Image 66: N](https://s0.wp.com/latex.php?latex=N&bg=ffffff&fg=333333&s=0&c=20201002) such that we can achieve all of the values of ![Image 67: \mathcal{R}(h,k)](https://s0.wp.com/latex.php?latex=%5Cmathcal%7BR%7D%28h%2Ck%29&bg=ffffff&fg=333333&s=0&c=20201002) using ![Image 68: k](https://s0.wp.com/latex.php?latex=k&bg=ffffff&fg=333333&s=0&c=20201002)-element sets ![Image 69: A \subset \{0,1,2,\ldots,N\}](https://s0.wp.com/latex.php?latex=A+%5Csubset+%5C%7B0%2C1%2C2%2C%5Cldots%2CN%5C%7D&bg=ffffff&fg=333333&s=0&c=20201002). I spent last summer explicitly characterizing the set ![Image 70: \mathcal{R}(h,k)](https://s0.wp.com/latex.php?latex=%5Cmathcal%7BR%7D%28h%2Ck%29&bg=ffffff&fg=333333&s=0&c=20201002) for large ![Image 71: k](https://s0.wp.com/latex.php?latex=k&bg=ffffff&fg=333333&s=0&c=20201002), by constructing sets ![Image 72: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002) such that ![Image 73: |hA|](https://s0.wp.com/latex.php?latex=%7ChA%7C&bg=ffffff&fg=333333&s=0&c=20201002) achieves all sizes which I could not rule out as impossible. So, ![Image 74: N(h,k)](https://s0.wp.com/latex.php?latex=N%28h%2Ck%29&bg=ffffff&fg=333333&s=0&c=20201002) can be upper-bounded by optimizing my constructions.
I constructed these sets ![Image 75: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002) by combining smaller component sets which are simpler to analyze. Some of these components are the geometric series
![Image 76: \displaystyle S = \{0,1,m,m^2,\ldots,m^{\ell-2}\} \quad \hbox{and} \quad T = \{1,m,m^2,\ldots,m^{\ell-1}\} \qquad (1)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+S+%3D+%5C%7B0%2C1%2Cm%2Cm%5E2%2C%5Cldots%2Cm%5E%7B%5Cell-2%7D%5C%7D+%5Cquad+%5Chbox%7Band%7D+%5Cquad+T+%3D+%5C%7B1%2Cm%2Cm%5E2%2C%5Cldots%2Cm%5E%7B%5Cell-1%7D%5C%7D+%5Cqquad+%281%29&bg=ffffff&fg=333333&s=0&c=20201002)
for various values of ![Image 77: 2 \leq m \leq h](https://s0.wp.com/latex.php?latex=2+%5Cleq+m+%5Cleq+h&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 78: 2 \leq \ell \leq k](https://s0.wp.com/latex.php?latex=2+%5Cleq+%5Cell+%5Cleq+k&bg=ffffff&fg=333333&s=0&c=20201002). Unfortunately, the elements of ![Image 79: S](https://s0.wp.com/latex.php?latex=S&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 80: T](https://s0.wp.com/latex.php?latex=T&bg=ffffff&fg=333333&s=0&c=20201002) are exponentially large in terms of ![Image 81: k](https://s0.wp.com/latex.php?latex=k&bg=ffffff&fg=333333&s=0&c=20201002). So, I asked ChatGPT (through Tim) whether there exist sets of ![Image 82: \ell](https://s0.wp.com/latex.php?latex=%5Cell&bg=ffffff&fg=333333&s=0&c=20201002) elements which have similar sumset sizes to these geometric series, but contain only numbers of polynomial size in ![Image 83: \ell](https://s0.wp.com/latex.php?latex=%5Cell&bg=ffffff&fg=333333&s=0&c=20201002): I had no idea if this was possible, or how to begin constructing such sets. ChatGPT came back with an answer, constructing sets ![Image 84: G](https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 85: H](https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=333333&s=0&c=20201002) which behave like “half a geometric series squeezed into a polynomial interval,” which is counterintuitive. Before I discuss the construction of ![Image 86: G](https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 87: H](https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=333333&s=0&c=20201002), I will explain the important properties of the sumset sizes of ![Image 88: S](https://s0.wp.com/latex.php?latex=S&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 89: T](https://s0.wp.com/latex.php?latex=T&bg=ffffff&fg=333333&s=0&c=20201002) which they recreate.
For ![Image 90: h > 0](https://s0.wp.com/latex.php?latex=h+%3E+0&bg=ffffff&fg=333333&s=0&c=20201002), a set ![Image 91: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002) is called a ![Image 92: B_h](https://s0.wp.com/latex.php?latex=B_h&bg=ffffff&fg=333333&s=0&c=20201002) set if the only solutions to
![Image 93: \displaystyle x_1+\cdots+x_h = y_1+\cdots+y_h](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+x_1%2B%5Ccdots%2Bx_h+%3D+y_1%2B%5Ccdots%2By_h&bg=ffffff&fg=333333&s=0&c=20201002)
with ![Image 94: x_i,y_i](https://s0.wp.com/latex.php?latex=x_i%2Cy_i&bg=ffffff&fg=333333&s=0&c=20201002) in ![Image 95: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002) are the “trivial” solutions, by which I mean that one side of the equation is a reordering of the other side. If ![Image 96: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002) is a ![Image 97: B_h](https://s0.wp.com/latex.php?latex=B_h&bg=ffffff&fg=333333&s=0&c=20201002) set of size ![Image 98: \ell](https://s0.wp.com/latex.php?latex=%5Cell&bg=ffffff&fg=333333&s=0&c=20201002), then elements of ![Image 99: hA](https://s0.wp.com/latex.php?latex=hA&bg=ffffff&fg=333333&s=0&c=20201002) correspond exactly to choices of ![Image 100: h](https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=333333&s=0&c=20201002) elements of ![Image 101: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002), with repetition allowed. Using “stars and bars,” one can see that ![Image 102: |hA| = \binom{h+\ell - 1}{h}](https://s0.wp.com/latex.php?latex=%7ChA%7C+%3D+%5Cbinom%7Bh%2B%5Cell+-+1%7D%7Bh%7D&bg=ffffff&fg=333333&s=0&c=20201002) and this is the maximum possible value of ![Image 103: |hA|](https://s0.wp.com/latex.php?latex=%7ChA%7C&bg=ffffff&fg=333333&s=0&c=20201002) among sets of size ![Image 104: \ell](https://s0.wp.com/latex.php?latex=%5Cell&bg=ffffff&fg=333333&s=0&c=20201002). So, another definition is that ![Image 105: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002) is a ![Image 106: B_h](https://s0.wp.com/latex.php?latex=B_h&bg=ffffff&fg=333333&s=0&c=20201002) set if ![Image 107: |hA| = \binom{h+|A| - 1}{h}](https://s0.wp.com/latex.php?latex=%7ChA%7C+%3D+%5Cbinom%7Bh%2B%7CA%7C+-+1%7D%7Bh%7D&bg=ffffff&fg=333333&s=0&c=20201002). Sidon sets, which Tim discussed, are exactly ![Image 108: B_2](https://s0.wp.com/latex.php?latex=B_2&bg=ffffff&fg=333333&s=0&c=20201002) sets.
To make things more concrete, let us assume that ![Image 109: m = 4](https://s0.wp.com/latex.php?latex=m+%3D+4&bg=ffffff&fg=333333&s=0&c=20201002) in (1). Then, ![Image 110: S](https://s0.wp.com/latex.php?latex=S&bg=ffffff&fg=333333&s=0&c=20201002) is a ![Image 111: B_3](https://s0.wp.com/latex.php?latex=B_3&bg=ffffff&fg=333333&s=0&c=20201002) set, but it is not a ![Image 112: B_4](https://s0.wp.com/latex.php?latex=B_4&bg=ffffff&fg=333333&s=0&c=20201002) set because of the relations
![Image 113: \displaystyle 4^{a} + 4^a + 4^a + 4^a = 4^{a+1} + 0 + 0 + 0 \qquad (2)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+4%5E%7Ba%7D+%2B+4%5Ea+%2B+4%5Ea+%2B+4%5Ea+%3D+4%5E%7Ba%2B1%7D+%2B+0+%2B+0+%2B+0+%5Cqquad+%282%29&bg=ffffff&fg=333333&s=0&c=20201002)
for any choice of ![Image 114: a](https://s0.wp.com/latex.php?latex=a&bg=ffffff&fg=333333&s=0&c=20201002) in ![Image 115: \{0,1,2,\ldots, \ell-3\}](https://s0.wp.com/latex.php?latex=%5C%7B0%2C1%2C2%2C%5Cldots%2C+%5Cell-3%5C%7D&bg=ffffff&fg=333333&s=0&c=20201002). In particular, ![Image 116: \binom{\ell+3}{4} - |4S| = \ell-2](https://s0.wp.com/latex.php?latex=%5Cbinom%7B%5Cell%2B3%7D%7B4%7D+-+%7C4S%7C+%3D+%5Cell-2&bg=ffffff&fg=333333&s=0&c=20201002), as these ![Image 117: \ell-2](https://s0.wp.com/latex.php?latex=%5Cell-2&bg=ffffff&fg=333333&s=0&c=20201002) relations are the only ones preventing ![Image 118: S](https://s0.wp.com/latex.php?latex=S&bg=ffffff&fg=333333&s=0&c=20201002) from being a ![Image 119: B_4](https://s0.wp.com/latex.php?latex=B_4&bg=ffffff&fg=333333&s=0&c=20201002) set. ![Image 120: T](https://s0.wp.com/latex.php?latex=T&bg=ffffff&fg=333333&s=0&c=20201002) lacks the relations in (2) because ![Image 121: 0](https://s0.wp.com/latex.php?latex=0&bg=ffffff&fg=333333&s=0&c=20201002) is not in ![Image 122: T](https://s0.wp.com/latex.php?latex=T&bg=ffffff&fg=333333&s=0&c=20201002). So, ![Image 123: T](https://s0.wp.com/latex.php?latex=T&bg=ffffff&fg=333333&s=0&c=20201002) is a ![Image 124: B_4](https://s0.wp.com/latex.php?latex=B_4&bg=ffffff&fg=333333&s=0&c=20201002) set, but it is not a ![Image 125: B_5](https://s0.wp.com/latex.php?latex=B_5&bg=ffffff&fg=333333&s=0&c=20201002) set because of the relations
![Image 126: \displaystyle 4^{a} + 4^a + 4^a + 4^a + 4^{b+1} = 4^{a+1} + 4^b + 4^b + 4^b + 4^b \qquad (3)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+4%5E%7Ba%7D+%2B+4%5Ea+%2B+4%5Ea+%2B+4%5Ea+%2B+4%5E%7Bb%2B1%7D+%3D+4%5E%7Ba%2B1%7D+%2B+4%5Eb+%2B+4%5Eb+%2B+4%5Eb+%2B+4%5Eb+%5Cqquad+%283%29&bg=ffffff&fg=333333&s=0&c=20201002)
for any choices of ![Image 127: a \neq b](https://s0.wp.com/latex.php?latex=a+%5Cneq+b&bg=ffffff&fg=333333&s=0&c=20201002) in ![Image 128: \{0,1,2,\ldots, \ell-2\}](https://s0.wp.com/latex.php?latex=%5C%7B0%2C1%2C2%2C%5Cldots%2C+%5Cell-2%5C%7D&bg=ffffff&fg=333333&s=0&c=20201002). This gives ![Image 129: \binom{\ell-1}{2}](https://s0.wp.com/latex.php?latex=%5Cbinom%7B%5Cell-1%7D%7B2%7D&bg=ffffff&fg=333333&s=0&c=20201002) relations, and one can check that ![Image 130: \binom{\ell+4}{5} - |5T| = \binom{\ell-1}{2}](https://s0.wp.com/latex.php?latex=%5Cbinom%7B%5Cell%2B4%7D%7B5%7D+-+%7C5T%7C+%3D+%5Cbinom%7B%5Cell-1%7D%7B2%7D&bg=ffffff&fg=333333&s=0&c=20201002). To summarize, we have seen that
(a) ![Image 131: S](https://s0.wp.com/latex.php?latex=S&bg=ffffff&fg=333333&s=0&c=20201002) is a ![Image 132: B_{m-1}](https://s0.wp.com/latex.php?latex=B_%7Bm-1%7D&bg=ffffff&fg=333333&s=0&c=20201002) set.
(b) ![Image 133: \binom{m+\ell-1}{m} - |mS| = \ell -2](https://s0.wp.com/latex.php?latex=%5Cbinom%7Bm%2B%5Cell-1%7D%7Bm%7D+-+%7CmS%7C+%3D+%5Cell+-2&bg=ffffff&fg=333333&s=0&c=20201002) is a linear function of ![Image 134: \ell](https://s0.wp.com/latex.php?latex=%5Cell&bg=ffffff&fg=333333&s=0&c=20201002).
(c) ![Image 135: T](https://s0.wp.com/latex.php?latex=T&bg=ffffff&fg=333333&s=0&c=20201002) is a ![Image 136: B_{m}](https://s0.wp.com/latex.php?latex=B_%7Bm%7D&bg=ffffff&fg=333333&s=0&c=20201002) set.
(d) ![Image 137: \binom{m+\ell}{m+1} - |(m+1)T| = \binom{\ell-1}{2}](https://s0.wp.com/latex.php?latex=%5Cbinom%7Bm%2B%5Cell%7D%7Bm%2B1%7D+-+%7C%28m%2B1%29T%7C+%3D+%5Cbinom%7B%5Cell-1%7D%7B2%7D&bg=ffffff&fg=333333&s=0&c=20201002) is a quadratic function of ![Image 138: \ell](https://s0.wp.com/latex.php?latex=%5Cell&bg=ffffff&fg=333333&s=0&c=20201002).
