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. 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. 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 $A$ is a set of integers, then its sumset $A+A$ is defined to be $\{a+b:a,b\in A\}$ . For a positive integer $h$ , the $h$ –fold sumset, denoted $hA$ , is defined to be $\{a_1+\dots+a_h: a_1,\dots,a_h\in A\}$ . Nathanson is interested in the possible sizes of $hA$ given the size of $A$ . To that end one can define a set $\mathcal R(h,k)$ to be the set of all $t$ such that there exists a set $A$ with $|A|=k$ and $|hA|=t$ .

An obvious first question to ask is simply “What is $\mathcal R(h,k)$ ?” When $h=2$ , the answer is the set of all integers between $2k-1$ and $\binom{k+1}2$ . It is an easy exercise to show that if $|A|=k$ , then $2k-1\leq|A+A|\leq\binom{k+1}2$ , so this result is saying that all sizes in between can be realized. However, it is not true in general that $hA$ can take every size between its minimum and maximum possibilities, and we do not currently have a complete description of $\mathcal R(h,k)$ .

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 $A$ with $A$ and $hA$ having prescribed sizes. (Of course, the size of $hA$ must belong to $\mathcal R(h,k)$ .) Nathanson showed that for every $t\in[2k-1,\binom{k+1}2]$ there is a subset $A$ of $\{0,1,2,\dots,2^k-1\}$ with $|A|=k$ and $|A+A|=t$ , and asked whether the bound $2^k-1$ 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), 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 $\{a+b:a,b\in A, a\ne b\}$ . 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.

I then asked what it could do for general $h$ . I was much less optimistic that it would manage to do anything interesting, because the proof for $h=2$ 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 $\mathcal R(h,k)$ is, then it seems that we are forced to start with a hypothetical set $A$ with $|A|=k$ and $|hA|=t$ 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 $\mathcal R(h,k)$ on $k$ for each fixed $h$ .

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 $\mathcal R(h,k)$ when $k$ is sufficiently large, and if one wants to prove a polynomial dependence for fixed $h$ , then assuming that $k$ 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 $h$ , 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.

After 16 minutes and 41 seconds, it came back with an argument that claimed to have improved the upper bound from exponential in $k$ to exponential in $k^\alpha$ for any $\alpha>1/2$ .
I asked it to write that in preprint form too, which took it a further 47 minutes and 39 seconds.
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.
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.
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.
I asked it to check them.
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.
After 31 minutes and 40 seconds the “preprint” was ready. Here it is.
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 $N(h,k)$ (which I will define very soon) from exponential in $k$ to polynomial in $k$ . While its first improvement of the bound, from exponential in $k$ to exponential in $k^{\frac{1}{2} + \varepsilon}$ , was a routine modification of my work, the improvement to polynomial in $k$ 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 $N(h,k)$ is closely related to a problem I worked on at the Duluth REU (Research Experience for Undergrads) program, of determining $\mathcal{R}(h,k)$ . In particular, $\mathcal{R}(h,k)$ is the set of possible $h$ -fold sumset sizes $|hA|$ , where $A$ can be chosen to be any set of $k$ integers. $N(h,k)$ is the minimal $N$ such that we can achieve all of the values of $\mathcal{R}(h,k)$ using $k$ -element sets $A \subset \{0,1,2,\ldots,N\}$ . I spent last summer explicitly characterizing the set $\mathcal{R}(h,k)$ for large $k$ , by constructing sets $A$ such that $|hA|$ achieves all sizes which I could not rule out as impossible. So, $N(h,k)$ can be upper-bounded by optimizing my constructions.

I constructed these sets $A$ by combining smaller component sets which are simpler to analyze. Some of these components are the geometric series

$\displaystyle S = \{0,1,m,m^2,\ldots,m^{\ell-2}\} \quad \hbox{and} \quad T = \{1,m,m^2,\ldots,m^{\ell-1}\} \qquad (1)$

for various values of $2 \leq m \leq h$ and $2 \leq \ell \leq k$ . Unfortunately, the elements of $S$ and $T$ are exponentially large in terms of $k$ . So, I asked ChatGPT (through Tim) whether there exist sets of $\ell$ elements which have similar sumset sizes to these geometric series, but contain only numbers of polynomial size in $\ell$ : I had no idea if this was possible, or how to begin constructing such sets. ChatGPT came back with an answer, constructing sets $G$ and $H$ which behave like “half a geometric series squeezed into a polynomial interval,” which is counterintuitive. Before I discuss the construction of $G$ and $H$ , I will explain the important properties of the sumset sizes of $S$ and $T$ which they recreate.

For $h > 0$ , a set $A$ is called a $B_h$ set if the only solutions to

$\displaystyle x_1+\cdots+x_h = y_1+\cdots+y_h$

with $x_i,y_i$ in $A$ are the “trivial” solutions, by which I mean that one side of the equation is a reordering of the other side. If $A$ is a $B_h$ set of size $\ell$ , then elements of $hA$ correspond exactly to choices of $h$ elements of $A$ , with repetition allowed. Using “stars and bars,” one can see that $|hA| = \binom{h+\ell - 1}{h}$ and this is the maximum possible value of $|hA|$ among sets of size $\ell$ . So, another definition is that $A$ is a $B_h$ set if $|hA| = \binom{h+|A| - 1}{h}$ . Sidon sets, which Tim discussed, are exactly $B_2$ sets.

To make things more concrete, let us assume that $m = 4$ in (1). Then, $S$ is a $B_3$ set, but it is not a $B_4$ set because of the relations

$\displaystyle 4^{a} + 4^a + 4^a + 4^a = 4^{a+1} + 0 + 0 + 0 \qquad (2)$

for any choice of $a$ in $\{0,1,2,\ldots, \ell-3\}$ . In particular, $\binom{\ell+3}{4} - |4S| = \ell-2$ , as these $\ell-2$ relations are the only ones preventing $S$ from being a $B_4$ set. $T$ lacks the relations in (2) because $0$ is not in $T$ . So, $T$ is a $B_4$ set, but it is not a $B_5$ set because of the relations

$\displaystyle 4^{a} + 4^a + 4^a + 4^a + 4^{b+1} = 4^{a+1} + 4^b + 4^b + 4^b + 4^b \qquad (3)$

for any choices of $a \neq b$ in $\{0,1,2,\ldots, \ell-2\}$ . This gives $\binom{\ell-1}{2}$ relations, and one can check that $\binom{\ell+4}{5} - |5T| = \binom{\ell-1}{2}$ . To summarize, we have seen that

(a) $S$ is a $B_{m-1}$ set.

