Just two days ago, ChatGPT was ridiculed by the group, but then it solved a difficult math problem.

A few days ago, OpenAI researchers claimed that GPT-5 had "discovered" solutions to 10 long-standing mathematical problems with cash prizes. The public mistakenly thought that GPT-5 had provided the solutions. However, it was later found that it had only retrieved existing literature, which triggered ridicule from academic giants and intense discussions about the exaggerated publicity in the AI field and AI's retrieval capabilities. Refer to the report "Did OpenAI 'Solve' 10 Mathematical Problems? Demis Hassabis Called It 'Embarrassing,' and Yann LeCun Gave a Caustic Comment."

Ironically, while people were still debating whether AI was a qualified "literature researcher," a real mathematical discovery had quietly taken place.

AI Achieves a Research Breakthrough

Ernest Ryu, a mathematics professor at the University of California, Los Angeles (UCLA), tweeted, "I used ChatGPT to solve an unsolved problem in convex optimization."

Subsequently, he introduced his joint achievements with ChatGPT through a series of tweets.

First, let's take a look at the problem he studied:

Well, it's hard to understand, but we can let AI help us (AI scores another great achievement!):

This mathematical problem explores a well - known dynamic system in optimization theory. We can use a vivid physical analogy to understand it: the rolling process of a ball in a bowl. In this analogy, the so - called "convex function" f represents a perfectly shaped bowl with a smooth interior. The slope from the edge to the bottom of the bowl gradually decreases without any depressions or small hills. The bottom of the bowl may be a sharp point or a wide flat area, which is called argmin f in mathematics. And X(t) describes the position of a ball in the bowl at time t. The core differential equation in the screenshot, Ẍ(t) + (3/t)Ẋ(t) + ∇f (X (t)) = 0, is the "physical law" that controls how the ball rolls. Here, ∇f (X (t)) plays the role of "gravity," constantly pulling the ball down the steepest slope; while (3/t)Ẋ(t) is a very special "friction force." Its peculiarity lies in that it gradually weakens over time. At the beginning, the friction force is strong and can effectively slow down the ball. But as time t gets larger and larger, this friction effect becomes weaker and weaker. The whole problem is to release the ball from rest at an initial position X₀ on the bowl wall and then observe how it moves under this unique set of physical rules.

The real core and challenge of this problem lie in the need to strictly prove that the rolling ball will not only reach the bottom of the bowl but also come to a complete stop at a specific point on the bottom of the bowl. On the surface, this seems taken for granted, but it is a profound problem in mathematics. Mathematicians have already proved that the "height" f (X (t)) of the ball will inevitably approach the lowest height at the bottom of the bowl as time goes by. In other words, we are 100% sure that the ball will eventually enter the lowest area at the bottom of the bowl and will not stop halfway up the slope. But this is only "function value convergence." The real "outstanding problem" is whether the "position" X (t) of the ball will also converge. If the bottom of the bowl is a wide flat area, after the ball reaches this area, will it slide, oscillate, or circle endlessly due to inertia, just like a spinning top on a smooth surface? This problem requires proof that precisely because of the special time - varying friction force of 3/t, it can exhaust all the kinetic energy of the ball in just the right way and finally guide it to park at a fixed position instead of drifting eternally in the lowest energy state. This has been an open problem that has attracted a lot of research for a long time because it touches on the cornerstone of the convergence theory of optimization algorithms.

Below is ChatGPT's proof, which has also been sorted out by Professor Ernest Ryu:

He also shared the original interaction record: https://chatgpt.com/share/68f805f2-b8fc-8010-8df6-20a46bc1df44

From this record, we can see that the model he used is GPT - 5 Pro, and the model performed 22 minutes of reasoning for this problem.

Similarly, the analysis given by AI based on this is that the solution X (t) of the Nesterov ODE (ordinary differential equation) will eventually converge to a minimum point X∞ of the function f.

We can also see in the proof that the distance between z₁ and z₂ is 0, which means that they must be the same point. This contradicts the initial "assumption that there are two different points." Therefore, the initial assumption is wrong, so the ball can only stop at one point.

Ernest Ryu also introduced his process and thoughts: "My reaction: ChatGPT did effectively accelerate my progress. This work took about 12 hours over 3 days. Looking back now, the proof process is actually quite simple."

He continued, "But I tried many other strategies without success, and ChatGPT was crucial in helping me quickly explore and eliminate these dead - ends. Moreover, the key successful step was also proposed by ChatGPT."

However, he also pointed out that ChatGPT's success was not achieved overnight: "ChatGPT did not give the proof all at once. The whole process was highly interactive. It put forward many arguments, about 80% of which were wrong. But some ideas were really novel to me. Whenever I realized a novel idea, whether it was correct or not, I would extract the key insights and prompt ChatGPT to further develop it."

Ryu also summarized the respective contributions of himself and ChatGPT:

Finally, he pointed out, "This result can already be published in an authoritative optimization theory journal. However, I still want to further improve it." In the future, he also plans to generalize the proof to ODEs with r > 0 and try to "transform this argument into a proof of the convergence of the discrete - time counterpart method (i.e., Nesterov's accelerated gradient method)."

He summarized, "ChatGPT is now at a level where it can solve some mathematical research problems, but it really needs an expert to guide it."

Interestingly, he mentioned that the biggest obstacle in his research process was "running out of ChatGPT Pro queries," and he was already using the "expensive Pro plan" and had to wait for the next month's refresh.

Of course, this was a quite good publicity opportunity, and an OpenAI staff member had contacted him and provided more credits.

AI Becomes the First Author of a Paper

Coincidentally, Paata Ivanisvili, a mathematics professor at the University of California, Irvine (UCI), also claimed a few days ago that GPT - 5 Pro had helped him discover a counter - example to a proposition.

What's even more interesting is that he has just announced that he will list ChatGPT as a co - author of his paper, and the first author!

Of course, this is not the first time that AI has appeared as an author in a serious academic paper. As early as 2023, ChatGPT was listed as the third author of a paper. Refer to the report "An Author of a Paper Has Become Famous. When Can Large Language Models Like ChatGPT Become Co - authors of Papers?" However, it is worth noting that ChatGPT no longer appears in the author list of the latest version of the paper.

Screenshot from 2023. Now ChatGPT no longer appears in the author list of the paper.

AI Assists in the Proof and Becomes the Second Author

After the so - called "Did OpenAI 'Solve' 10 Mathematical Problems?" incident a few days ago, two human researchers encountered a similar embarrassment. After announcing that they had successfully solved the #707 Erdos problem, they found that this problem had actually been solved 30 years ago!

However, they did not stop there. Instead, they continued to let GPT - 5 write a Lean formal proof, which was successfully verified. Of course, they also emphasized the importance of expert guidance and feedback.

In short, we can see that both ChatGPT and Lean are included in the author list of their paper.

Of course, the practice of listing AI as an author of a paper is still highly controversial.

Conclusion

Incidentally, in the comment section of the aforementioned relevant tweets, we can also see some other information about researchers making progress in their studies with the help of AI:

The story of Professor Ernest Ryu, together with the experiences of other researchers, reveals an upcoming new era: AI may no longer be just a tool; it is becoming a research partner.

This means that in the future, top - level scientific research may no longer be a heroic solo act but a deep dialogue and collaboration between human experts and powerful AI.

So, what about you in front of the screen? Have you used AI in your research work? How was your experience? Welcome to share your story.

Reference Links

https://x.com/ErnestRyu/status/198075952898468