ChatGPT was able to find sets ![Image 139: G](https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 140: H](https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=333333&s=0&c=20201002) of ![Image 141: \ell](https://s0.wp.com/latex.php?latex=%5Cell&bg=ffffff&fg=333333&s=0&c=20201002) elements which satisfy (a)-(d), but whose elements all have polynomial size in ![Image 142: \ell](https://s0.wp.com/latex.php?latex=%5Cell&bg=ffffff&fg=333333&s=0&c=20201002). The construction of ![Image 143: G](https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 144: H](https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=333333&s=0&c=20201002) uses ![Image 145: h^2](https://s0.wp.com/latex.php?latex=h%5E2&bg=ffffff&fg=333333&s=0&c=20201002)-dissociated sets, which are sets ![Image 146: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002) where the only solutions to
![Image 147: \displaystyle x_1+\cdots+x_s = y_1+\cdots+y_{s'} \qquad (4)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+x_1%2B%5Ccdots%2Bx_s+%3D+y_1%2B%5Ccdots%2By_%7Bs%27%7D+%5Cqquad+%284%29&bg=ffffff&fg=333333&s=0&c=20201002)
with ![Image 148: s,s' \leq h^2](https://s0.wp.com/latex.php?latex=s%2Cs%27+%5Cleq+h%5E2&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 149: x_i,y_i](https://s0.wp.com/latex.php?latex=x_i%2Cy_i&bg=ffffff&fg=333333&s=0&c=20201002) in ![Image 150: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002) are the “trivial” solutions, i.e. ![Image 151: s = s'](https://s0.wp.com/latex.php?latex=s+%3D+s%27&bg=ffffff&fg=333333&s=0&c=20201002) and one side of the equation is a reordering of the other side. For ![Image 152: r > 0](https://s0.wp.com/latex.php?latex=r+%3E+0&bg=ffffff&fg=333333&s=0&c=20201002), it is possible to construct an ![Image 153: h^2](https://s0.wp.com/latex.php?latex=h%5E2&bg=ffffff&fg=333333&s=0&c=20201002)-dissociated set ![Image 154: U = \{u_1,\ldots,u_r\} \subseteq \{0,1,2,\ldots,N\}](https://s0.wp.com/latex.php?latex=U+%3D+%5C%7Bu_1%2C%5Cldots%2Cu_r%5C%7D+%5Csubseteq+%5C%7B0%2C1%2C2%2C%5Cldots%2CN%5C%7D&bg=ffffff&fg=333333&s=0&c=20201002), where ![Image 155: N](https://s0.wp.com/latex.php?latex=N&bg=ffffff&fg=333333&s=0&c=20201002) is approximately ![Image 156: r^{h^2}](https://s0.wp.com/latex.php?latex=r%5E%7Bh%5E2%7D&bg=ffffff&fg=333333&s=0&c=20201002), and in particular polynomial in ![Image 157: r](https://s0.wp.com/latex.php?latex=r&bg=ffffff&fg=333333&s=0&c=20201002). Constructions of such a ![Image 158: U](https://s0.wp.com/latex.php?latex=U&bg=ffffff&fg=333333&s=0&c=20201002) using finite fields date back to Singer (1938) and Bose–Chowla (1963) and are described in Appendix 1. Define
![Image 159: \displaystyle G = \{0, u_1,u_2,\ldots,u_r,mu_1,mu_2,\ldots, mu_r\}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+G+%3D+%5C%7B0%2C+u_1%2Cu_2%2C%5Cldots%2Cu_r%2Cmu_1%2Cmu_2%2C%5Cldots%2C+mu_r%5C%7D&bg=ffffff&fg=333333&s=0&c=20201002)
and
![Image 160: \displaystyle H= \{u_1,u_2,\ldots,u_r,mu_1,mu_2,\ldots, mu_r\}. \qquad (5)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+H%3D+%5C%7Bu_1%2Cu_2%2C%5Cldots%2Cu_r%2Cmu_1%2Cmu_2%2C%5Cldots%2C+mu_r%5C%7D.+%5Cqquad+%285%29&bg=ffffff&fg=333333&s=0&c=20201002)
In hindsight, I have good intuition for the construction of ![Image 161: G](https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 162: H](https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=333333&s=0&c=20201002). All of the relations in (2) and (3) are formed by combining one or two relations of the form ![Image 163: 4x = y](https://s0.wp.com/latex.php?latex=4x+%3D+y&bg=ffffff&fg=333333&s=0&c=20201002). There are approximately ![Image 164: \ell](https://s0.wp.com/latex.php?latex=%5Cell&bg=ffffff&fg=333333&s=0&c=20201002) relations of the form ![Image 165: mx = y](https://s0.wp.com/latex.php?latex=mx+%3D+y&bg=ffffff&fg=333333&s=0&c=20201002) in ![Image 166: S](https://s0.wp.com/latex.php?latex=S&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 167: T](https://s0.wp.com/latex.php?latex=T&bg=ffffff&fg=333333&s=0&c=20201002), and approximately ![Image 168: \ell/2](https://s0.wp.com/latex.php?latex=%5Cell%2F2&bg=ffffff&fg=333333&s=0&c=20201002) such relations in ![Image 169: G](https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 170: H](https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=333333&s=0&c=20201002). There are few other low-order relations in ![Image 171: S](https://s0.wp.com/latex.php?latex=S&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 172: T](https://s0.wp.com/latex.php?latex=T&bg=ffffff&fg=333333&s=0&c=20201002), and similarly in ![Image 173: G](https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 174: H](https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=333333&s=0&c=20201002) because ![Image 175: U](https://s0.wp.com/latex.php?latex=U&bg=ffffff&fg=333333&s=0&c=20201002) is ![Image 176: h^2](https://s0.wp.com/latex.php?latex=h%5E2&bg=ffffff&fg=333333&s=0&c=20201002)-dissociated. So, ![Image 177: G](https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 178: H](https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=333333&s=0&c=20201002) manage to contain half as many ![Image 179: mx = y](https://s0.wp.com/latex.php?latex=mx+%3D+y&bg=ffffff&fg=333333&s=0&c=20201002)-relations as their geometric series counterparts, while also containing few low-order relations.
We now see why (a)-(d) hold with ![Image 180: S](https://s0.wp.com/latex.php?latex=S&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 181: T](https://s0.wp.com/latex.php?latex=T&bg=ffffff&fg=333333&s=0&c=20201002) replaced by ![Image 182: G](https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 183: H](https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=333333&s=0&c=20201002), respectively. For concreteness, we assume that ![Image 184: m = 4](https://s0.wp.com/latex.php?latex=m+%3D+4&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 185: h>4](https://s0.wp.com/latex.php?latex=h%3E4&bg=ffffff&fg=333333&s=0&c=20201002), so ![Image 186: U](https://s0.wp.com/latex.php?latex=U&bg=ffffff&fg=333333&s=0&c=20201002) contains no nontrivial relations as in (4) with ![Image 187: s,s' \leq 25 \leq h^2](https://s0.wp.com/latex.php?latex=s%2Cs%27+%5Cleq+25+%5Cleq+h%5E2&bg=ffffff&fg=333333&s=0&c=20201002). Then, ![Image 188: G](https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=333333&s=0&c=20201002) is a ![Image 189: B_3](https://s0.wp.com/latex.php?latex=B_3&bg=ffffff&fg=333333&s=0&c=20201002) set, but it is not a ![Image 190: B_4](https://s0.wp.com/latex.php?latex=B_4&bg=ffffff&fg=333333&s=0&c=20201002) set because of the relations
![Image 191: \displaystyle u_i + u_i + u_i + u_i = 4u_i + 0 + 0 + 0](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+u_i+%2B+u_i+%2B+u_i+%2B+u_i+%3D+4u_i+%2B+0+%2B+0+%2B+0&bg=ffffff&fg=333333&s=0&c=20201002)
for any choice of ![Image 192: i](https://s0.wp.com/latex.php?latex=i&bg=ffffff&fg=333333&s=0&c=20201002) in ![Image 193: \{1,2,\ldots, r\}](https://s0.wp.com/latex.php?latex=%5C%7B1%2C2%2C%5Cldots%2C+r%5C%7D&bg=ffffff&fg=333333&s=0&c=20201002). If we let ![Image 194: \ell = |G| = 2r+1](https://s0.wp.com/latex.php?latex=%5Cell+%3D+%7CG%7C+%3D+2r%2B1&bg=ffffff&fg=333333&s=0&c=20201002), we can check that ![Image 195: \binom{\ell + 3}{4} - |4G| = r = \frac{\ell-1}{2}](https://s0.wp.com/latex.php?latex=%5Cbinom%7B%5Cell+%2B+3%7D%7B4%7D+-+%7C4G%7C+%3D+r+%3D+%5Cfrac%7B%5Cell-1%7D%7B2%7D&bg=ffffff&fg=333333&s=0&c=20201002) is linear in ![Image 196: \ell](https://s0.wp.com/latex.php?latex=%5Cell&bg=ffffff&fg=333333&s=0&c=20201002). In particular, (a) and (b) hold with ![Image 197: S](https://s0.wp.com/latex.php?latex=S&bg=ffffff&fg=333333&s=0&c=20201002) replaced by ![Image 198: G](https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=333333&s=0&c=20201002), and the linear function ![Image 199: \ell-2](https://s0.wp.com/latex.php?latex=%5Cell-2&bg=ffffff&fg=333333&s=0&c=20201002) replaced by ![Image 200: \frac{\ell-1}{2}](https://s0.wp.com/latex.php?latex=%5Cfrac%7B%5Cell-1%7D%7B2%7D&bg=ffffff&fg=333333&s=0&c=20201002). We can also see that ![Image 201: H](https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=333333&s=0&c=20201002) is a ![Image 202: B_4](https://s0.wp.com/latex.php?latex=B_4&bg=ffffff&fg=333333&s=0&c=20201002) set, but it is not a ![Image 203: B_5](https://s0.wp.com/latex.php?latex=B_5&bg=ffffff&fg=333333&s=0&c=20201002) set because of the relations
![Image 204: \displaystyle u_i + u_i + u_i + u_i + 4u_j = 4u_i + u_j + u_j +u_j + u_j](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+u_i+%2B+u_i+%2B+u_i+%2B+u_i+%2B+4u_j+%3D+4u_i+%2B+u_j+%2B+u_j+%2Bu_j+%2B+u_j&bg=ffffff&fg=333333&s=0&c=20201002)
for any ![Image 205: i\neq j](https://s0.wp.com/latex.php?latex=i%5Cneq+j&bg=ffffff&fg=333333&s=0&c=20201002) in ![Image 206: \{1,2,\ldots, r\}](https://s0.wp.com/latex.php?latex=%5C%7B1%2C2%2C%5Cldots%2C+r%5C%7D&bg=ffffff&fg=333333&s=0&c=20201002). If we let ![Image 207: \ell = |H| = 2r](https://s0.wp.com/latex.php?latex=%5Cell+%3D+%7CH%7C+%3D+2r&bg=ffffff&fg=333333&s=0&c=20201002), we can check that ![Image 208: \binom{\ell + 4}{5} - |5H| = \binom{r}{2} = \binom{\ell/2}{2}](https://s0.wp.com/latex.php?latex=%5Cbinom%7B%5Cell+%2B+4%7D%7B5%7D+-+%7C5H%7C+%3D+%5Cbinom%7Br%7D%7B2%7D+%3D+%5Cbinom%7B%5Cell%2F2%7D%7B2%7D&bg=ffffff&fg=333333&s=0&c=20201002) is quadratic in ![Image 209: \ell](https://s0.wp.com/latex.php?latex=%5Cell&bg=ffffff&fg=333333&s=0&c=20201002). In a similar manner, (c) and (d) hold with ![Image 210: T](https://s0.wp.com/latex.php?latex=T&bg=ffffff&fg=333333&s=0&c=20201002) replaced by ![Image 211: H](https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=333333&s=0&c=20201002), and the quadratic function ![Image 212: \binom{\ell-1}{2}](https://s0.wp.com/latex.php?latex=%5Cbinom%7B%5Cell-1%7D%7B2%7D&bg=ffffff&fg=333333&s=0&c=20201002) replaced by ![Image 213: \binom{\ell/2}{2}](https://s0.wp.com/latex.php?latex=%5Cbinom%7B%5Cell%2F2%7D%7B2%7D&bg=ffffff&fg=333333&s=0&c=20201002).
Even though I can motivate it in retrospect, ChatGPT’s idea to use ![Image 214: h^2](https://s0.wp.com/latex.php?latex=h%5E2&bg=ffffff&fg=333333&s=0&c=20201002)-dissociated sets to control relations of order at most ![Image 215: h](https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=333333&s=0&c=20201002) feels quite ingenious. As far as I can tell, this idea is completely original.
ChatGPT’s proof that its construction produces the desired values of ![Image 216: |hA|](https://s0.wp.com/latex.php?latex=%7ChA%7C&bg=ffffff&fg=333333&s=0&c=20201002) is very similar to my proof that the sets ![Image 217: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002) which I construct achieve all possible values of ![Image 218: |hA|](https://s0.wp.com/latex.php?latex=%7ChA%7C&bg=ffffff&fg=333333&s=0&c=20201002), after replacing ![Image 219: S](https://s0.wp.com/latex.php?latex=S&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 220: T](https://s0.wp.com/latex.php?latex=T&bg=ffffff&fg=333333&s=0&c=20201002) by ![Image 221: G](https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 222: H](https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=333333&s=0&c=20201002), respectively. Properties (a)-(d) capture many of the important properties of ![Image 223: S](https://s0.wp.com/latex.php?latex=S&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 224: T](https://s0.wp.com/latex.php?latex=T&bg=ffffff&fg=333333&s=0&c=20201002) (or ![Image 225: G](https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 226: H](https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=333333&s=0&c=20201002)) which are used in this proof. The final constructions involve combining the sets ![Image 227: G](https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 228: H](https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=333333&s=0&c=20201002) (or ![Image 229: S](https://s0.wp.com/latex.php?latex=S&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 230: T](https://s0.wp.com/latex.php?latex=T&bg=ffffff&fg=333333&s=0&c=20201002) in my paper) for each value of ![Image 231: m](https://s0.wp.com/latex.php?latex=m&bg=ffffff&fg=333333&s=0&c=20201002) between ![Image 232: 2](https://s0.wp.com/latex.php?latex=2&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 233: h](https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=333333&s=0&c=20201002) with another set which is the union of an arithmetic progression and a point. Intuitively, ![Image 234: G](https://s0.wp.com/latex.php?latex=G&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 235: H](https://s0.wp.com/latex.php?latex=H&bg=ffffff&fg=333333&s=0&c=20201002) (or ![Image 236: S](https://s0.wp.com/latex.php?latex=S&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 237: T](https://s0.wp.com/latex.php?latex=T&bg=ffffff&fg=333333&s=0&c=20201002)) have large sumsets, while arithmetic progressions have small sumsets, so it is plausible that one could get sets which achieve all the medium-sized sumsets by combining them. However, the proof of this is quite involved, and it occupies Section 4 of [my paper](https://arxiv.org/pdf/2510.23022) and the entirety of the ChatGPT preprint. In Appendix 2, I work out the details of the ChatGPT construction to show that for ![Image 238: k](https://s0.wp.com/latex.php?latex=k&bg=ffffff&fg=333333&s=0&c=20201002) sufficiently large,
![Image 239: \displaystyle N(h,k) \leq O\left(k^{10h^3}\right).](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+N%28h%2Ck%29+%5Cleq+O%5Cleft%28k%5E%7B10h%5E3%7D%5Cright%29.&bg=ffffff&fg=333333&s=0&c=20201002)
For comparison, it is easy to see that ![Image 240: N(h,k)](https://s0.wp.com/latex.php?latex=N%28h%2Ck%29&bg=ffffff&fg=333333&s=0&c=20201002) is at least on the order of ![Image 241: k^{h}](https://s0.wp.com/latex.php?latex=k%5E%7Bh%7D&bg=ffffff&fg=333333&s=0&c=20201002), and it is unknown what the real value is. In Appendix 3, I give details of the correspondence between my paper and the ChatGPT preprint, which will be helpful for those who want to read either.