(b) $\binom{m+\ell-1}{m} - |mS| = \ell -2$ is a linear function of $\ell$ .

(d) $\binom{m+\ell}{m+1} - |(m+1)T| = \binom{\ell-1}{2}$ is a quadratic function of $\ell$ .

ChatGPT was able to find sets $G$ and $H$ of $\ell$ elements which satisfy (a)-(d), but whose elements all have polynomial size in $\ell$ . The construction of $G$ and $H$ uses $h^2$ -dissociated sets, which are sets $A$ where the only solutions to

$\displaystyle x_1+\cdots+x_s = y_1+\cdots+y_{s'} \qquad (4)$

with $s,s' \leq h^2$ and $x_i,y_i$ in $A$ are the “trivial” solutions, i.e. $s = s'$ and one side of the equation is a reordering of the other side. For $r > 0$ , it is possible to construct an $h^2$ -dissociated set $U = \{u_1,\ldots,u_r\} \subseteq \{0,1,2,\ldots,N\}$ , where $N$ is approximately $r^{h^2}$ , and in particular polynomial in $r$ . Constructions of such a $U$ using finite fields date back to Singer (1938) and Bose–Chowla (1963) and are described in Appendix 1. Define

$\displaystyle G = \{0, u_1,u_2,\ldots,u_r,mu_1,mu_2,\ldots, mu_r\}$

and

$\displaystyle H= \{u_1,u_2,\ldots,u_r,mu_1,mu_2,\ldots, mu_r\}. \qquad (5)$

In hindsight, I have good intuition for the construction of $G$ and $H$ . All of the relations in (2) and (3) are formed by combining one or two relations of the form $4x = y$ . There are approximately $\ell$ relations of the form $mx = y$ in $S$ and $T$ , and approximately $\ell/2$ such relations in $G$ and $H$ . There are few other low-order relations in $S$ and $T$ , and similarly in $G$ and $H$ because $U$ is $h^2$ -dissociated. So, $G$ and $H$ manage to contain half as many $mx = y$ -relations as their geometric series counterparts, while also containing few low-order relations.

We now see why (a)-(d) hold with $S$ and $T$ replaced by $G$ and $H$ , respectively. For concreteness, we assume that $m = 4$ and $h>4$ , so $U$ contains no nontrivial relations as in (4) with $s,s' \leq 25 \leq h^2$ . Then, $G$ is a $B_3$ set, but it is not a $B_4$ set because of the relations

$\displaystyle u_i + u_i + u_i + u_i = 4u_i + 0 + 0 + 0$

for any choice of $i$ in $\{1,2,\ldots, r\}$ . If we let $\ell = |G| = 2r+1$ , we can check that $\binom{\ell + 3}{4} - |4G| = r = \frac{\ell-1}{2}$ is linear in $\ell$ . In particular, (a) and (b) hold with $S$ replaced by $G$ , and the linear function $\ell-2$ replaced by $\frac{\ell-1}{2}$ . We can also see that $H$ is a $B_4$ set, but it is not a $B_5$ set because of the relations

$\displaystyle u_i + u_i + u_i + u_i + 4u_j = 4u_i + u_j + u_j +u_j + u_j$

for any $i\neq j$ in $\{1,2,\ldots, r\}$ . If we let $\ell = |H| = 2r$ , we can check that $\binom{\ell + 4}{5} - |5H| = \binom{r}{2} = \binom{\ell/2}{2}$ is quadratic in $\ell$ . In a similar manner, (c) and (d) hold with $T$ replaced by $H$ , and the quadratic function $\binom{\ell-1}{2}$ replaced by $\binom{\ell/2}{2}$ .

Even though I can motivate it in retrospect, ChatGPT’s idea to use $h^2$ -dissociated sets to control relations of order at most $h$ 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 $|hA|$ is very similar to my proof that the sets $A$ which I construct achieve all possible values of $|hA|$ , after replacing $S$ and $T$ by $G$ and $H$ , respectively. Properties (a)-(d) capture many of the important properties of $S$ and $T$ (or $G$ and $H$ ) which are used in this proof. The final constructions involve combining the sets $G$ and $H$ (or $S$ and $T$ in my paper) for each value of $m$ between $2$ and $h$ with another set which is the union of an arithmetic progression and a point. Intuitively, $G$ and $H$ (or $S$ and $T$ ) 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 and the entirety of the ChatGPT preprint. In Appendix 2, I work out the details of the ChatGPT construction to show that for $k$ sufficiently large,

$\displaystyle N(h,k) \leq O\left(k^{10h^3}\right).$

For comparison, it is easy to see that $N(h,k)$ is at least on the order of $k^{h}$ , 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 $h$ -dissociated set $U = \{u_1,\ldots,u_r\} \subseteq \{0,1,2,\ldots,N\}$ , where $N$ is approximately $r^{h}$ . This construction is a very minor modification of Bose–Chowla (1963)’s construction of a $B_h$ set, which I learned about from this paper. For whatever reason, the GPT preprint (Lemma 3.1) uses a different, less efficient construction using moment curves.