Finally, I want to express my deep gratitude to Tim for allowing me to contribute to this blog. I am still stunned by the coincidence that the problem he chose to put into ChatGPT 5.5 Pro led him to my paper on the arXiv.
## Tim on what this means for mathematical research
I would judge the level of the result that ChatGPT found in under two hours to be that of a perfectly reasonable chapter in a combinatorics PhD. It wouldn’t be considered an amazing result, since it leant very heavily on Isaac’s ideas, but it was definitely a non-trivial extension of those ideas, and for a PhD student to find that extension it would be necessary to invest quite a bit of time digesting Isaac’s paper, looking for places where it might not be optimal, familiarizing oneself with various algebraic techniques that he used, and so on.
It seems to me that training beginning PhD students to do research, which has always been hard (unless one is lucky enough, as I have often been, to have a student who just seems to get it and therefore doesn’t need in any sense to be trained), has just got harder, since one obvious way to help somebody get started is to give them a problem that looks as though it might be a relatively gentle one. If LLMs are at the point where they can solve “gentle problems”, then that is no longer an option. The lower bound for contributing to mathematics will now be to prove something that LLMs can’t prove, rather than simply to prove something that nobody has proved up to now and that at least somebody finds interesting.
I would qualify that statement in two ways though. First, there is the obvious point that a beginning PhD student has the option of using LLMs. So the task is potentially easier than proving something that LLMs can’t prove: it is proving something _in collaboration with LLMs_ that LLMs cannot manage on their own. I have done quite a lot of such collaboration recently and found that LLMs have made useful contributions without (yet) having game-changing ideas.
A second point is that I don’t know how much of what I have said generalizes to other areas of mathematics. Combinatorics tends to be quite focused on problems: you start with a question and you reason back from the question or if you reason forwards you do so very much with the question in mind. In other areas there can be much more of an emphasis on forwards reasoning: you start with a circle of ideas and see where it leads. To do it successfully, you need to have some way of discriminating between interesting observations and uninteresting ones, and it isn’t obvious to me what LLMs would be like at that.
Of course, everything I am saying concerns LLMs as they are right now. But they are developing so fast that it seems almost certain that my comments will go out of date in a matter of months. It is also almost certain that these developments will have a profoundly disruptive effect on how we go about mathematical research, and especially on how we introduce newcomers to it. Somebody starting a PhD next academic year will be finishing it in 2029 at the earliest, and my guess is that by then what it means to undertake research in mathematics will have changed out of all recognition.
I sometimes get emails from people who are interested in doing mathematical research but are not sure whether that makes sense any more as an aspiration. I have a view on that question, but it may very well change in response to further developments. That view is that there is still a great deal of value in struggling with a mathematics problem, but that the era where you could enjoy the thrill of having your name forever associated with a particular theorem or definition may well be close to its end. So if your aim in doing mathematics is to achieve some kind of immortality, so to speak, then you should understand that that won’t necessarily be possible for much longer — not just for you, but for anybody. Here’s a thought experiment: suppose that a mathematician solved a major problem by having a long exchange with an LLM in which the mathematician played a useful guiding role but the LLM did all the technical work and had the main ideas. Would we regard that as a major achievement of the mathematician? I don’t think we would.
So what is the point of struggling with a difficult mathematics problem? One answer is that it can be very satisfying to solve a problem even if the answer is already known, but I don’t think that is a sufficient reason to spend several years of your life on this peculiar activity. A better answer is that by solving hard problems you get an insight into the problem-solving process itself, at least in your area of expertise, in a way that you simply don’t if all you do is read other people’s solutions. One consequence of this is that people who have themselves solved difficult problems are likely to be significantly better at using solving problems with the help of AI, just as very good coders are better at vibe coding than not such good coders, or people who have a solid grasp of how to do basic arithmetic are likely to be more skilled at using calculators (and especially at noticing when an answer feels off). Mathematics is a highly transferable skill, and that applies to research-level mathematics as well. By doing research in mathematics, you may not get the same rewards as your equivalents a generation ago, but there is a good chance that you will be equipping yourself very well for the world we are about to experience.
## Appendix 1 (Isaac)
We will construct an ![Image 242: h](https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=333333&s=0&c=20201002)-dissociated set ![Image 243: U = \{u_1,\ldots,u_r\} \subseteq \{0,1,2,\ldots,N\}](https://s0.wp.com/latex.php?latex=U+%3D+%5C%7Bu_1%2C%5Cldots%2Cu_r%5C%7D+%5Csubseteq+%5C%7B0%2C1%2C2%2C%5Cldots%2CN%5C%7D&bg=ffffff&fg=333333&s=0&c=20201002), where ![Image 244: N](https://s0.wp.com/latex.php?latex=N&bg=ffffff&fg=333333&s=0&c=20201002) is approximately ![Image 245: r^{h}](https://s0.wp.com/latex.php?latex=r%5E%7Bh%7D&bg=ffffff&fg=333333&s=0&c=20201002). This construction is a very minor modification of Bose–Chowla (1963)’s construction of a ![Image 246: B_h](https://s0.wp.com/latex.php?latex=B_h&bg=ffffff&fg=333333&s=0&c=20201002) set, which I learned about from [this paper](https://arxiv.org/abs/2308.12406). For whatever reason, the GPT preprint (Lemma 3.1) uses a different, less efficient construction using moment curves.
Let ![Image 247: p > r](https://s0.wp.com/latex.php?latex=p+%3E+r&bg=ffffff&fg=333333&s=0&c=20201002) be a prime, let ![Image 248: N = p^{h+1}-2](https://s0.wp.com/latex.php?latex=N+%3D+p%5E%7Bh%2B1%7D-2&bg=ffffff&fg=333333&s=0&c=20201002), let ![Image 249: K](https://s0.wp.com/latex.php?latex=K&bg=ffffff&fg=333333&s=0&c=20201002) be the finite field with ![Image 250: p^{h+1}](https://s0.wp.com/latex.php?latex=p%5E%7Bh%2B1%7D&bg=ffffff&fg=333333&s=0&c=20201002) elements and fix a generator ![Image 251: \theta](https://s0.wp.com/latex.php?latex=%5Ctheta&bg=ffffff&fg=333333&s=0&c=20201002) of ![Image 252: K^\times](https://s0.wp.com/latex.php?latex=K%5E%5Ctimes&bg=ffffff&fg=333333&s=0&c=20201002), so that ![Image 253: K^\times](https://s0.wp.com/latex.php?latex=K%5E%5Ctimes&bg=ffffff&fg=333333&s=0&c=20201002) is equal to ![Image 254: \{\theta^0,\theta^1,\ldots, \theta^N\}](https://s0.wp.com/latex.php?latex=%5C%7B%5Ctheta%5E0%2C%5Ctheta%5E1%2C%5Cldots%2C+%5Ctheta%5EN%5C%7D&bg=ffffff&fg=333333&s=0&c=20201002). Define a set of ![Image 255: p](https://s0.wp.com/latex.php?latex=p&bg=ffffff&fg=333333&s=0&c=20201002) elements
![Image 256: \displaystyle U = \{a \in \{0,1,2,\ldots,N\}: \theta^a - \theta \in \mathbb{F}_p\}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+U+%3D+%5C%7Ba+%5Cin+%5C%7B0%2C1%2C2%2C%5Cldots%2CN%5C%7D%3A+%5Ctheta%5Ea+-+%5Ctheta+%5Cin+%5Cmathbb%7BF%7D_p%5C%7D.&bg=ffffff&fg=333333&s=0&c=20201002)
Then, each element ![Image 257: a \in U](https://s0.wp.com/latex.php?latex=a+%5Cin+U&bg=ffffff&fg=333333&s=0&c=20201002) corresponds to a unique value of ![Image 258: \tilde{a} \in \mathbb{F}_p](https://s0.wp.com/latex.php?latex=%5Ctilde%7Ba%7D+%5Cin+%5Cmathbb%7BF%7D_p&bg=ffffff&fg=333333&s=0&c=20201002), by taking ![Image 259: \tilde{a} = \theta^a - \theta](https://s0.wp.com/latex.php?latex=%5Ctilde%7Ba%7D+%3D+%5Ctheta%5Ea+-+%5Ctheta&bg=ffffff&fg=333333&s=0&c=20201002). Now an additive relation of the form in (4) with ![Image 260: s,s' \leq h](https://s0.wp.com/latex.php?latex=s%2Cs%27+%5Cleq+h&bg=ffffff&fg=333333&s=0&c=20201002) can be reframed by taking powers of ![Image 261: \theta](https://s0.wp.com/latex.php?latex=%5Ctheta&bg=ffffff&fg=333333&s=0&c=20201002) as
![Image 262: \displaystyle (\theta + \tilde{x_1})(\theta + \tilde{x_2})\cdots (\theta + \tilde{x_s}) = (\theta + \tilde{y_1})(\theta + \tilde{y_2})\cdots (\theta + \tilde{y_{s'}}). \qquad (6)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%28%5Ctheta+%2B+%5Ctilde%7Bx_1%7D%29%28%5Ctheta+%2B+%5Ctilde%7Bx_2%7D%29%5Ccdots+%28%5Ctheta+%2B+%5Ctilde%7Bx_s%7D%29+%3D+%28%5Ctheta+%2B+%5Ctilde%7By_1%7D%29%28%5Ctheta+%2B+%5Ctilde%7By_2%7D%29%5Ccdots+%28%5Ctheta+%2B+%5Ctilde%7By_%7Bs%27%7D%7D%29.+%5Cqquad+%286%29&bg=ffffff&fg=333333&s=0&c=20201002)
As ![Image 263: K](https://s0.wp.com/latex.php?latex=K&bg=ffffff&fg=333333&s=0&c=20201002) is a degree-![Image 264: h+1](https://s0.wp.com/latex.php?latex=h%2B1&bg=ffffff&fg=333333&s=0&c=20201002) extension of ![Image 265: \mathbb{F}\sb{p}](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BF%7D%5Csb%7Bp%7D&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 266: \theta](https://s0.wp.com/latex.php?latex=%5Ctheta&bg=ffffff&fg=333333&s=0&c=20201002) is a generator of ![Image 267: K](https://s0.wp.com/latex.php?latex=K&bg=ffffff&fg=333333&s=0&c=20201002) as an ![Image 268: \mathbb{F}\sb{p}](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BF%7D%5Csb%7Bp%7D&bg=ffffff&fg=333333&s=0&c=20201002)-extension, this means that ![Image 269: \theta](https://s0.wp.com/latex.php?latex=%5Ctheta&bg=ffffff&fg=333333&s=0&c=20201002) does not satisfy any nonzero polynomials in ![Image 270: \mathbb{F}\sb{p}[x]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BF%7D%5Csb%7Bp%7D%5Bx%5D&bg=ffffff&fg=333333&s=0&c=20201002) of degree ![Image 271: \leq h](https://s0.wp.com/latex.php?latex=%5Cleq+h&bg=ffffff&fg=333333&s=0&c=20201002). So, both sides of (6) are identical as polynomials in ![Image 272: \mathbb{F}_{p}[\theta]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BF%7D_%7Bp%7D%5B%5Ctheta%5D&bg=ffffff&fg=333333&s=0&c=20201002) and thus the additive relation in (4) is trivial. So, ![Image 273: U](https://s0.wp.com/latex.php?latex=U&bg=ffffff&fg=333333&s=0&c=20201002) is ![Image 274: h](https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=333333&s=0&c=20201002)-dissociated, and of course one can prune a few elements to reduce ![Image 275: U](https://s0.wp.com/latex.php?latex=U&bg=ffffff&fg=333333&s=0&c=20201002) to size ![Image 276: r](https://s0.wp.com/latex.php?latex=r&bg=ffffff&fg=333333&s=0&c=20201002).
## Appendix 2 (Isaac)
Fix constants ![Image 277: \alpha,\beta,\gamma](https://s0.wp.com/latex.php?latex=%5Calpha%2C%5Cbeta%2C%5Cgamma&bg=ffffff&fg=333333&s=0&c=20201002) such that ![Image 278: 0.5 < \beta\gamma < \beta < \alpha < 1](https://s0.wp.com/latex.php?latex=0.5+%3C+%5Cbeta%5Cgamma+%3C+%5Cbeta+%3C+%5Calpha+%3C+1&bg=ffffff&fg=333333&s=0&c=20201002) (in my paper I arbitrarily chose ![Image 279: (\alpha,\beta,\gamma) = (0.9,0.8,0.7)](https://s0.wp.com/latex.php?latex=%28%5Calpha%2C%5Cbeta%2C%5Cgamma%29+%3D+%280.9%2C0.8%2C0.7%29&bg=ffffff&fg=333333&s=0&c=20201002)). Let the two sets in (5) be called ![Image 280: G_{m,r}](https://s0.wp.com/latex.php?latex=G_%7Bm%2Cr%7D&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 281: H_{m,r}](https://s0.wp.com/latex.php?latex=H_%7Bm%2Cr%7D&bg=ffffff&fg=333333&s=0&c=20201002). Let ![Image 282: [a,b]](https://s0.wp.com/latex.php?latex=%5Ba%2Cb%5D&bg=ffffff&fg=333333&s=0&c=20201002) denote the set of integers ![Image 283: x](https://s0.wp.com/latex.php?latex=x&bg=ffffff&fg=333333&s=0&c=20201002) satisfying ![Image 284: a \leq x \leq b](https://s0.wp.com/latex.php?latex=a+%5Cleq+x+%5Cleq+b&bg=ffffff&fg=333333&s=0&c=20201002). Similarly to my paper, the constructions of ![Image 285: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002) such that ![Image 286: hA](https://s0.wp.com/latex.php?latex=hA&bg=ffffff&fg=333333&s=0&c=20201002) achieves the desired sizes will combine sets of the following four types:
*   ![Image 287: B_{j,b} := [0,b-2] \cup \{b-2+j\}](https://s0.wp.com/latex.php?latex=B_%7Bj%2Cb%7D+%3A%3D+%5B0%2Cb-2%5D+%5Ccup+%5C%7Bb-2%2Bj%5C%7D&bg=ffffff&fg=333333&s=0&c=20201002) with choices of ![Image 288: b \in [3, k-k^\gamma]](https://s0.wp.com/latex.php?latex=b+%5Cin+%5B3%2C+k-k%5E%5Cgamma%5D&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 289: j \in [1,hb]](https://s0.wp.com/latex.php?latex=j+%5Cin+%5B1%2Chb%5D&bg=ffffff&fg=333333&s=0&c=20201002).
*   ![Image 290: G_{m,r_m}](https://s0.wp.com/latex.php?latex=G_%7Bm%2Cr_m%7D&bg=ffffff&fg=333333&s=0&c=20201002) for each value of ![Image 291: m \in [3, h]](https://s0.wp.com/latex.php?latex=m+%5Cin+%5B3%2C+h%5D&bg=ffffff&fg=333333&s=0&c=20201002), with choices of ![Image 292: r_m \in [0, (k-b)^\alpha]](https://s0.wp.com/latex.php?latex=r_m+%5Cin+%5B0%2C+%28k-b%29%5E%5Calpha%5D&bg=ffffff&fg=333333&s=0&c=20201002).
*   ![Image 293: H_{m,u_m}](https://s0.wp.com/latex.php?latex=H_%7Bm%2Cu_m%7D&bg=ffffff&fg=333333&s=0&c=20201002) for each value of ![Image 294: m \in [2,h-1]](https://s0.wp.com/latex.php?latex=m+%5Cin+%5B2%2Ch-1%5D&bg=ffffff&fg=333333&s=0&c=20201002), with choices of ![Image 295: u_m \in [0, (k-b)^\beta]](https://s0.wp.com/latex.php?latex=u_m+%5Cin+%5B0%2C+%28k-b%29%5E%5Cbeta%5D&bg=ffffff&fg=333333&s=0&c=20201002).
*   A ![Image 296: B_h](https://s0.wp.com/latex.php?latex=B_h&bg=ffffff&fg=333333&s=0&c=20201002) set of the correct size so that ![Image 297: |A| = k](https://s0.wp.com/latex.php?latex=%7CA%7C+%3D+k&bg=ffffff&fg=333333&s=0&c=20201002).
One reason that this construction needs to be complicated is that we need to create at least ![Image 298: \Omega(k^h)](https://s0.wp.com/latex.php?latex=%5COmega%28k%5Eh%29&bg=ffffff&fg=333333&s=0&c=20201002) many sets. To do this, we vary ![Image 299: 2h-4](https://s0.wp.com/latex.php?latex=2h-4&bg=ffffff&fg=333333&s=0&c=20201002) parameters ![Image 300: r_m](https://s0.wp.com/latex.php?latex=r_m&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 301: u_m](https://s0.wp.com/latex.php?latex=u_m&bg=ffffff&fg=333333&s=0&c=20201002) in the domain ![Image 302: [0,k^\alpha]](https://s0.wp.com/latex.php?latex=%5B0%2Ck%5E%5Calpha%5D&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 303: 2](https://s0.wp.com/latex.php?latex=2&bg=ffffff&fg=333333&s=0&c=20201002) parameters ![Image 304: b](https://s0.wp.com/latex.php?latex=b&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 305: j](https://s0.wp.com/latex.php?latex=j&bg=ffffff&fg=333333&s=0&c=20201002) in the domain ![Image 306: [1,hk]](https://s0.wp.com/latex.php?latex=%5B1%2Chk%5D&bg=ffffff&fg=333333&s=0&c=20201002). We can choose ![Image 307: \alpha](https://s0.wp.com/latex.php?latex=%5Calpha&bg=ffffff&fg=333333&s=0&c=20201002) to be slightly bigger than ![Image 308: 1/2](https://s0.wp.com/latex.php?latex=1%2F2&bg=ffffff&fg=333333&s=0&c=20201002), and then the above construction gives us ![Image 309: O(k^{\alpha(2h-4)+ 2})=O(k^{h + \delta})](https://s0.wp.com/latex.php?latex=O%28k%5E%7B%5Calpha%282h-4%29%2B+2%7D%29%3DO%28k%5E%7Bh+%2B+%5Cdelta%7D%29&bg=ffffff&fg=333333&s=0&c=20201002) different sets where ![Image 310: \delta >0](https://s0.wp.com/latex.php?latex=%5Cdelta+%3E0&bg=ffffff&fg=333333&s=0&c=20201002) can be made arbitrarily small. So, if we were to remove any of the above parameters from the construction, and not change the others, this construction would no longer create ![Image 311: \Omega(k^h)](https://s0.wp.com/latex.php?latex=%5COmega%28k%5Eh%29&bg=ffffff&fg=333333&s=0&c=20201002) many sets. In comparison, Nathanson’s construction when ![Image 312: h=2](https://s0.wp.com/latex.php?latex=h%3D2&bg=ffffff&fg=333333&s=0&c=20201002) only needs to create ![Image 313: \Omega(k^2)](https://s0.wp.com/latex.php?latex=%5COmega%28k%5E2%29&bg=ffffff&fg=333333&s=0&c=20201002) sets. He does this by combining a Sidon set, an arithmetic progression, and one extra value, and varying the size of the arithmetic progression and the extra value in ranges of size ![Image 314: O(k)](https://s0.wp.com/latex.php?latex=O%28k%29&bg=ffffff&fg=333333&s=0&c=20201002).
We want to combine ![Image 315: q = 2h-2](https://s0.wp.com/latex.php?latex=q+%3D+2h-2&bg=ffffff&fg=333333&s=0&c=20201002) sets ![Image 316: A_1,\ldots,A_q](https://s0.wp.com/latex.php?latex=A_1%2C%5Cldots%2CA_q&bg=ffffff&fg=333333&s=0&c=20201002), which are given by ![Image 317: B_{j,b}](https://s0.wp.com/latex.php?latex=B_%7Bj%2Cb%7D&bg=ffffff&fg=333333&s=0&c=20201002), ![Image 318: G_{m,r_m}](https://s0.wp.com/latex.php?latex=G_%7Bm%2Cr_m%7D&bg=ffffff&fg=333333&s=0&c=20201002) for the ![Image 319: h-2](https://s0.wp.com/latex.php?latex=h-2&bg=ffffff&fg=333333&s=0&c=20201002) values of ![Image 320: m \in [3,h]](https://s0.wp.com/latex.php?latex=m+%5Cin+%5B3%2Ch%5D&bg=ffffff&fg=333333&s=0&c=20201002), ![Image 321: H_{m,u_m}](https://s0.wp.com/latex.php?latex=H_%7Bm%2Cu_m%7D&bg=ffffff&fg=333333&s=0&c=20201002) for the ![Image 322: h-2](https://s0.wp.com/latex.php?latex=h-2&bg=ffffff&fg=333333&s=0&c=20201002) values of ![Image 323: m \in [2,h-1]](https://s0.wp.com/latex.php?latex=m+%5Cin+%5B2%2Ch-1%5D&bg=ffffff&fg=333333&s=0&c=20201002), and a ![Image 324: B_h](https://s0.wp.com/latex.php?latex=B_h&bg=ffffff&fg=333333&s=0&c=20201002) set. By Appendix 1, for all ![Image 325: r \leq k](https://s0.wp.com/latex.php?latex=r+%5Cleq+k&bg=ffffff&fg=333333&s=0&c=20201002), there exists a ![Image 326: h^2](https://s0.wp.com/latex.php?latex=h%5E2&bg=ffffff&fg=333333&s=0&c=20201002)-dissociated set ![Image 327: {u_{1},\ldots,u_{r}}](https://s0.wp.com/latex.php?latex=%7Bu_%7B1%7D%2C%5Cldots%2Cu_%7Br%7D%7D&bg=ffffff&fg=333333&s=0&c=20201002) of diameter ![Image 328: M \leq r^{2h^2} \leq k^{2h^2}](https://s0.wp.com/latex.php?latex=M+%5Cleq+r%5E%7B2h%5E2%7D+%5Cleq+k%5E%7B2h%5E2%7D&bg=ffffff&fg=333333&s=0&c=20201002). By the constructions of ![Image 329: G_{m,r_m}](https://s0.wp.com/latex.php?latex=G_%7Bm%2Cr_m%7D&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 330: H_{m,u_m}](https://s0.wp.com/latex.php?latex=H_%7Bm%2Cu_m%7D&bg=ffffff&fg=333333&s=0&c=20201002), we can take each ![Image 331: A_i \subseteq [0,M]](https://s0.wp.com/latex.php?latex=A_i+%5Csubseteq+%5B0%2CM%5D&bg=ffffff&fg=333333&s=0&c=20201002), where ![Image 332: M \leq hk^{2h^2}](https://s0.wp.com/latex.php?latex=M+%5Cleq+hk%5E%7B2h%5E2%7D&bg=ffffff&fg=333333&s=0&c=20201002). Let ![Image 333: \mathbb{Z}^{2q}](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BZ%7D%5E%7B2q%7D&bg=ffffff&fg=333333&s=0&c=20201002) have basis vectors ![Image 334: e_1,\ldots,e_{2q}](https://s0.wp.com/latex.php?latex=e_1%2C%5Cldots%2Ce_%7B2q%7D&bg=ffffff&fg=333333&s=0&c=20201002). To combine ![Image 335: A_1,\ldots,A_q](https://s0.wp.com/latex.php?latex=A_1%2C%5Cldots%2CA_q&bg=ffffff&fg=333333&s=0&c=20201002), we can define ![Image 336: A \subseteq \mathbb{Z}^{2q}](https://s0.wp.com/latex.php?latex=A+%5Csubseteq+%5Cmathbb%7BZ%7D%5E%7B2q%7D&bg=ffffff&fg=333333&s=0&c=20201002) as
![Image 337: \displaystyle A = \bigcup_{i=1}^q (A_i e_i + e_{q+i}) \subseteq \{0,1,2,\ldots,M\}^{2q} \subseteq \mathbb{Z}^{2q}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+A+%3D+%5Cbigcup_%7Bi%3D1%7D%5Eq+%28A_i+e_i+%2B+e_%7Bq%2Bi%7D%29+%5Csubseteq+%5C%7B0%2C1%2C2%2C%5Cldots%2CM%5C%7D%5E%7B2q%7D+%5Csubseteq+%5Cmathbb%7BZ%7D%5E%7B2q%7D.&bg=ffffff&fg=333333&s=0&c=20201002)
Similarly to my Lemma 4.9, this construction ensures that the generating function product ![Image 338: \mathcal{F}_{A}(z) = \prod_{i=1}^q \mathcal{F}_{A_i}(z)](https://s0.wp.com/latex.php?latex=%5Cmathcal%7BF%7D_%7BA%7D%28z%29+%3D+%5Cprod_%7Bi%3D1%7D%5Eq+%5Cmathcal%7BF%7D_%7BA_i%7D%28z%29&bg=ffffff&fg=333333&s=0&c=20201002) holds, which is the identity that both my paper and the GPT preprint use (see either paper for a definition of these generating functions). By (the standard) Lemma 2.3 of the GPT preprint, ![Image 339: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002) is Freiman-isomorphic of order ![Image 340: h](https://s0.wp.com/latex.php?latex=h&bg=ffffff&fg=333333&s=0&c=20201002) to a subset of ![Image 341: [0,2qM(2hM)^{2q-1}]](https://s0.wp.com/latex.php?latex=%5B0%2C2qM%282hM%29%5E%7B2q-1%7D%5D&bg=ffffff&fg=333333&s=0&c=20201002). Therefore, for ![Image 342: k](https://s0.wp.com/latex.php?latex=k&bg=ffffff&fg=333333&s=0&c=20201002) sufficiently large (the whole construction relies on this for the same reasons as in my paper),
![Image 343: \displaystyle N(h,k) \leq 2qM(2hM)^{2q-1} \leq 2\left(2h^2k^{2h^2}\right)^{2(2h-2)} \leq k^{10 h^3}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+N%28h%2Ck%29+%5Cleq+2qM%282hM%29%5E%7B2q-1%7D+%5Cleq+2%5Cleft%282h%5E2k%5E%7B2h%5E2%7D%5Cright%29%5E%7B2%282h-2%29%7D+%5Cleq+k%5E%7B10+h%5E3%7D.&bg=ffffff&fg=333333&s=0&c=20201002)
## Appendix 3 (Isaac)
In Section 4.2 of my paper, I use a different, simpler construction to construct sets ![Image 344: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002) achieving the values in ![Image 345: \mathcal{R}(h,k)](https://s0.wp.com/latex.php?latex=%5Cmathcal%7BR%7D%28h%2Ck%29&bg=ffffff&fg=333333&s=0&c=20201002) which have ![Image 346: |hA| < \varepsilon k^h](https://s0.wp.com/latex.php?latex=%7ChA%7C+%3C+%5Cvarepsilon+k%5Eh&bg=ffffff&fg=333333&s=0&c=20201002), for some small ![Image 347: \varepsilon](https://s0.wp.com/latex.php?latex=%5Cvarepsilon&bg=ffffff&fg=333333&s=0&c=20201002). These sets ![Image 348: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002) are subsets of ![Image 349: {0,1,2,\ldots,k^h}](https://s0.wp.com/latex.php?latex=%7B0%2C1%2C2%2C%5Cldots%2Ck%5Eh%7D&bg=ffffff&fg=333333&s=0&c=20201002), meaning that all elements have polynomial size in ![Image 350: k](https://s0.wp.com/latex.php?latex=k&bg=ffffff&fg=333333&s=0&c=20201002). This is observed in Section 5 of the GPT preprint.