Let $p > r$ be a prime, let $N = p^{h+1}-2$ , let $K$ be the finite field with $p^{h+1}$ elements and fix a generator $\theta$ of $K^\times$ , so that $K^\times$ is equal to $\{\theta^0,\theta^1,\ldots, \theta^N\}$ . Define a set of $p$ elements

$\displaystyle U = \{a \in \{0,1,2,\ldots,N\}: \theta^a - \theta \in \mathbb{F}_p\}.$

Then, each element $a \in U$ corresponds to a unique value of $\tilde{a} \in \mathbb{F}_p$ , by taking $\tilde{a} = \theta^a - \theta$ . Now an additive relation of the form in (4) with $s,s' \leq h$ can be reframed by taking powers of $\theta$ as

$\displaystyle (\theta + \tilde{x_1})(\theta + \tilde{x_2})\cdots (\theta + \tilde{x_s}) = (\theta + \tilde{y_1})(\theta + \tilde{y_2})\cdots (\theta + \tilde{y_{s'}}). \qquad (6)$

As $K$ is a degree- $h+1$ extension of $\mathbb{F}\sb{p}$ and $\theta$ is a generator of $K$ as an $\mathbb{F}\sb{p}$ -extension, this means that $\theta$ does not satisfy any nonzero polynomials in $\mathbb{F}\sb{p}[x]$ of degree $\leq h$ . So, both sides of (6) are identical as polynomials in $\mathbb{F}_{p}[\theta]$ and thus the additive relation in (4) is trivial. So, $U$ is $h$ -dissociated, and of course one can prune a few elements to reduce $U$ to size $r$ .

Appendix 2 (Isaac)

Fix constants $\alpha,\beta,\gamma$ such that $0.5 < \beta\gamma < \beta < \alpha < 1$ (in my paper I arbitrarily chose $(\alpha,\beta,\gamma) = (0.9,0.8,0.7)$ ). Let the two sets in (5) be called $G_{m,r}$ and $H_{m,r}$ . Let $[a,b]$ denote the set of integers $x$ satisfying $a \leq x \leq b$ . Similarly to my paper, the constructions of $A$ such that $hA$ achieves the desired sizes will combine sets of the following four types:

$B_{j,b} := [0,b-2] \cup \{b-2+j\}$ with choices of $b \in [3, k-k^\gamma]$ and $j \in [1,hb]$ .
$G_{m,r_m}$ for each value of $m \in [3, h]$ , with choices of $r_m \in [0, (k-b)^\alpha]$ .
$H_{m,u_m}$ for each value of $m \in [2,h-1]$ , with choices of $u_m \in [0, (k-b)^\beta]$ .
A $B_h$ set of the correct size so that $|A| = k$ .

One reason that this construction needs to be complicated is that we need to create at least $\Omega(k^h)$ many sets. To do this, we vary $2h-4$ parameters $r_m$ and $u_m$ in the domain $[0,k^\alpha]$ and $2$ parameters $b$ and $j$ in the domain $[1,hk]$ . We can choose $\alpha$ to be slightly bigger than $1/2$ , and then the above construction gives us $O(k^{\alpha(2h-4)+ 2})=O(k^{h + \delta})$ different sets where $\delta >0$ 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 $\Omega(k^h)$ many sets. In comparison, Nathanson’s construction when $h=2$ only needs to create $\Omega(k^2)$ 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 $O(k)$ .

We want to combine $q = 2h-2$ sets $A_1,\ldots,A_q$ , which are given by $B_{j,b}$ , $G_{m,r_m}$ for the $h-2$ values of $m \in [3,h]$ , $H_{m,u_m}$ for the $h-2$ values of $m \in [2,h-1]$ , and a $B_h$ set. By Appendix 1, for all $r \leq k$ , there exists a $h^2$ -dissociated set ${u_{1},\ldots,u_{r}}$ of diameter $M \leq r^{2h^2} \leq k^{2h^2}$ . By the constructions of $G_{m,r_m}$ and $H_{m,u_m}$ , we can take each $A_i \subseteq [0,M]$ , where $M \leq hk^{2h^2}$ . Let $\mathbb{Z}^{2q}$ have basis vectors $e_1,\ldots,e_{2q}$ . To combine $A_1,\ldots,A_q$ , we can define $A \subseteq \mathbb{Z}^{2q}$ as

$\displaystyle A = \bigcup_{i=1}^q (A_i e_i + e_{q+i}) \subseteq \{0,1,2,\ldots,M\}^{2q} \subseteq \mathbb{Z}^{2q}.$

Similarly to my Lemma 4.9, this construction ensures that the generating function product $\mathcal{F}_{A}(z) = \prod_{i=1}^q \mathcal{F}_{A_i}(z)$ 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, $A$ is Freiman-isomorphic of order $h$ to a subset of $[0,2qM(2hM)^{2q-1}]$ . Therefore, for $k$ sufficiently large (the whole construction relies on this for the same reasons as in my paper),

$\displaystyle N(h,k) \leq 2qM(2hM)^{2q-1} \leq 2\left(2h^2k^{2h^2}\right)^{2(2h-2)} \leq k^{10 h^3}.$

Appendix 3 (Isaac)

In Section 4.2 of my paper, I use a different, simpler construction to construct sets $A$ achieving the values in $\mathcal{R}(h,k)$ which have $|hA| < \varepsilon k^h$ , for some small $\varepsilon$ . These sets $A$ are subsets of ${0,1,2,\ldots,k^h}$ , meaning that all elements have polynomial size in $k$ . 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 $S$ and $T$ . 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 $\mathcal{F}_A(z)$ as a bookkeeping tool to keep track of the sumset sizes of a set $A$ . 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 $\mathcal{F}_S(z)$ (Lemma 4.15) and $\mathcal{F}_T(z)$ (Lemma 4.17). This corresponds to Section 3 and Section 6.1 of the GPT preprint. In particular, $\mathcal{F}_{G}(z)$ is computed in Lemma 3.3 and $\mathcal{F}_{H}(z)$ 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 $A$ which I have constructed, the values of $|hA|$ will assume all of the elements of ${\lceil\varepsilon k^h\rceil, \lceil\varepsilon k^h\rceil+1,\ldots ,\binom{h+k-1}{h} }$ . The key idea is to show that the set of all values of $|hA|$ forms an interval, and contains numbers both smaller than $\varepsilon k^h$ and equal to $\binom{h+k-1}{h}$ .