Section 4.3 of my paper carries out the construction which combines many components including ![Image 351: S](https://s0.wp.com/latex.php?latex=S&bg=ffffff&fg=333333&s=0&c=20201002) and ![Image 352: T](https://s0.wp.com/latex.php?latex=T&bg=ffffff&fg=333333&s=0&c=20201002). This corresponds to Sections 2, 3, 4, and 6 of the GPT preprint. This section has a lot of moving parts; I give an outline in Section 4.3.1.
In Section 4.3.2, I describe how the different components will be combined, using a construction which I call the disjoint union, and introduce generating functions ![Image 353: \mathcal{F}_A(z)](https://s0.wp.com/latex.php?latex=%5Cmathcal%7BF%7D_A%28z%29&bg=ffffff&fg=333333&s=0&c=20201002) as a bookkeeping tool to keep track of the sumset sizes of a set ![Image 354: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002). This corresponds to Section 2 and Section 4 of the GPT preprint.
In Section 4.3.3, I compute the generating function of each of the component sets, including ![Image 355: \mathcal{F}_S(z)](https://s0.wp.com/latex.php?latex=%5Cmathcal%7BF%7D_S%28z%29&bg=ffffff&fg=333333&s=0&c=20201002) (Lemma 4.15) and ![Image 356: \mathcal{F}_T(z)](https://s0.wp.com/latex.php?latex=%5Cmathcal%7BF%7D_T%28z%29&bg=ffffff&fg=333333&s=0&c=20201002) (Lemma 4.17). This corresponds to Section 3 and Section 6.1 of the GPT preprint. In particular, ![Image 357: \mathcal{F}_{G}(z)](https://s0.wp.com/latex.php?latex=%5Cmathcal%7BF%7D_%7BG%7D%28z%29&bg=ffffff&fg=333333&s=0&c=20201002) is computed in Lemma 3.3 and ![Image 358: \mathcal{F}_{H}(z)](https://s0.wp.com/latex.php?latex=%5Cmathcal%7BF%7D_%7BH%7D%28z%29&bg=ffffff&fg=333333&s=0&c=20201002) is computed in Lemma 3.4. Once these generating functions have been computed, the remainder of the proof is almost identical in my paper and in the GPT preprint.
In Section 4.3.4, I put all the pieces together to show that as we range over the sets ![Image 359: A](https://s0.wp.com/latex.php?latex=A&bg=ffffff&fg=333333&s=0&c=20201002) which I have constructed, the values of ![Image 360: |hA|](https://s0.wp.com/latex.php?latex=%7ChA%7C&bg=ffffff&fg=333333&s=0&c=20201002) will assume all of the elements of ![Image 361: {\lceil\varepsilon k^h\rceil, \lceil\varepsilon k^h\rceil+1,\ldots ,\binom{h+k-1}{h} }](https://s0.wp.com/latex.php?latex=%7B%5Clceil%5Cvarepsilon+k%5Eh%5Crceil%2C+%5Clceil%5Cvarepsilon+k%5Eh%5Crceil%2B1%2C%5Cldots+%2C%5Cbinom%7Bh%2Bk-1%7D%7Bh%7D+%7D&bg=ffffff&fg=333333&s=0&c=20201002). The key idea is to show that the set of all values of ![Image 362: |hA|](https://s0.wp.com/latex.php?latex=%7ChA%7C&bg=ffffff&fg=333333&s=0&c=20201002) forms an interval, and contains numbers both smaller than ![Image 363: \varepsilon k^h](https://s0.wp.com/latex.php?latex=%5Cvarepsilon+k%5Eh&bg=ffffff&fg=333333&s=0&c=20201002) and equal to ![Image 364: \binom{h+k-1}{h}](https://s0.wp.com/latex.php?latex=%5Cbinom%7Bh%2Bk-1%7D%7Bh%7D&bg=ffffff&fg=333333&s=0&c=20201002).
### Share this:
*   [Share on X (Opens in new window)X](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?share=twitter)
*   [Share on Facebook (Opens in new window)Facebook](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?share=facebook)
Like Loading...
[](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/)
### _Related_
Tags: [ai](https://gowers.wordpress.com/tag/ai/), [mathematics](https://gowers.wordpress.com/tag/mathematics/)
This entry was posted on May 8, 2026 at 4:40 pm and is filed under [Computing](https://gowers.wordpress.com/category/computing/), [Straight maths](https://gowers.wordpress.com/category/straight-maths/). You can follow any responses to this entry through the [RSS 2.0](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/feed/) feed. You can [leave a response](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#respond), or [trackback](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/trackback/) from your own site.
### 74 Responses to “A recent experience with ChatGPT 5.5 Pro”
1.   ![Image 365: Moses Charikar's avatar](https://1.gravatar.com/avatar/1ad27471953d667adeb9d370b4ef51c7fe9880b2a43b441e34f3007389764fe6?s=32&d=identicon)[Moses Charikar](https://profiles.stanford.edu/moses-charikar) Says: 
[May 8, 2026 at 5:14 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540038) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540038#respond)
Tim, in Terry Tao’s recent talk at the Future of Mathematics symposium at Stanford, he also suggested that perhaps we ought to have different venues for AI generated mathematics versus human mathematics, making an analogy to a highway vs a pedestrian walkway: [https://www.youtube.com/live/tN4hsT5t0nw?si=cIQj2Di6sNdZHr7P&t=6330](https://www.youtube.com/live/tN4hsT5t0nw?si=cIQj2Di6sNdZHr7P&t=6330)
2.   ![Image 366: Manu's avatar](https://1.gravatar.com/avatar/d684d9cd7ee82b9c4ead9ce1c11e6945533a040713a0fff7698d8afb03c3c699?s=32&d=identicon)[Manu](http://emanueleviola.wordpress.com/) Says: 
[May 8, 2026 at 5:39 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540039) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540039#respond)
Very interesting post, it will be fun to look back at it in 2029. To some of your points, there’s a famous Italian quote (but not so famous that chatgpt knows who said it): “Chi meglio combina meglio crea.” The literal translation is “Who better combines better creates.” Personally I don’t think there is anything “special” in human intelligence or insight, and like you suggest I feel that a very vast amount of results in math (but also literature etc.) are “banal” in the sense that they are basically a combination of known idea; they can be obtained by tediously trying one idea after the other in the “obvious way.” Papers (in math) are often written (and talks given) to give the opposite impression of phenomenal and inexplicable deus ex machina insight of the author, but in many (most) cases the ideas can be presented in a much more pednatic way (I think you expressed a similar view that ideas always come from somewhere, with the exception of Razborov’s ;-). LLMs obviously excel at this type of combination. Personally I don’t think anyone has a clear idea of the extent to which they will be able to produce without guidance research or art that we humans are interested in, and I am open to various scenarios. What seems clear is that being able to harness these tools is already a key factor. But so far, at the high level this is not very different than Google, or mathematical software. The ability to do quick searches online or use mathematical software has been a key advantage. I’ll add that I have often wondered how to define “banality.” In some sense Kolmogorov complexity seems relevant, if something has a short description given available data, it is banal. Time-bounded Kolmogorov complexity is a better idea. One issue is how to capture “available data.” Trained LLMs seem to give us just that.
3.   ![Image 367: Unknown's avatar](https://0.gravatar.com/avatar/?s=32&d=identicon)Anonymous Says: 
[May 8, 2026 at 7:51 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540040) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540040#respond)
The question of how best to introduce beginning PhD students to research in an LLM-era feels extremely important to think about. I want to highlight that, while it’s true in theory that such students have the option of using LLMs, the top models are currently quite expensive to get access to, and there are internal models at various companies to which only a select few have access. If one goes down the route of ‘PhD students are also allowed to use LLMs’, then it can quickly become a game of ‘which student has access to the best LLMs’, which seems to me extremely unfortunate. Is it an issue that can be gotten round on a global scale?
    *   ![Image 368: jovial95facf5d33's avatar](https://2.gravatar.com/avatar/28683ac6b2c4db4d984c75980eb5ebddb5f78a9dd7f16b402aa69113940166d1?s=32&d=identicon)jovial95facf5d33 Says: 
[May 8, 2026 at 7:52 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540041)
Sorry, I didn’t realise that was anonymous! Best wishes, Olof (Sisask)
    *   ![Image 369: domotorp's avatar](https://2.gravatar.com/avatar/bc8ff7e8d93e0482820a5c53c7d273b536559c176c2a98d90a26c70869a8987c?s=32&d=identicon)[domotorp](https://domotorp.web.elte.hu/) Says: 
[May 9, 2026 at 5:32 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540049)
This raises a very important issue that is relevant to all researchers, not just PhD students. Until now, unlike in most other sciences, to do research level math having access to expensive resources gave almost no advantage (except, of course, having prior access to a good education). That is gone now. I don’t know what will happen in the future, but at this moment the age of equality, in the communist sense, is sadly over in research math.
4.   ![Image 370: gowers's avatar](https://1.gravatar.com/avatar/df33029a240838a35b04d057c5f44700d023485414deef2830a0f6262fe11493?s=32&d=identicon)[gowers](https://gowers.wordpress.com/) Says: 
[May 8, 2026 at 8:01 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540042) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540042#respond)
A quick comment to say that I’m having annoying compilation problems with LaTeX subscripts and superscripts, which have affected Isaac’s appendixes. I will try to sort them out soon, but if anyone has any idea what to do then that would be helpful.
    *   ![Image 371: old-bielefelder's avatar](https://0.gravatar.com/avatar/3dd7258760cfd31c2571662f4926c98fbf1ddffb01e0fc3904837135816d5445?s=32&d=identicon)hopefula0ac40f0f2 Says: 
[May 9, 2026 at 5:54 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540050)
_My standard is to ask the AI (in particular ChatGPT): Write your finding/proof in texfile, also output as pdf. It works savely._
    *   ![Image 372: ateixeira's avatar](https://1.gravatar.com/avatar/794074213a1a611b448ef8dc0623a60a847e74d36b5b4efe4f248b54ee8be80a?s=32&d=identicon)[ateixeira](https://climbingthemountain.wordpress.com/) Says: 
[May 9, 2026 at 7:02 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540082)
I always use Luca Trevisan’s script to convert LaTeX files into friendly wordpress.com html files ([https://lucatrevisan.wordpress.com/latex-to-wordpress/](https://lucatrevisan.wordpress.com/latex-to-wordpress/)). This have some limitations though because wordpress.com LaTeX lacks some packages. Anoter option is to install one LaTeX plugin for wordpress.com (I don’t use any) because usually these allow the the use of other LaTeX packages.
5.   ![Image 373: Bruce Smith's avatar](https://1.gravatar.com/avatar/a898f4a3a078cd5dc63334d6d56225a33a71720754c78693f096098b50678472?s=32&d=identicon)[Bruce Smith](http://oresmus.wordpress.com/) Says: 
[May 8, 2026 at 9:13 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540043) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540043#respond)
> one obvious way to help somebody get started is to give them a problem that looks as though it might be a relatively gentle one. If LLMs are at the point where they can solve “gentle problems”, then that is no longer an option.
I don’t see how this follows. If the student wants to learn, and if you as their advisor suggest it, they will refrain from using the LLM for such an exercise. This won’t produce an “equally original/publishable result” as it would have before, but it should in principle be just as educational as if the LLM didn’t exist. It doesn’t seem too different from how the student in the past would have refrained from asking you for detailed help with the same problem.
> Here’s a thought experiment: suppose that a mathematician solved a major problem by having a long exchange with an LLM in which the mathematician played a useful guiding role but the LLM did all the technical work and had the main ideas. Would we regard that as a major achievement of the mathematician? I don’t think we would.
I think this depends on whether that guidance was also a significant contribution. This will sometimes be hard to judge. But if the problem had been open and interesting, and the LLMs had been generally available, for awhile, that would be evidence in favor. It seems similar to one coauthor playing an important guiding role in a joint work with another one.
6.   ![Image 374: Unknown's avatar](https://0.gravatar.com/avatar/?s=32&d=identicon)Anonymous Says: 
[May 8, 2026 at 9:17 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540044) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540044#respond)
My view on this is really pessimistic. The way things progress, the value of thinking and having deep ideas seems to be lower and lower. Even before AI, institutions questioned if mathematics research was worth it. I wouldn’t recommend anyone to start a PhD now in pure maths.
    *   ![Image 375: old-bielefelder's avatar](https://0.gravatar.com/avatar/3dd7258760cfd31c2571662f4926c98fbf1ddffb01e0fc3904837135816d5445?s=32&d=identicon)[old-bielefelder](https://althofer.de/) Says: 
[May 9, 2026 at 12:10 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540068)
>I wouldn’t recommend anyone to start a PhD now in pure maths.
I see it more positively, but the young candidate would need an open-minded supervisor and the courage to use AI systems full throttle. – Likely, Mathe departments should install new procedures for PhD projects.
Cheers, Ingo.
7.   [最近使用ChatGPT 5.5 Pro的经历 - 偏执的码农](https://geek.ds3783.com/2026/05/%e6%9c%80%e8%bf%91%e4%bd%bf%e7%94%a8chatgpt-5-5-pro%e7%9a%84%e7%bb%8f%e5%8e%86/) Says: 
[May 8, 2026 at 9:22 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540045) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540045#respond)
[…] 详情参考 […]
8.   ![Image 376: Unknown's avatar](https://0.gravatar.com/avatar/?s=32&d=identicon)Anonymous Says: 
[May 8, 2026 at 10:51 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540046) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540046#respond)
You wrote “I understand that arXiv has a policy against accepting AI-written content, which makes good sense to me. So maybe there should be a different repository where AI-produced results can live.” You may find [https://arxiv.org/abs/2604.16476](https://arxiv.org/abs/2604.16476) a step in this direction.
9.   ![Image 377: Phillip's avatar](https://1.gravatar.com/avatar/d0cf18ce757efa61bbd1d597e8b7ca69e374fa673551271e85ae789c1aa8746e?s=32&d=identicon)Phillip Says: 
[May 8, 2026 at 10:54 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540047) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540047#respond)
It’s sad, but really Mathematics is just at the leading edge of a wider phenomenon. We’re going to see similar questions raised for most intellectually fulfilling activities.