Tags: ai, mathematics

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110 Responses to “A recent experience with ChatGPT 5.5 Pro”

Moses Charikar Says:
May 8, 2026 at 5:14 pm | Reply
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
Manu Says:
May 8, 2026 at 5:39 pm | Reply
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.
Anonymous Says:
May 8, 2026 at 7:51 pm | Reply
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?
- jovial95facf5d33 Says:
  May 8, 2026 at 7:52 pm
  Sorry, I didn’t realise that was anonymous! Best wishes, Olof (Sisask)
- domotorp Says:
  May 9, 2026 at 5:32 am
  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.
gowers Says:
May 8, 2026 at 8:01 pm | Reply
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.
- hopefula0ac40f0f2 Says:
  May 9, 2026 at 5:54 am
  My standard is to ask the AI (in particular ChatGPT): Write your finding/proof in texfile, also output as pdf. It works savely.
- ateixeira Says:
  May 9, 2026 at 7:02 pm
  I always use Luca Trevisan’s script to convert LaTeX files into friendly wordpress.com html files (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.
Bruce Smith Says:
May 8, 2026 at 9:13 pm | Reply

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.
Anonymous Says:
May 8, 2026 at 9:17 pm | Reply
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.
- old-bielefelder Says:
  May 9, 2026 at 12:10 pm
  > 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.
最近使用ChatGPT 5.5 Pro的经历 - 偏执的码农 Says:
May 8, 2026 at 9:22 pm | Reply
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Anonymous Says:
May 8, 2026 at 10:51 pm | Reply
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 a step in this direction.
Phillip Says:
May 8, 2026 at 10:54 pm | Reply
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.
antilli Says:
May 9, 2026 at 12:25 am | Reply
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
- hopefula0ac40f0f2 Says:
  May 9, 2026 at 6:01 am
  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.
- antilli Says:
  May 9, 2026 at 11:51 pm
  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
- Anonymous Says:
  May 9, 2026 at 11:08 am
  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/, click the “select model” dropdown menu. You do need to pay $200 first.
- antilli Says:
  May 9, 2026 at 11:55 pm
  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”
A recent experience with ChatGPT 5.5 Pro / 最近使用ChatGPT 5.5 Pro的体验 – OpenClawLog Says:
May 9, 2026 at 6:00 am | Reply
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Anonymous Says:
May 9, 2026 at 6:20 am | Reply
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…
- gowers Says:
  May 9, 2026 at 12:46 pm
  Corrected — thanks.
ChatGPT 5.5 Proが数学研究の新たな扉を開く！ – ainewsfeed Says:
May 9, 2026 at 6:52 am | Reply
[…] 元記事: https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/ […]
Anonymous Says:
May 9, 2026 at 7:12 am | Reply
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.
Anonymous Says:
May 9, 2026 at 7:27 am | Reply
Why do you get access to ChatGPT 5.5 Pro but not everyone else?
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Anonymous Says:
May 9, 2026 at 8:49 am | Reply
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
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May 9, 2026 at 9:00 am | Reply
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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 Says:
May 9, 2026 at 9:15 am | Reply
[…] 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。 […]
John Baez Says:
May 9, 2026 at 10:03 am | Reply
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.)
- Anonymous Says:
  May 10, 2026 at 10:07 am
  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”.
- Anonymous Says:
  May 11, 2026 at 4:58 am
  
  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’
- Anonymous Says:
  May 11, 2026 at 5:00 am
  
  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.
Anonymous Says:
May 9, 2026 at 11:59 am | Reply

It is an easy exercise to show that if

$|A|=k$

, then

$2k-1\leq|A|\leq\binom{k+1}2$

“2k-1 <= k” typo?

Thanks — corrected now.
Alexander Poddiakov Says:
May 9, 2026 at 12:45 pm | Reply
Can LLM not only solve but pose new math problems worthy of attention? It can be an interesting study.
- Anonymous Says:
  May 9, 2026 at 3:04 pm
  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.
Anonymous Says:
May 9, 2026 at 1:31 pm | Reply
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
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Anonymous Says:
May 9, 2026 at 6:13 pm | Reply
lean+paper of infinite twins cocreated with gpt 5.4/5.5 pro

https://github.com/alegator-cs/infinite_twin_primes
mwildon Says:
May 9, 2026 at 7:00 pm | Reply
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.
Daniel Says:
May 9, 2026 at 8:02 pm | Reply
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.
Anonymous Says:
May 9, 2026 at 8:19 pm | Reply
This reminds me of Richard Dawkins’ recent thoughts on LLMs
- Anonymous Says:
  May 17, 2026 at 10:18 pm
  his religious belief that his sycophantic chatbot is sentient?
On the mathematical abilities of LLMs - Logic Matters Says:
May 9, 2026 at 8:30 pm | Reply
[…] A very interesting new blog post by Tim Gowers on his recent experiences with ChatGPT 5.5 Pro. Well worth reading. […]
Из экспоненты в полином за два часа: что GPT-5.5 Pro сделала с задачей по теории чисел — Bukvomat Says:
May 9, 2026 at 9:02 pm | Reply
[…] Филдсовской премии и кембриджский профессор — опубликовал в блоге отчет о своем эксперименте с GPT-5.5 Pro: за неполных два […]
adolfont Says:
May 9, 2026 at 10:55 pm | Reply
Shouldn’t the fact that this technology is based on such unethical principles make us think very carefully before using it?
- Anonymous Says:
  May 11, 2026 at 4:55 am
  My understanding is that mathematicians like gowers don’t really care.
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� Says:
May 9, 2026 at 11:08 pm | Reply
[…] Timothy Gowers Blog – A Recent Experience with ChatGPT 5.5 Pro […]
2026年5月10日科技简报 VC程序员，找资源找VC程序员 Says:
May 10, 2026 at 2:00 am | Reply
[…] 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。 […]
Anonymous Says:
May 10, 2026 at 6:34 am | Reply
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.
- gowers Says:
  May 10, 2026 at 9:23 am
  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.
adarshsad8 Says:
May 10, 2026 at 9:50 am | Reply
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.
John Baez Says:
May 10, 2026 at 9:54 am | Reply
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.
- fengyuling Says:
  May 10, 2026 at 12:15 pm
  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
- JollyJoker Says:
  May 10, 2026 at 7:44 pm
  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.
- fengyuling Says:
  May 11, 2026 at 7:31 am
  @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?
AI Дайджест • 10 мая 2026 — airecap.ru Says:
May 10, 2026 at 10:31 am | Reply
[…] Источник […]
fengyuling Says:
May 10, 2026 at 12:21 pm | Reply
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
- Anonymous Says:
  May 12, 2026 at 10:52 am
  
  AI itself admits it cannot solve this problem because it lacks the capacity for causal logic.
  
  Really? I asked an LLM if it has the capacity for causal logic and it said “Yes” and gave an example which I verified.
  
  If you think that probabilistic paradigms are limitations to work on Gödel’s Incompleteness Theorems (not “Proposition”) then you should consider the non-determinism and quantum basis of human cognition.
  
  I would further recommend that you read the essay “Your LLM-assisted scientific breakthrough probably isn’t real” (2025), as it contains practical advice for someone making bold claims like yours.
Anonymous Says:
May 10, 2026 at 3:44 pm | Reply
Prof. Gowers, I found your piece thought-provoking. My response here:

https://chadtopaz.com/essays/gowers-response

“Cheap production, scarce judgment”
- gowers Says:
  May 11, 2026 at 9:32 am
  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.
- fengyuling Says:
  May 13, 2026 at 5:15 pm
  “Really? I asked an LLM if it has the capacity for causal logic and it said ‘Yes’…”
  
  Since Mr. Gowers has enabled comment moderation, I am unsure whether you will be able to see this text.
  
  In response to your skepticism, I would like to point out a core paradigmatic fallacy: 1. Simulation Does Not Equal Execution An LLM is, in essence, a high-dimensional probabilistic predictor trained on a massive corpus of data. When it claims to possess “causal reasoning capabilities,” it is merely statistically matching the linguistic patterns of logical inference—it is not actually performing logical operations. It selects “Yes” because, within its predictive distribution, “Yes” constitutes the statistically optimal response to that specific query. Just as probability cannot logically deduce the inevitable single-track trajectory of the Collatz Conjecture, a probabilistic model cannot generate a genuine causal closed loop. 2. The Disconnect Between Statistical Emergence and Axiomatic Logic The “LLM logic” you mentioned reveals its inherently stochastic nature when confronted with rigorous, closed-system mathematical proofs. In my own proofs—whether involving the arithmetic gradient of -0.2877 or the topological analysis of Gödel’s incompleteness theorems—the logical pathways demand 100% determinacy. However, the underlying architecture of an LLM dictates that it cannot escape the phenomenon of “probabilistic hallucination.” 3. On Cognition and Quantum Theory Equating the inherent indeterminacy of human cognition with the probabilistic choices made by an AI constitutes a fundamental attribution error. Human thought possesses the capacity for “Causal Settlement”—a concept that serves as the very bedrock of my FT Theory (Fengyuling Theory). I suggest that, rather than simply asking an LLM “if it has the capability,” you instead attempt to have it audit the logic presented in my Zenodo publication (DOI: 10.5281/zenodo.19847203). You will discover that, while it can certainly regurgitate my conclusions, it will never be able to truly comprehend *why*—from a logical standpoint—there is “absolutely no possibility of divergence.” https://x.com/FengTheory/status/2049544550369378604
artisanmysticffef4245fa Says:
May 10, 2026 at 6:48 pm | Reply
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?
- Alex Wright Says:
  May 10, 2026 at 6:51 pm
  (I didn’t intend that to be anonymous. This was written by Alex Wright.)
- Daniel Says:
  May 10, 2026 at 7:20 pm
  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.
Anonymous Says:
May 10, 2026 at 11:46 pm | Reply
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)
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 Says:
May 11, 2026 at 5:26 am | Reply
[…] Nguồn: The Decoder / Gowers’s Weblog […]
Anonymous Says:
May 11, 2026 at 6:34 am | Reply
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 ].

Dialogue is a dualistic animal – thence means of teaching. Looking forward to the first letting the letter be.
Anonymous Says:
May 11, 2026 at 12:51 pm | Reply
How would preparing for this look like for math departments at universities?
Alex Wright Says:
May 11, 2026 at 12:54 pm | Reply
I think one thing we can do is suggest to our students that they read this excellent
ChatGPT 5.5 Proが2時間で博士論文級の数学研究 | AIフレンズ Says:
May 11, 2026 at 3:17 pm | Reply
[…] A recent experience with ChatGPT 5.5 Pro（Gowers’s Weblog、2026年5月8日） […]
Math Has Entered the Production Stack – Post Reboot Says:
May 11, 2026 at 7:54 pm | Reply
[…] claim would have sounded inflated not long ago. It sounds much less inflated after mathematician Timothy Gowers described getting PhD-level mathematical work from ChatGPT 5.5 Pro in about an hour with little serious […]
Anonymous Says:
May 11, 2026 at 8:12 pm | Reply
Interesting read, you raise some important issues.

Just one comment. You say “[…] ChatGPT 5.5 Pro, to which I am fortunate to have been given access”.

The fortunate ones here are OpenAI, because no amount of money can buy the publicity that you’re giving them. That is, a Fields medallist waxing lyrical about their software. Knowing a little about how silicon valley works, I promise your blog post is being circulated among investors as we speak.

Also, I can promise that if the access you’ve been provided is linked to an email identified with you, Tim Gowers, then you’re not getting the regular ChatGPT 5.5 Pro, they’re providing you orders-of-magnitude more parallelized compute than a normal Pro account, but disguised as a normal account, to enhance your experience. If you log out and use an anonymous account, your experience will certainly be different.

Of course this doesn’t mean the capabilities aren’t there, clearly they are as you and others have observed, but high-profile academics are being manipulated, and are easy targets. Those accustomed to university culture are often naive to the cut-throat workings of silicon valley culture.

So be careful, even if these tools are capable, don’t give them the satisfaction.
- mwildon Says:
  May 11, 2026 at 10:00 pm
  I subscribe to the Pro account that lets me use ChatGPT 5.5 ‘Extended Thinking’ and it regularly reports 30 minutes or more of thinking time on tough prompts, often with results that impressive me. I do not see any evidence for the claim that Prof. Gowers is getting anything more more than the ‘regular ChatGPT 5.5 Pro’.
ChatGPT 5.5 Proが博士課程レベルの数学研究を1時間で実行、数学者が「人間の研究の最低ラインが変わる」と指摘 | News Everyday Says:
May 11, 2026 at 9:50 pm | Reply
[…] 　A recent experience with ChatGPT 5.5 Pro | Gowers’s Weblog 　https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/ […]
quasihumanist Says:
May 12, 2026 at 6:54 am | Reply
What I’m really worried about with LLMs is the impact on the lower levels of mathematics research – the research done by people in developing countries or at teaching-oriented institutions that does not and is not intended to have impact on the big story of mathematics. Doing this research is a form of professional development that makes the researcher better teachers and mentors, and this research also enriches to the body of mathematics, helping make mathematics a healthy rather than purely elitist subject.