10.   ![Image 378: antilli's avatar](https://1.gravatar.com/avatar/d12adc40cbd822096e9583c308ea3bc6b254b46fce71cad2438e5622fbe46c00?s=32&d=identicon)[antilli](http://antilli.wordpress.com/) Says: 
[May 9, 2026 at 12:25 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540048) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540048#respond)
Dear Professor, Respectfully, it would be remarkable if this Chatgpt model were some evolutionary result of the (free) model I have consulted from time to time; it tells me today, when I submit a very brief source, rudimentary arithmetic, and ask for an evaluation of the conclusion, “The conclusion is unproven since, conditionally, the result of a CRT set may be smaller than one of the set of strictly positive base residues”, and sticks to its guns when challenged; the source a demonstration of an Archimedean obstruction which prevents the addition of some divisor th2 to the singleton CRT set under some divisor th1, a candidate “non Brauer Manin obstruction” (Katherine Stange); I have found a method which corrects in at least 6 cases the (free) LLMs’ mishandling of reductio arguments, but not Chatgpt’s; on the off chance you have the time / interest to put the source to this 5.5 Pro version, my email address is registered; Regards, Davide
    *   ![Image 379: old-bielefelder's avatar](https://0.gravatar.com/avatar/3dd7258760cfd31c2571662f4926c98fbf1ddffb01e0fc3904837135816d5445?s=32&d=identicon)hopefula0ac40f0f2 Says: 
[May 9, 2026 at 6:01 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540052)
Often, it helps to use different AIs in pingpong mode: AI 1 thinks to have proved something. Ask it to give output in texfile. This becomes input for AI 2 with the prompt: “Check this proof carefully for correctness. List all errors, gaps, and weaknesses. Ouput in tex file.” If this feedback claims to have found errors or gaps, ask AI 2 for a repair, or give ints answer file back to AI 1, asking: “Here is feed back to your proof attempt…” It works really very often in my research.
    *   ![Image 380: antilli's avatar](https://1.gravatar.com/avatar/d12adc40cbd822096e9583c308ea3bc6b254b46fce71cad2438e5622fbe46c00?s=32&d=identicon)[antilli](http://antilli.wordpress.com/) Says: 
[May 9, 2026 at 11:51 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540089)
Yes: I first talked an LLM through its objections, Agent 1; then opened a chat with the same LLM in a different browser, Agent 2, and pasted in the same source; I copied Agent 2’s objections into Agent 1, asked for rebuttals, copied those into Agent 2, and so on until Agent 1 had guided Agent 2 through a (very easy) reductio; I then had Agent 1 write an LLM guide, mandatory reading, its instructions to anticipate known LLM tendencies; the objections gathered from say a total of 3 independent LLM platforms were sufficient to keep 6 independent LLMs on track, ie at least two had played no part in the “consultation process”; the exercise was to test “Set a thief to catch a thief”, ie set a confabulator/hallucinator to forestall the confabulations/hallucinations of a peer Agent and so neutralise those aspects of LLM workings which cannot be useful, to leave the useful aspects in possession of the field; only a very wee test but at least a satisfactory one; this seems to me rather like your own experience
    *   ![Image 381: Unknown's avatar](https://0.gravatar.com/avatar/?s=32&d=identicon)Anonymous Says: 
[May 9, 2026 at 11:08 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540065)
You don’t need to ask Gowers if he can put the source to this 5.5 Pro version as you phrase it. You can access it yourself at [https://chatgpt.com/](https://chatgpt.com/), click the “select model” dropdown menu. You do need to pay $200 first.
    *   ![Image 382: antilli's avatar](https://1.gravatar.com/avatar/d12adc40cbd822096e9583c308ea3bc6b254b46fce71cad2438e5622fbe46c00?s=32&d=identicon)[antilli](http://antilli.wordpress.com/) Says: 
[May 9, 2026 at 11:55 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540090)
Aha, pay $200 to The People Who Brought Us, “Conditionally, if we have coprime t1, t2, the magnitude of t1^2 – t2*x may be absolute zero”
11.   ![Image 383: Unknown's avatar](https://openclawlog.com/wp-content/uploads/2026/02/cropped-OpenClawLogLogo-1.png?w=32)[A recent experience with ChatGPT 5.5 Pro / 最近使用ChatGPT 5.5 Pro的体验 – OpenClawLog](https://openclawlog.com/2026/05/09/a-recent-experience-with-chatgpt-5-5-pro-%e6%9c%80%e8%bf%91%e4%bd%bf%e7%94%a8chatgpt-5-5-pro%e7%9a%84%e4%bd%93%e9%aa%8c/) Says: 
[May 9, 2026 at 6:00 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540051) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540051#respond)
[…] A recent experience with ChatGPT 5.5 Pro 🔥 12 […]
12.   [最近使用 ChatGPT 5.5 Pro 的經驗 - AI 資訊](https://ai.jiayun.info/%e6%9c%80%e8%bf%91%e4%bd%bf%e7%94%a8-chatgpt-5-5-pro-%e7%9a%84%e7%b6%93%e9%a9%97/) Says: 
[May 9, 2026 at 6:08 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540053) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540053#respond)
[…] [https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/) […]
13.   ![Image 384: Unknown's avatar](https://0.gravatar.com/avatar/?s=32&d=identicon)Anonymous Says: 
[May 9, 2026 at 6:20 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540054) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540054#respond)
I think you must have a typo after “It is an easy exercise to show that”, because it reads “if |A| = k, then 2k-1 <= |A|…” which isn’t even a true statement, let alone an easy exercise…
    *   ![Image 385: gowers's avatar](https://1.gravatar.com/avatar/df33029a240838a35b04d057c5f44700d023485414deef2830a0f6262fe11493?s=32&d=identicon)[gowers](https://gowers.wordpress.com/) Says: 
[May 9, 2026 at 12:46 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540071)
Corrected — thanks.
14.   [ChatGPT 5.5 Proが数学研究の新たな扉を開く！ – ainewsfeed](https://ainewsfeed.net/archives/2570) Says: 
[May 9, 2026 at 6:52 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540055) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540055#respond)
[…] 元記事: [https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/) […]
15.   ![Image 386: Unknown's avatar](https://0.gravatar.com/avatar/?s=32&d=identicon)Anonymous Says: 
[May 9, 2026 at 7:12 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540056) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540056#respond)
I have found interacting with these models to be a rather frustrating experience, though I have only been trying to get it to solve my favorite problems or give me new ideas as to how to solve it. I’ve found myself either trying to filter my way through nonsense or excitedly trying an idea it has, only to be let down for the idea is either trivial or hopeless.
If this is the future of mathematics research-endlessly trying to filter through an LLM’s output looking for something sensible, I’m really not looking forward to this.
16.   ![Image 387: Unknown's avatar](https://0.gravatar.com/avatar/?s=32&d=identicon)Anonymous Says: 
[May 9, 2026 at 7:27 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540057) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540057#respond)
Why do you get access to ChatGPT 5.5 Pro but not everyone else?
17.   [Daily Trend Signal - May 9, 2026 - Daily Trend Signal](https://dailytrendsignal.com/daily/daily-trend-signal-2026-05-09/) Says: 
[May 9, 2026 at 7:51 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540058) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540058#respond)
[…] A recent experience with ChatGPT 5.5 Pro […]
18.   [Hacker News 每日精選 – 2026-05-09 – 小丁的家](https://blog.tomting.com/2026/05/09/hacker-news-%e6%af%8f%e6%97%a5%e7%b2%be%e9%81%b8-2026-05-09/) Says: 
[May 9, 2026 at 8:02 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540059) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540059#respond)
[…] 🔗 閱讀原文 […]
19.   ![Image 388: Unknown's avatar](https://0.gravatar.com/avatar/?s=32&d=identicon)Anonymous Says: 
[May 9, 2026 at 8:49 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540060) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540060#respond)
hello! I am a numberphobe, my friend sent this article, and I skipped the math-y parts but definitely got the main message of this article so good job you
20.   [A recent experience with ChatGPT 5.5 Pro - CodeGurus - CodeGurus](https://codegurus.eu/a-recent-experience-with-chatgpt-5-5-pro/) Says: 
[May 9, 2026 at 9:00 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540061) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540061#respond)
[…] Share on X (Opens in new window) X […]
21.   [AI 攻克博士級數學難題？菲爾茲獎得主 Timothy Gowers 與 ChatGPT 5.5 Pro 的震撼接觸 – CyberQ 賽博客](https://cyberq.tw/2026/05/09/can-ai-solve-doctoral-level/) Says: 
[May 9, 2026 at 9:15 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540062) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540062#respond)
[…] 知名菲爾茲獎得主、數學家 Timothy Gowers 日前在其個人部落格發表了一篇文章《A recent experience with ChatGPT 5.5 Pro》，引起不少關注和討論。他表示，在幾乎沒有人類數學提示的情況下，ChatGPT 5.5 Pro 僅用了短短一小時左右，就產出了一份達到博士生研究水準的數學證明。 […]
22.   ![Image 389: John Baez's avatar](https://1.gravatar.com/avatar/7a940c155bd1e3fbddbf5dff9b93df0588e66d7121832d395ffec41b6e91792b?s=32&d=identicon)[John Baez](http://math.ucr.edu/home/baez/) Says: 
[May 9, 2026 at 10:03 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540064) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540064#respond)
Anonymous wrote:
“My view on this is really pessimistic. The way things progress, the value of thinking and having deep ideas seems to be lower and lower.”
Where does the value of thinking and having deep ideas come from? We need to think about this now. If it comes primarily from their _scarcity_ – the fact that having certain ideas is _hard_– then indeed this value may drop precipitously when the manufacture of ideas can be automated. But if the value comes from the _utility_ of the ideas – the benefit that the idea brings – then the story changes: perhaps creating more good ideas is actually better, not worse. Here I’m using “utility” in a broad sense, not just in the sense of what people often call applied mathematics.
In other words, mathematicians may need to adjust to a transformation from a scarcity economy to an abundance economy.
In a society where food is cheap to produce, people still get paid to make nice food.
Maybe mathematicians will need to pay more attention to convincing people that their work is not only difficult, but good. If it’s truly good, it doesn’t get its value mainly from being difficult.
Then, looking ahead another step, maybe we should think about how good AI is at convincing people that the mathematics it creates is actually good. When AI becomes better at this than humans, that’s another thing we don’t need mathematicians for. But at this point we may be wondering what we need humans for at all. (It’s mainly humans who need humans.)
    *   ![Image 390: Unknown's avatar](https://0.gravatar.com/avatar/?s=32&d=identicon)Anonymous Says: 
[May 10, 2026 at 10:07 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540097)
I agree that the value comes from utility. However, I would say that the utility of pure maths comes from organising, abstracting and refining various pieces of applied (and other pure) maths.
It’s a big line and as a pure mathematician, the only justification you can give for your work is to point at the next person in line who does a bit less abstract thing and say “my research is useful, because that other thing is useful”. I think this line also goes the other way, people look into how the ideas are organized on a bit more abstract level, engage with the problems (whose beauty might only be apparent if you are already high enough on the line).
I think this is fine, and it was fine historically. I also do this, every grant application I write starts super far, pointing a few steps down the line to give it some grounding. And I also look up the beautiful work top pure mathematicians put out (I subscribe to this blog for a reason) on top of following the work in my area.
The main issue is that this line eventually reaches people who do the actually applied maths (various engineering, statistics, coding, finance), where technical problems don’t require the most beautiful abstractions and deep insights. If AI is good enough to automate that, then at one point a big chunk of the actual value generating segment of the line will be redundant and you have nowhere to point.
Now back to this “In a society where food is cheap to produce, people still get paid to make nice food.”
You suggest a redirection, where mathematics is elevated to a form of art. However, art needs audience. It is not hard to eat a fancy food, or look at a painting, everyone can do that and appreciate some part of it.
But if an entire section of the line erodes, who will look at mathematics and appreciate the beauty? My estimate is that my articles reach <200 views, and probably <10 actual reads who might appreciate the actual content. Even if they contained an insanely beautiful idea, I would be in a trouble financing my life from these readers and their donations for this piece of “art”.
    *   ![Image 391: Unknown's avatar](https://0.gravatar.com/avatar/?s=32&d=identicon)Anonymous Says: 
[May 11, 2026 at 4:58 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540111)
> But at this point we may be wondering what we need humans for at all. (It’s mainly humans who need humans.)
Are you reading what you wrote? Are you actually reading what you wrote? Or do you pretend to function completely from a position of pure rationality?
How different is this from ‘Let’s lobotomize the poor’
    *   ![Image 392: Unknown's avatar](https://0.gravatar.com/avatar/?s=32&d=identicon)Anonymous Says: 
[May 11, 2026 at 5:00 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540112)
> Of course, everything I am saying concerns LLMs as they are right now. But they are developing so fast that it seems almost certain that my comments will go out of date in a matter of months. It is also almost certain that these developments will have a profoundly disruptive effect on how we go about mathematical research, and especially on how we introduce newcomers to it.
Is it really clear that LLMs are developing fast? 
Is the development of LLMs beyond all control actually a good thing?
These questions are not addressed and we are expected to not ask these questions. This is extremely annoying on your part.
Let’s relax from the hype.
23.   ![Image 393: Unknown's avatar](https://0.gravatar.com/avatar/?s=32&d=identicon)Anonymous Says: 
[May 9, 2026 at 11:59 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540067) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540067#respond)
> It is an easy exercise to show that if
> 
> ![Image 394: |A|=k](https://s0.wp.com/latex.php?latex=%7CA%7C%3Dk&bg=ffffff&fg=333333&s=0&c=20201002)
> , then
> 
> ![Image 395: 2k-1\leq|A|\leq\binom{k+1}2](https://s0.wp.com/latex.php?latex=2k-1%5Cleq%7CA%7C%5Cleq%5Cbinom%7Bk%2B1%7D2&bg=ffffff&fg=333333&s=0&c=20201002)
“2k-1 <= k” typo?
_Thanks — corrected now._
24.   ![Image 396: Alexander Poddiakov's avatar](https://2.gravatar.com/avatar/50fad0d68c07602bdb87855b14f58dff5390e17f165d2515b09b2c3f9973e0a9?s=32&d=identicon)Alexander Poddiakov Says: 
[May 9, 2026 at 12:45 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540070) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540070#respond)
Can LLM not only solve but pose new math problems worthy of attention? It can be an interesting study.