A lot of this research has become a sideline precisely because it is perceived (usually but not always accurately) to be routine given what is known. That makes it susceptible to solution by LLM. What happens to this research when someone cleans out a whole subject rapidly with the AI help? Do we have to resign ourselves to having fewer (marginally) research-active mathematicians? What happens to those grad students who find themselves not able to do better than an LLM, even with LLM help? Can they still earn PhDs?
- gowers Says:
  May 13, 2026 at 5:34 pm
  I’m worried about that too.
ChatGPT 5.5 Pro 1 小時解組合數學難題　專家：人類研究最低標準將改變 – 香港網絡 Says:
May 12, 2026 at 9:09 am | Reply
[…] 來源: Gowers […]
IA Revoluciona Matemática: GPT-5.5 Proger Pesquisa Científica Says:
May 12, 2026 at 6:33 pm | Reply
[…] A recent experience with ChatGPT 5.5 Pro | Gowers’s Weblog – Fonte primária do próprio Timothy Gowers sobre a experiência […]
Anonymous Says:
May 13, 2026 at 4:16 am | Reply
Dear Professor,

As always, your formulation is remarkably elegant, though I must admit that it is not entirely clear to me what emotions, if any, you experience when you write about “a profoundly disruptive effect” on your job.

After all, you have always been among those at the forefront of hoping that computers would one day completely replace humans in proving theorems, and you have taken concrete steps to help bring that future about as quickly as possible. Yet now, it seems that you may not be entirely pleased with what is happening.

I would be very interested to understand your view on this!
flippiefanus Says:
May 13, 2026 at 5:02 am | Reply
Dear Prof. Grower

In a previous comment, J Baez conveyed a comment that I’ve posted on mathstodon, somewhat misrepresenting my intentions (I’m not implying that mathematicians are being duped). So, please allow me to explain.

According to some explanation I read (not sure where), LLMs work by doing pattern matching. In other words, when a question is posed to an LLM, it effectively searches a vast database to find an approach that can address that question. From what I understand, the training set of any LLM is much larger than volume of literature that any single or handful of humans can ever absorb. So, it is not unreasonable that the LLM would be able to identify approaches in its training set (some paper perhaps) that solves the problem, and which the person asking the question is unaware of.

So, I propose a little experiment: when the LLM comes up with some unexpected solution for a give step in the process, perhaps one can ask it: “where did you see that?” or “how do you know that?” The hope is that the LLM would then provide the reference to literature where that approach has been used. What such an experiment be sensible?

If all surprising steps provided by the LLM can be related to literature, then the role of the LLM is one of a highly effective literature search tool. In such a case, one does not need to consider the LLM as some collaborator in the research process. If I’ve used Maple to perform a calculation, I don’t consider in necessary to mention it in the manuscript.
- Anonymous Says:
  May 17, 2026 at 11:24 pm
  However, if Doron Zeilberger uses Maple and it results in a publication, then Ekhad B, Shalosh would be a coauthor of that paper.
Wind Flash Says:
May 13, 2026 at 5:02 pm | Reply
Seeing AI tools like CodeX carry out complex data research is genuinely inspiring. Lately I’ve already replaced Claude Code with CodeX for all sorts of experiments — from development work to curiosity‑driven research. I have a strong feeling that LLMs and agents could see some unexpected breakthroughs in 2026.
IA Gera Pesquisa de Doutorado em Menos de Duas Horas Says:
May 13, 2026 at 8:56 pm | Reply
[…] A recent experience with ChatGPT 5.5 Pro — Timothy Gowers (blog pessoal) […]
Anonymous Says:
May 14, 2026 at 7:38 pm | Reply
Ian Stewart, prophetically, in 1975, in a book on modern mathematics (before turning up your nose at him because he now writes popular science and entertainment books, remember that his book on Galois Theory is recommended by Saunders Mac Lane among the references in his Algebra textbook):

“Even in pure mathematics the computer has scored some notable triumphs, especially in the study of finite groups. However, very few problems are suitable for computation; and even some of those that are would take too long to perform, even for today’s very fast machines (or tomorrow’s, for that matter).

The uses of computers are not confined to numerical problems. Computers have been programmed to play draughts (well) and chess (badly), to translate from one language to another (execrably), to compose music (of sorts) and poetry. Some of the recent advances in producing ‘intelligent’ machines are quite remarkable.

This brings me naturally to the oft-asked question ‘Can computers think?’ As Joad would have said, it all depends what you mean by ‘think’. As yet, the computer can perform some of the functions of the human brain faster and more accurately; others it cannot perform at all. But if we ask, ‘Is there something special about the way in which human beings think which in principle can never be performed by some kind of machine?’ then my personal opinion is that the answer is ‘No’. Certainly we cannot duplicate the functions of the brain at the present time; and it is fairly certain that the resemblance between the brain and existing computers is about as close as that between a cow and a milk-lorry. Our technology may well never get anywhere near making a truly ‘intelligent’ machine: the human brain may well be too stupid. But I don’t think there is any obstacle to the production of a machine which performs the functions of the human brain; not any logical obstacle such as prevents √2 from being rational or a man from lifting himself by his bootstraps; for the following reason: the human body is visibly a machine, in the sense that it composed out of matter and the components obey the same laws as other matter. It is a very complicated and wonderful machine which we don’t understand. If there were in principle an obstacle to the construction of machines which behaved like people, then there would be no people.

This is not to reduce humanity to the level of a can-opener. Many people insist that the complexities of human behaviour, the emotional, creative, and spiritual attributes, must be consequences of something ‘greater’ than physical laws. This is a wonderful concept. How much more wonderful it would be, however, if these very attributes were consequences of physical laws. Far from demeaning humanity, this would elevate physics!”