    *   ![Image 397: Unknown's avatar](https://0.gravatar.com/avatar/?s=32&d=identicon)Anonymous Says: 
[May 9, 2026 at 3:04 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540075)
From my experience, (GPT 5.5 Pro) yes and no. The main problem is that the model seems to have a fuzzy understanding of where the “solvable” frontier (given the current theory) is located. I have tried to rank problems in the Erdos problems to see if it is able to “predict” which problems are solvable, or eventually close to be solved, with relative success. Then when asking to reevaluate the problems with e.g. Deep Research, this prediction or “difficulty score” can drastically go either up or down. Since it doesn’t have a defined internal criteria/”representation” of this frontier, when creating new problems, often they are solvable in easy or trivial ways, or are too strong, and outside the technology of the theory. Longer prompting and user’s mathematical knowledge (including trends and importance) can help to define a good frontier problem by “collaborating” with the LLM, but I haven’t got a new interesting question solely from the LLM in one-shot.
25.   ![Image 398: Unknown's avatar](https://0.gravatar.com/avatar/?s=32&d=identicon)Anonymous Says: 
[May 9, 2026 at 1:31 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540072) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540072#respond)
Interesting. While you’re debating whether LLMs are truly intelligent, I built a deterministic execution framework that forces consistent outputs regardless of the underlying reasoning mechanism.The question isn’t whether it’s ‘real’ intelligence. The question is whether you can engineer reliable outcomes.If you want to move past probabilistic outputs and into structural control, here’s the system:
[https://www.skool.com/trans-sentient-intelligence-8186/about?ref=8aeedb072d4b4d7fb98cc2238610f2f4](https://www.skool.com/trans-sentient-intelligence-8186/about?ref=8aeedb072d4b4d7fb98cc2238610f2f4)
26.   ![Image 399: Unknown's avatar](https://www.rootclub.it/wp-content/uploads/2018/10/cropped-logo14c-1.png?w=32)[Quando l’IA ruba il lavoro (e pure il merito) ai matematici – Associazione ROOT APS](https://www.rootclub.it/quando-lia-ruba-il-lavoro-e-pure-il-merito-ai-matematici/) Says: 
[May 9, 2026 at 2:14 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540073) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540073#respond)
[…] Source: A recent experience with ChatGPT 5.5 Pro […]
27.   [Datasphere Dispatch #62 | May 9, 2026 | Trust Friction, AI Guardrails, and the Physical Bottlenecks – Datasphere Labs LLC](https://dataspheredata.com/datasphere-dispatch-62-may-9-2026-trust-friction-ai-guardrails-and-the-physical-bottlenecks) Says: 
[May 9, 2026 at 3:03 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540074) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540074#respond)
[…] for de-googled Android usersTrust and access are increasingly mediated by platform identity. A recent experience with ChatGPT 5.5 ProPower users are now benchmarking models by workflow reliability, not demo quality. Using Claude […]
28.   ![Image 400: Unknown's avatar](https://virentanews.com/wp-content/uploads/2026/04/cropped-e24ca6a4-5da6-4612-ac42-7c8c0e699b58-e1775756887585.jpeg?w=32)[How ChatGPT 5.5 Pro Works – VirentaNews](https://virentanews.com/2026/05/09/how-chatgpt-55-pro-works/) Says: 
[May 9, 2026 at 5:11 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540076) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540076#respond)
[…] Source: Gowers […]
29.   [ChatGPT 5.5 Pro: Исследование уровня PhD за 2 часа](https://milyutin-codes.ru/medalist-fildsa-govorit/) Says: 
[May 9, 2026 at 5:30 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540077) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540077#respond)
[…] Источник: Гауэрс […]
30.   [Hacker News 今日TOP 20| 2026-05-09 - 出海掘金，无限可能。为独立开发者、跨境电商从业者、海外自媒体提供最新出海资讯和资源-出海掘金，无限可能。为独立开发者、跨境电商从业者、海外自媒体�](https://www.chuhaix.com/hackernews-daily-2026-05-09/) Says: 
[May 9, 2026 at 5:38 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540078) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540078#respond)
[…] 网站: gowers.wordpress.com HN评论: […]
31.   ![Image 401: Unknown's avatar](https://0.gravatar.com/avatar/?s=32&d=identicon)Anonymous Says: 
[May 9, 2026 at 6:13 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540080) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540080#respond)
lean+paper of infinite twins cocreated with gpt 5.4/5.5 pro
[https://github.com/alegator-cs/infinite_twin_primes](https://github.com/alegator-cs/infinite_twin_primes)
32.   ![Image 402: mwildon's avatar](https://0.gravatar.com/avatar/32e7063f065aec24ae7dcef58d504926333033641dd58b34afe3a6598b19f1cf?s=32&d=identicon)[mwildon](http://wildonblog.wordpress.com/) Says: 
[May 9, 2026 at 7:00 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540081) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540081#respond)
Thank you, I enjoyed reading your write-up of your interactions with ChatGPT. I really wish more mathematicians would do this.
Thinking about $R(2,k)$ was interesting for me: I knew that the minimum value of $2k-1$ was attained for cosets of subgroups, but I did not expect the result that every value between the minimum $2k-1$ and the obvious maximum of the number of 2-multisubsets of $\{1,\ldots, k\}$ would be attained. And I think I learned something by asking ‘if that’s true, then how do we get $k^{3/2}$’. The construction I came up with is $\{1,\ldots, r\}, \{2^s ,2^{s+1}, \ldots, 2^{s+t-1} \}$ where $2^s$ is bigger than $2r$; then by thinking about binary representations of the numbers it’s easy to see that $|A + A| = 2r-1 + rt + t^2$, and by taking $t = \sqrt{r}$, we get $|A| \approx r$ and $|A+A| \approx r^{3/2}$. On closer reading, I saw this has some of the flavour of the sets the LLM found.
To prove I’m not an LLM, let me give a completely off-the-wall analogy. The chain decompositions e.g. for $A = \{a < b < c < d < e\}$ that $a+a < a+b < 2b < b+c < 2c < c+d < 2d < d+e < 2e$ and $a+c < a+d < b+d < b + e < c+e$ and $a+e$ (on its own) show that $9 \le |A+A| \le 9 + 5 + 1 = 15 = \binom{5}{2} + \binom{5}{1}$. These numbers are familiar to me from the decomposition of the $\mathrm{SL}_2(\mathbb{C})$-representation $\mathrm{Sym}^2 \mathrm{Sym}^4 \mathbb{C}^2$ as a direct sum of the irreducible representations $\mathrm{Sym}^8 \mathbb{C}^2$, $\mathrm{Sym}^4 \mathbb{C}^2$ and $\mathrm{Sym}^0 \mathbb{C}^2 \cong \mathbb{C}$. I don’t think the analogy goes any further: the algebraic side has just too much structure, but it would be rather wonderful if one could ‘categorify’ some aspect of arithmetic combinatorics.
Finally, a brief response to your thought experiment:
> Here’s a thought experiment: suppose that a mathematician solved a major problem by having a long exchange with an LLM in which the mathematician played a useful guiding role but the LLM did all the technical work and had the main ideas. Would we regard that as a major achievement of the mathematician? I don’t think we would.
I think you are right for ‘we’ as the community stands at the moment, but it is perhaps interesting to reread this quote substituting ‘computer algebra system’ in place of ‘LLM’ … My tentative hope is that the community learns to use LLMs as the tools they are, and that the more adventurous of us start to credit LLMs in the acknowledgements of our papers. Or even as coauthors?! Going back to my analogy with computer algebra, Doron Zeilberger has set a precedent here with his frequent coauthor Shalosh B. Ekhad.
33.   ![Image 403: Daniel's avatar](https://2.gravatar.com/avatar/ead20c4b018f6c8b2f6933852b9399010f193f649c0c6996325a8b5fd5e0dbea?s=32&d=identicon)[Daniel](http://woodhouse/) Says: 
[May 9, 2026 at 8:02 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540083) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540083#respond)
I think this is going to dramatically affect how mathematical research is valued. A lot of PhD supervisors are going to be in for a rude shock when their new graduate students turn up to meetings solving their problems in short order using these tools. There will be a lot of “cope”, as the kids like to say. I have seen how new areas of research have attracted lots of interest simply because there were plenty of accessible problems ready to be solved. Papers were there to be written, and early careers to be forged. That will cease to be a thing if such problems can be cleared out within a few months.
The value of the mathematics and the problems that are *really* interesting is going to increase dramatically. There are proofs that people are actually interested in reading, where the provenance of the proof will be a secondary concern. More people will try to solve the harder problems when you don’t have to spend a lot of time struggling to understand new techniques and can instead get an AI to power through technical details. I, like many mathematicians, have spent a huge amount of mathematical energy writing and rewriting long and technical papers for results that were, in the grand scheme of mathematical research, unremarkable. The economics of mathematical research change if those details can be farmed out to machines, while the sophisticated researcher can spend more time reading and imagining novel directions to send the Agents.
Mathematics is deeply rooted in its traditions. (We fetishize the chalkboard!) The disruption to these traditions might be utterly devastating. We are at an interesting point.
Maybe the models will never be potent enough. But if there is some fundamental obstruction to these models doing mathematics as we are supposing, that obstruction itself will become one of the most interesting problems in neuroscience and mathematics.
34.   ![Image 404: Unknown's avatar](https://0.gravatar.com/avatar/?s=32&d=identicon)Anonymous Says: 
[May 9, 2026 at 8:19 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540084) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540084#respond)
This reminds me of Richard Dawkins’ recent thoughts on LLMs
35.   ![Image 405: Unknown's avatar](https://www.logicmatters.net/wp-content/uploads/2022/06/cropped-From-Clipboard.jpg?w=32)[On the mathematical abilities of LLMs - Logic Matters](https://www.logicmatters.net/2026/05/09/on-the-mathematical-abilities-of-llms/) Says: 
[May 9, 2026 at 8:30 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540085) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540085#respond)
[…] A very interesting new blog post by Tim Gowers on his recent experiences with ChatGPT 5.5 Pro. Well worth reading. […]
36.   [Из экспоненты в полином за два часа: что GPT-5.5 Pro сделала с задачей по теории чисел — Bukvomat](https://bukvomat.ru/telekom/iz-eksponenty-v-polinom-za-dva-chasa-chto-gpt-5-5-pro-sdelala-s-zadachej-po-teorii-chisel/?doing_wp_cron=1778356937.6565120220184326171875) Says: 
[May 9, 2026 at 9:02 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540086) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540086#respond)
[…] Филдсовской премии и кембриджский профессор —опубликовал в блоге отчет о своем эксперименте с GPT-5.5 Pro: за неполных два […]
37.   ![Image 406: adolfont's avatar](https://1.gravatar.com/avatar/76c186a663bb1a86d697df31e4c419c814160e80b8ea395a8a0c496f9291b9e9?s=32&d=identicon)adolfont Says: 
[May 9, 2026 at 10:55 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540087) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540087#respond)
Shouldn’t the fact that this technology is based on such unethical principles make us think very carefully before using it?
    *   ![Image 407: Unknown's avatar](https://0.gravatar.com/avatar/?s=32&d=identicon)Anonymous Says: 
[May 11, 2026 at 4:55 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540110)
My understanding is that mathematicians like gowers don’t really care.
38.   [菲尔兹奖得主、剑桥大学数学家Timothy Gowers近日在个人博客上分享了他使用ChatGPT 5.5 Pro的体验。令他惊讶的是，这款模型在一小时内连续攻克了多个博士级别的数学难题，展现出远超前代的推理�](https://feng.cx/662.html) Says: 
[May 9, 2026 at 11:08 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540088) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540088#respond)
[…] Timothy Gowers Blog – A Recent Experience with ChatGPT 5.5 Pro […]
39.   [2026年5月10日 科技简报 VC程序员，找资源找VC程序员](https://www.vccoder.com/2548/) Says: 
[May 10, 2026 at 2:00 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540091) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540091#respond)
[…] ChatGPT 5.5 Pro 一小时攻克博士级数学难题 为什么重要：菲尔兹奖得主陶哲轩等顶尖数学家参与评估，证实 AI 在高阶逻辑推理与数学证明上实现重大突破。这标志着 LLM 正从“文本生成器”向“科研副驾驶”发生质变。 […]
40.   ![Image 408: Unknown's avatar](https://0.gravatar.com/avatar/?s=32&d=identicon)Anonymous Says: 
[May 10, 2026 at 6:34 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540092) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540092#respond)
I, too, am expecting LLM proof capabilities to improve rapidly in the near future. So, what are we researchers in poor countries to do? Mathematics was one of the few disciplines that didn’t require vast financial resources to conduct research in (at least in principle…).
Now, mathematicians in affluent countries can afford LLM co-authors who can accelerate their output by orders of magnitude, while poorer colleagues will be left in the dust. Normal, free LLMs are in no way competitive in reasoning capabilities with top end, closed, paid models, and I don’t see a way to change this any time soon.
So, sucks to be us I guess.
    *   ![Image 409: gowers's avatar](https://1.gravatar.com/avatar/df33029a240838a35b04d057c5f44700d023485414deef2830a0f6262fe11493?s=32&d=identicon)[gowers](https://gowers.wordpress.com/) Says: 
[May 10, 2026 at 9:23 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540094)
I agree that that is potentially a very bad aspect of the current situation, and we should think about what to do about it. For example, a small thing that companies like OpenAI could do right now is make sure that they offer as many free subscriptions to people from less affluent countries (however one wants to define that) as they do to people from more affluent countries.
41.   ![Image 410: adarshsad8's avatar](https://2.gravatar.com/avatar/2342bf9aabdbb8b8ece1367355ae12858cb4507081dff51645ed86abebb8cbad?s=32&d=identicon)adarshsad8 Says: 
[May 10, 2026 at 9:50 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540095) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540095#respond)
I’m yet to come across any similar comments from mathematicians working in a field which is heavily theory-oriented (for instance Quantum Groups, Langlands Correspondences or Geometric Representation Theory).
It is hard to judge whether the absence is due to the genuine inability of AI models to make significant contribution in these fields, or simply blissful ignorance of the experts in the field. It might also be practical issue. For an example, it is very time consuming to reject incorrect formulations made by AI on how a particular theory should develop further (as compared to checking the solution of a problem), so one would refrain from using this tool for such applications at the moment.