Concepts of Modern Mathematics, pp. 267-268. Dover Books.
Anonymous Says:
May 15, 2026 at 6:02 pm | Reply
A question for which I have no answer is the following: in an era of proof abundance (as nicely characterized by Terence Tao), where novelty (‘I proved this first’) must cede ground to understanding (‘Here’s how this result works, and why’) – what’s the case for taxpayers to fund mathematics research?

As a follow-up how do we restructure assessment for such grants as will still be available? Currently our proposals are about our future plans, and we’re certainly heavily assessed on ‘novelty’. There are next to no points awarded to ‘I plan to thoroughly understand this set of results’.

As a community we’ve made the case to governments by pointing to applications of mathematics, and stating that research in ‘pure’ mathematics is the seed corn which is critical for future developments. In an era of proof abundance, I am uncertain about the arguments for pure research we may present.

I appreciated very much your essay, which is both exciting and disquieting. It seems indeed we’re rapidly approaching a time when AI-assisted creation of mathematics is the norm, but exposition and conveying understanding about a mathematical result will perhaps be the task of humans. The latter task seems closer, in my view, to the relationship that art historians and art critics have to art, as compared to that of artists. Art history and criticism are really important, but are orders of magnitude less well-funded by taxpayers.

As concerns go, funding is a small one. But it will almost surely impact younger researchers in a big way.
- Anonymous Says:
  May 15, 2026 at 6:07 pm
  (This comment was by Nilima Nigam. I did not intend to comment anonymously, but don’t have a social-media login.)
Anonymous Says:
May 17, 2026 at 3:31 pm | Reply
This is really remarkable. And it ought to be an occasion for reflecting on what’s the point of education and of research.

It’s still been claimed that education is useful for getting a job and doing research is a job.

That’s true, but I’d submit that practically anyone doing research does it for understanding something for oneself. So, sure, it’s possible to use AI to solve some problem, or even to explore new directions; but this statement conflates means with ends, for human beings.

So, indeed, AI programs can solve problems that would deserve a PhD; but, normally, awarding a PhD is a means, not an end in itself, for a human being.

It just highlights that how we teach and how we do research must take the existence of these tools into account; but we’re doing research for ourselves, I’d submit, not, simply, because we’re paid for it. It’s the other way around: We’re doing it to figure things out and this activity, of itself, is of value to humans and that’s why those that engage ini it deserve to mak a living out of it.
Anonymous Says:
May 17, 2026 at 10:13 pm | Reply
Do you see the issue with having a publicly accessible therefore AI readable archive of unchecked unvetted mathematical proofs? Get one piece of slop in there and you have an undetected error which will then propagate through further proofs and good luck when you end up with a proof that pi is an even integer. Somebody will have to work backwards through a sea of proofs to find a proof on planar geometry that had an obvious or even subtle error that chat didn’t notice and nobody bothered to vet but other chat find very useful in building further proofs.
Anonymous Says:
May 22, 2026 at 10:56 am | Reply
sit tf down tim. you ain’t decide the future.
Anonymous Says:
May 22, 2026 at 5:33 pm | Reply
Nobody works through any sea of proofs and won’t, because there’s no need for it. There are, already, many wrong papers, so AI won’t do much to increase the noise and may, indeed, help in increasing the signal (cf. the Univalent program). In any case, reducing “slop” isn’t a technical issue, it’s a social one.

However, I’d remark that, while AI can solve, in hours, problems that humans would need years, this doesn’t mean that humans can understand the results in hours.

If the godess Athena had handed Thales Maxwell’s equations, that do describe the motions of electric charges and of magnets, Thales wouldn’t have been able to understand them!

So there’s even more work to be done by humans to make sense of the answers that an AI program is providing.

Indeed, if such a program can do a year’s work in an hour, it’s possible to imagine how much further one could advance in the space of a PhD.
Johan Aspegren Says:
May 23, 2026 at 10:43 am | Reply
Are you still trying that old school rule-based approach? It would be really hard to make it scale.
Anonymous Says:
May 24, 2026 at 8:19 am | Reply
Is using LLM’s not an accecptable type of Plagarism?

We are already heading in this direction where majority of Universities in India (that i know well) have as a Policy stated that 20% of Ph.D thesis can incorporate AI generated ideas and results. Now this 20% can actually be the KEY around which a wraper of 80% is done. Going further it is easy to make this look like 20% (by “altering” 60%-80%).

Also it has always been a Curse to be Poor to afford Equipment for doing cutting edge or accecptable research (like LIGO, LHC etc) . Mathematics was so far isolated from this paradigmn (you use your brain and blackboard or at worse a P-5 processor). But now Mathematics is also consumed by “being rich only enables you to do Maths”. So people like srinivsa ramanujan have no place anymore??
- Anonymous Says:
  May 24, 2026 at 9:49 am
  No, it’s not, anymore than using a calculator to perform arithmetic calculations is plagiarism.
  
  Or looking up information in a library.
  
  The information that’s needed to undertake interesting tasks is less and less localized in one place, that’s all. It’s spread out. The challenge is to find it, to establish the useful connections. Something that Ramanujan was exceptionally good at, incidentally.
- Anonymous Says:
  May 24, 2026 at 12:05 pm
  What is true is that the infrastructure that is now required to perform certain tasks comes at a cost that can’t be borne by any individual, or, even, any country.
  
  That’s not, necessarily, a bad thing. The infrastructure can allow people anywhere it reaches to participate in activities that were hitherto limited to fewer people in fewer places. The tradeoff is that its reliability and maintenance must be assured, something people tend to neglect.
So, Where Does Next-Token Prediction Leave Us? - POP RDI; RET; Says:
May 27, 2026 at 4:29 am | Reply
[…] Fields medalist Tim Gowers wrote about his experience with ChatGPT 5.5 Pro ($30 per M input, $180 per M output, mind you!). He […]
Science and AI | More Quantum Says:
May 28, 2026 at 11:18 am | Reply
[…] a little after I wrote this post, Gowers posted hyping up a minor result in combinatorics found by ChatGPT. Seriously, why would you ever want to use AI for […]
Adolfo Neto Says:
May 28, 2026 at 9:11 pm | Reply
Where is the link to your chat? You know you can share it, don’t you?
From today, science is dead. Or maybe not! – 〈Abbas| K |Rizi〉 Says:
May 29, 2026 at 3:58 pm | Reply
[…] Language Models (LLMs) can already solve “gentle problems.” The Fields Medalist Timothy Gowers recently used ChatGPT 5.5 Pro to solve a combinatorics problem, and he got a correct solution! He […]
HN 热点评论速览 · 2026年5月12日 – DataPipe Says:
June 24, 2026 at 6:04 am | Reply
[…] 原文链接：https://gowers.wordpress.com/2026/05/08/a-recent-experience-with-chatgpt-5-5-pro/ […]
Anonymous Says:
June 26, 2026 at 10:32 pm | Reply
Dear Professor Gowers (and the entire community),
I am one of the beginners in research you are talking about, having started my PhD in structural graph theory in November 2025.

First of all, given that a year ago the most powerful (common) LLMs were Claude Opus 4 and GPT-4.1, which are now both listed as retired and which were far from contributing much to a PhD-thesis, one can only imagine how strong the next generation (Mythos 5.something, GPT-6.something) will be, if the current state of the art is to produce “a perfectly reasonable chapter in a combinatorics PhD”. Such a chapter might be even considered good/excellent at a university, where people are not used to having as great PhD-students as there are in Cambridge.

With that, I am not optimistic that “proving something in collaboration with LLMs” will last too long. Besides that, AI will (with high probability) change mathematics forever, how are we going to grade different PhD-students, if collaboration with LLMs is allowed? Maybe one of them was just luckier in their collaboration with the LLM to get stronger results or is able to pay for an expensive subscription, while the other is not (as Olof Sisask pointed out).

Another question that I would like to raise is: What will we award the degree of a PhD for in the future?
According to the current PhD-regulations of the University of Hamburg (and certainly similar for other universities), the objective of a PhD is the following.

“To prove the ability of being able to carry out in-depth academic work through one’s own research achievements.”
(German: Durch die Promotion wird die Befähigung zu vertiefter wissenschaftlicher Arbeit durch eigene Forschungsleistungen nachgewiesen.)

But does it count as one’sown research achievement, if one just writes the right prompts for a LLM (as Gowers did when improving the upper bound of N(h,k))?

Probably not. If we allow the full use of AI in a PhD-thesis, we cannot expect every PhD-student to have “the sort of idea (an MIT-student) would be very proud to come up with after a week or two of pondering”. So, it seems like we have to change the objective of the PhD.
But if we weaken its objective, where is the difference between a PhD-thesis and a long master thesis?
And for a bachelor/master thesis, there are already declarations of authorship that try to prevent students from completely writing their thesis (or a draft of it) with AI. Such declarations contain for example: “I hereby declare in lieu of oath that I have written this thesis without assistance.” (University of Hamburg)

When universities realize that it is not too unrealistic that in the near future one can obtain the major results of a PhD-thesis by using LLMs, what does universities prevent from applying such declarations of authorship also to a PhD-thesis?

On the one-hand, this makes sense, since a PhD is just an additional qualification “on top” of a M.Sc. and we don’t want people writing their master thesis using (too much) AI.

On the other hand, this will hinder PhD-students to produce competitive mathematical results.

But there is also a twist to it: What if my peers and I will use LLMs, writing assisted proofs, declaring them correctly and in two years before submitting our PhD-theses, our university decides to change its regulations for AI-support in a PhD-thesis, since there have been too many heavily AI-assisted PhD-theses in the past. Rather an unrealistic scenario, but still possible.

Of course, none of this is set in stone yet, but even critics must admit that there is a (small but) realistic chance that the LLMs will beat us in “mathematical intelligence” like engines have beaten chess players at some point.

So, what will the future of a PhD look like? How do we differentiate between good and better PhD-students/researchers?
Given that LLMs have beaten us (eventually): What will the criterions be for hiring someone for a postdoc/permanent position in academia?

Thank you for reading that far, I’d like to hear your speculations on this 🙂

Best, a slightly too worried PhD-student,
Henri Ortmüller
AI不到两小时搞定一章博士论文，菲尔兹奖还能撑到2030年吗？ -区块周刊BlockWeeks Says:
July 24, 2026 at 4:22 am | Reply
[…] A recent experience with ChatGPT 5.5 Pro […]
Anonymous Says:
August 12, 2026 at 3:22 pm | Reply
Thank you for sharing this fascinating experience. It seems to raise questions not only about what AI can do in mathematics, but also about what academia should value most.

With great respect for inventors, discoverers, and scientists, perhaps AI invites us to reconsider what academia rewards. The lasting impact of inventions such as the bicycle or the car depended not only on who first created them, but also on how they were improved, explained, made accessible, and used in society.

Perhaps mathematics is facing something similar. If proofs and computations become cheaper to produce, human judgment may matter more than raw output volume. This includes choosing meaningful questions, verifying results, explaining them clearly, and connecting ideas across fields. This does not make originality or difficult proofs less valuable; it may simply mean that they should not be the only measures of value.

Rather than focusing mainly on restricting the use of AI, perhaps universities should also teach students how to use these language models wisely, critically, and responsibly. Students should learn not merely to obtain answers from such systems, but to test their claims, understand their limitations, detect errors, and use them to deepen rather than replace their own thinking.

Could this be a moment for universities to rethink how they evaluate mathematical work, valuing understanding, communication, responsible and skillful use of powerful tools, and long-term intellectual or social impact alongside novelty?
Thank you again for sharing your experience and for opening such an important discussion.

S. Shokrollahi
El gato y el ratón: computación cuántica – Wilbert Wusynth Says:
August 24, 2026 at 4:22 pm | Reply
[…] El matemático Timothy Gowers, medalla Fields, ha ofrecido la respuesta más serena a los jóvenes que ahora dudan si entrar en el campo: las matemáticas siguen siendo una habilidad extraordinariamente transferible, y quien investigue hoy quizá no reciba los mismos réditos que una generación atrás, pero se estará preparando muy bien para el mundo que viene. Gowers’s Weblog […]