I’ll be joining a PhD program this year and the lack such discussions in my field used to feel optimistic but it is now turning into a concern.
Also, on another note, I’d love to see if we can expect some fruitful outcome from attempting similar experiments in fields that lie at the intersection of theory-building and problem-solving* such as Ergodic Theory used in solving number theoretic problems, the intersection of combinatorics and representation theory or mathematical physics (e.g. arithmetic quantum unique ergodicity).
* : I understand some may not like my choice to use the phrase ‘theory-building and problem-solving’ but there was no better alternative to describe the commonality of the mentioned fields.
42.   ![Image 411: John Baez's avatar](https://1.gravatar.com/avatar/7a940c155bd1e3fbddbf5dff9b93df0588e66d7121832d395ffec41b6e91792b?s=32&d=identicon)[John Baez](http://math.ucr.edu/home/baez/) Says: 
[May 10, 2026 at 9:54 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540096) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540096#respond)
I find that, showing people this blog article, a certain number are convinced that Chat-GPT 5.5 could not have made up an “original and clever” idea (as Isaac Rajagopal describes it). They argue instead that LLMs are “stochastic parrots”, and say things like “The LLM can only improve some upper bound of some theory because it found a trick in some paper that allows it to do that.”
Now, the fact that Isaac and Tim are experts in this area reduces the chance that this trick was available in the literature unbeknownst to them, but I wonder how one would demonstrate to the skeptics that this is not what’s going on. Maybe it’s not a good use of time for mathematicians to put time into that demonstration. But it’s worth at least a thought. It’s a feature of our age that people have dramatically different views about what’s going on. In this case, there are people out there, watching mathematicians use LLMs, who think these mathematicians have been duped.
    *   ![Image 412: fengyuling's avatar](https://0.gravatar.com/avatar/3c4934c4b630b79f123c48beb33168f5e26fa216670e7975ffd49fbaea417eb8?s=32&d=identicon)[fengyuling](http://fengyuling.wordpress.com/) Says: 
[May 10, 2026 at 12:15 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540100)
Causal Logic vs. Institutional Gatekeeping
Mr. Baez, I sent my paper to you on May 4th. It is clear that the logic was shared with Mr. Gowers. If you are both so fascinated by these “original and clever” ideas in additive number theory, why refuse my arXiv endorsement while simultaneously hyping AI’s “PhD-level” capabilities on X?
The hypocrisy is staggering:
        1.   To protect your status, you claim AI cannot solve century-old problems.
        2.   To erase an independent researcher’s breakthrough, you hint it was “AI-generated.”
The truth is: LLMs operate on probabilistic prediction. They can never achieve the **“Arithmetic Settlement”** required to solve rigid problems like the Collatz Conjecture. My proof was published on **April 28th** (DOI: 10.5281/zenodo.19847203). Any “cleverness” appearing after that date is a reflection of my logic, not AI evolution.
**Feng Yuling (冯玉玲)**_Author of the LSG Framework and Collatz Theorem_
    *   ![Image 413: JollyJoker's avatar](https://0.gravatar.com/avatar/96a4a00e51733a37942426295a0fb243a9be3770353a1bcec85e0ac6412a3d62?s=32&d=identicon)JollyJoker Says: 
[May 10, 2026 at 7:44 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540107)
Ask them to define “original and clever”. Lots of people would have some magic thinking about how humans are special, but there’s probably a point to be made about how LLMS have very shallow and wide knowledge and very little actual reasoning ability.
    *   ![Image 414: fengyuling's avatar](https://0.gravatar.com/avatar/3c4934c4b630b79f123c48beb33168f5e26fa216670e7975ffd49fbaea417eb8?s=32&d=identicon)[fengyuling](http://fengyuling.wordpress.com/) Says: 
[May 11, 2026 at 7:31 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540115)
@JollyJoker, you’ve hit the nail on the head. If LLMs lack actual reasoning, then Gowers’ ‘AI-driven cleverness’ is a logical phantom. It’s not the AI that’s clever; it’s the original logic from my April 28 Deterministic Proof (DOI: 19847203) that was ‘fed’ into the system after I shared it with this elite circle on May 4. @wtgowers, as a mathematician, you know logic cannot arise from a shallow void. Where did the ‘cleverness’ come from, if not from my pre-recorded work?
43.   [AI Дайджест • 10 мая 2026 — airecap.ru](https://airecap.ru/ai-dajdzhest-10-maya-2026/) Says: 
[May 10, 2026 at 10:31 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540098) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540098#respond)
[…] Источник […]
44.   ![Image 415: fengyuling's avatar](https://0.gravatar.com/avatar/3c4934c4b630b79f123c48beb33168f5e26fa216670e7975ffd49fbaea417eb8?s=32&d=identicon)[fengyuling](http://fengyuling.wordpress.com/) Says: 
[May 10, 2026 at 12:21 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540101) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540101#respond)
Mr. Gowers,
It is intellectually dishonest to hype “AI solving PhD problems” (May 9) while ignoring the Deterministic Arithmetic Settlement delivered to your associate Mr. Baez on May 4.
My proof of the Collatz Theorem was published on April 28 (DOI: 10.5281/zenodo.19847203). AI itself admits it cannot solve this problem because it lacks the capacity for causal logic. Furthermore, I have released a second breakthrough: “On the Irreversibility of Transformation Ordinals in Peano Arithmetic: A Topological Resolution of Gödel’s Incompleteness Proposition.”DOI:10.5281/zenodo.20006060
This topological resolution of Gödel’s proposition is something no LLM can “build upon” because it fundamentally contradicts the probabilistic paradigms they are trained on.
Are you truly witnessing AI progress, or are you simply witnessing the arrival of FT Theory and choosing to mislabel its origin? Authority belongs to causality, not to those who use AI as a shield to bypass independent research.
Feng Yuling (冯玉玲)
 Author of the LSG Framework and the Collatz Theorem
45.   ![Image 416: Unknown's avatar](https://0.gravatar.com/avatar/?s=32&d=identicon)Anonymous Says: 
[May 10, 2026 at 3:44 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540103) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540103#respond)
Prof. Gowers, I found your piece thought-provoking. My response here:
[https://chadtopaz.com/essays/gowers-response](https://chadtopaz.com/essays/gowers-response)
“Cheap production, scarce judgment”
    *   ![Image 417: gowers's avatar](https://1.gravatar.com/avatar/df33029a240838a35b04d057c5f44700d023485414deef2830a0f6262fe11493?s=32&d=identicon)[gowers](https://gowers.wordpress.com/) Says: 
[May 11, 2026 at 9:32 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540116)
Thanks for that response, which I read with interest and broad agreement. Thanks also for reading my post carefully — I didn’t feel misrepresented at any point, which is not a given.
46.   ![Image 418: Alex Wright's avatar](https://2.gravatar.com/avatar/e217704a9a96dfe188100b23d5f537f419af5113ca68feb01ef2b5fb3b0b8460?s=32&d=identicon)artisanmysticffef4245fa Says: 
[May 10, 2026 at 6:48 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540104) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540104#respond)
All I have is questions:
    1.   Are there any math societies/departments already thinking about these issues? What about outside of math? Maybe CS departments? 
Universities must be making or at least considering policies on the extent AI generated content can be included in theses. Does anyone know what’s going on?
    2.   Should it be possible for a student to get a math PhD if some/all the ideas in the thesis came from AI? Should such ideas be acknowledged? Where should the line be drawn in the sand? 
    3.   Suppose a student chooses to work without AI to build on their skills, and they write a nice thesis. Before they graduate, someone notices that AI can one shot their whole thesis problem. Should they be able to get their PhD? What if they knew from the start that AI could solve the problem? What if AI couldn’t solve it at the start of the thesis, but three years latter at the conclusion it was doable for AI in half an hour? 
    4.   Should we be actively encouraging PhD students to use AI, or actively encouraging them to build up their own skills without AI, or some of each? 
    5.   Can we, and should we, try to position pure math education as true “learning how think” in an era where maybe fewer people really will learn how to think for themselves, and retrieving and combining information is seen as less valuable? Maybe fewer people will want to really learn how to think, but math could/should train a larger fraction of them?
    *   ![Image 419: Alex Wright's avatar](https://2.gravatar.com/avatar/e217704a9a96dfe188100b23d5f537f419af5113ca68feb01ef2b5fb3b0b8460?s=32&d=identicon)[Alex Wright](https://public.websites.umich.edu/~alexmw/) Says: 
[May 10, 2026 at 6:51 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540105)
(I didn’t intend that to be anonymous. This was written by Alex Wright.)
    *   ![Image 420: Daniel's avatar](https://2.gravatar.com/avatar/ead20c4b018f6c8b2f6933852b9399010f193f649c0c6996325a8b5fd5e0dbea?s=32&d=identicon)Daniel Says: 
[May 10, 2026 at 7:20 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540106)
imagine how quickly we could get referee reports back if Claude can find the horrible mistakes. The software engineers are getting good results with code reviews already.
47.   ![Image 421: Unknown's avatar](https://0.gravatar.com/avatar/?s=32&d=identicon)Anonymous Says: 
[May 10, 2026 at 11:46 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540108) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540108#respond)
A few points:
    *   re: “no mathematical input from me” not sure I fully agree – you provided it direction of search, which came from your intuition, and it searched along those lines to find the solution. I don’t know if this seems like much to you personally, but I feel like knowing the fruitful direction to look at requires genuine and deep understanding, which these models lack. I think LLMs are good at mechanistic aspects of math (or code) generation rather than bits that require leaps based on abstraction. 
    *   It would be interesting to know if you had trials where it did not go as well, how long it took, and what the results were
    *   Related to the above, 17 minutes and 49 minutes of thinking is crazy long for models of these types so cost needs to come down substantially over the years for this kind of approach to scale (it’s worth revisiting Rodney Brooks article on competence vs. performance)
    *   I think the large foundation models show the art of the possible (which *is* impressive), and so I think a really exciting avenue of collaboration between mathematicians and AI folks is to build math specific AI models that are much lower cost and transparent in their reasoning (perhaps some kind of neurosymoblic models or similar) 
48.   [AI News 11/05/2026: Từ hành vi tống tiền của Claude đến AI tự nhân bản và nghiên cứu toán học gốc – Blog — ZTO Labs](https://blog.ztolabs.com/ai-news-11-05-2026-tu-hanh-vi-tong-tien-cua-claude-den-ai-tu-nhan-ban-va-nghien-cuu-toan-hoc-goc/113) Says: 
[May 11, 2026 at 5:26 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540113) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540113#respond)
[…] Nguồn: The Decoder / Gowers’s Weblog […]
49.   ![Image 422: Unknown's avatar](https://0.gravatar.com/avatar/?s=32&d=identicon)Anonymous Says: 
[May 11, 2026 at 6:34 am](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540114) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540114#respond)
The input given resembles the best teaching: you have circumscribed the problem enough for a reader – a fast & uninhibited one, naturally unencumbered by ill-defined objects.
The letter character of purely mathematical text – uniquely, seems to assist pulling formalisms off the shelf [eg [https://x.com/ben_golub/status/1981344469032325152?s=20](https://x.com/ben_golub/status/1981344469032325152?s=20) ].
Dialogue is a dualistic animal – thence means of teaching. Looking forward to the first letting the letter be.
50.   ![Image 423: Unknown's avatar](https://0.gravatar.com/avatar/?s=32&d=identicon)Anonymous Says: 
[May 11, 2026 at 12:51 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540117) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540117#respond)
How would preparing for this look like for math departments at universities?
51.   ![Image 424: Alex Wright's avatar](https://2.gravatar.com/avatar/e217704a9a96dfe188100b23d5f537f419af5113ca68feb01ef2b5fb3b0b8460?s=32&d=identicon)[Alex Wright](https://public.websites.umich.edu/~alexmw/) Says: 
[May 11, 2026 at 12:54 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540118) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540118#respond)
I think one thing we can do is suggest to our students that they read this excellent
52.   ![Image 425: Unknown's avatar](https://aifriends.jp/wp-content/uploads/2024/11/cropped-7966bae78213db11914ad7af893f8431.png?w=32)[ChatGPT 5.5 Proが2時間で博士論文級の数学研究 | AIフレンズ](https://aifriends.jp/chatgpt-5-5-pro-gowers-phd-level-math-research-2026/) Says: 
[May 11, 2026 at 3:17 pm](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comment-540120) | [Reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/?replytocom=540120#respond)
[…] A recent experience with ChatGPT 5.5 Pro（Gowers’s Weblog、2026年5月8日） […]
### Leave a comment [Cancel reply](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#respond)
Δ
* * *
[Blog at WordPress.com.](https://wordpress.com/?ref=footer_blog)
[Entries (RSS)](https://gowers.wordpress.com/feed/) and [Comments (RSS)](https://gowers.wordpress.com/comments/feed/).
*   [Comment](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/#comments)
*   [Reblog](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/)
*   [Subscribe](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/)[Subscribed](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/)
    *   [![Image 426](https://s0.wp.com/i/logo/wpcom-gray-white.png?m=1479929237i) Gowers's Weblog](https://gowers.wordpress.com/)
Join 2,808 other subscribers
 Sign me up 
    *    Already have a WordPress.com account? [Log in now.](https://wordpress.com/log-in?redirect_to=https%3A%2F%2Fgowers.wordpress.com%2F2026%2F05%2F08%2Fa-recent-experience-with-chatgpt-5-5-pro%2F&signup_flow=account) 
*   
    *   [![Image 427](https://s0.wp.com/i/logo/wpcom-gray-white.png?m=1479929237i) Gowers's Weblog](https://gowers.wordpress.com/)
    *   [Subscribe](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/)[Subscribed](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/)
    *   [Sign up](https://wordpress.com/start/)
    *   [Log in](https://wordpress.com/log-in?redirect_to=https%3A%2F%2Fgowers.wordpress.com%2F2026%2F05%2F08%2Fa-recent-experience-with-chatgpt-5-5-pro%2F&signup_flow=account)
    *   [Copy shortlink](https://wp.me/p6XAf-1Jw)
    *   [Report this content](https://wordpress.com/abuse/?report_url=https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/)
    *   [View post in Reader](https://wordpress.com/reader/blogs/1659011/posts/6666)
    *   [Manage subscriptions](https://subscribe.wordpress.com/)
    *   [Collapse this bar](https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/)
%d