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[@ErnestRyu](/creator/twitter/ErnestRyu)
"I used ChatGPT to solve an open problem in convex optimization. *Part I* (1/N)"  
[X Link](https://x.com/ErnestRyu/status/1980759528984686715) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-21T22:14Z 7924 followers, 1.2M engagements


"@DocOctagonical ChatGPT generated many interesting leads that ended up being dead ends. I really appreciate those but I'm not showing the failed attempts"  
[X Link](https://x.com/ErnestRyu/status/1980767396865667256) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-21T22:45Z 7923 followers, 13.4K engagements


"@hyhieu226 @OpenAI Thanks Seb and Kevin reached out so I should be able to proceed with GPT5 Pro credits"  
[X Link](https://x.com/ErnestRyu/status/1980826243839979866) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-22T02:39Z 7923 followers, 15.4K engagements


"Yesterday I made a post about using ChatGPT to solve an open problem in mathematics. The post blew up and I got some fun comments. Heres one that I particularly liked: 2/N"  
[X Link](https://x.com/ErnestRyu/status/1981236024618725834) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-23T05:47Z 7923 followers, 1930 engagements


"For 1r3 there is some nuance. (i) The analysis for r=3 relied on a particular dissipative (i.e. decreasing) energy function. Similar energy functions need to be found for the 1r3 case. 4/N"  
[X Link](https://x.com/ErnestRyu/status/1981633393092960700) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-24T08:06Z 7924 followers, 1084 engagements


"However when prompted to solve the 1r3 case ChatGPT frequently made calculation errors and worked with an incorrect energy function that was not actually decreasing leading to invalid conclusions. 5/N"  
[X Link](https://x.com/ErnestRyu/status/1981633465100415310) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-24T08:06Z 7924 followers, 1034 engagements


"I used ChatGPT to solve an open problem in convex optimization. *Part III* 1/N"  
[X Link](https://x.com/ErnestRyu/status/1981977037276672040) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-25T06:52Z 7924 followers, 131.6K engagements


"Indeed translating the continuous-time proof into the discrete-time setup turned out to be relatively straightforward. With one piece of feedback ChatGPT was able to work out the proof. 4/N"  
[X Link](https://x.com/ErnestRyu/status/1981977491733356831) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-25T06:53Z 7924 followers, 2644 engagements


"ChatGPT did not produce the proof in a single prompt. The process was highly interactive. It generated many arguments roughly XX% of which were incorrect. (7/N)"  
[X Link](https://x.com/ErnestRyu/status/1980760255236821502) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-21T22:17Z 7924 followers, 38.2K engagements


"@SebastienBubeck @kevinweil ChatGPT is now at the level of solving some math research questions but you do need an expert guiding it. This exercise was a lot of fun and was highly productive. I also feel I'm getting better at prompting ChatGPT. I'll also try other open and unsolved problems. (16/N N=16)"  
[X Link](https://x.com/ErnestRyu/status/1980762374987821451) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-21T22:25Z 7923 followers, 35.4K engagements


"ChatGPT noticed this resolves the open problem. But instead of congratulating (paraphrased): Well. that doesnt *really* solve the open problem because this only works in finite dimensions. What if you were optimizing in an infinite-dimensional real Hilbert space 6/N"  
[X Link](https://x.com/ErnestRyu/status/1981236556963991622) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-23T05:49Z 7924 followers, 4660 engagements


"Thinking through conditions like these takes time since the possibility of non-convergence is subtle. However ChatGPT quickly produced a counterexample revealing that this line of reasoning could not succeed. This answer easily saved me XX minutes. 9/N"  
[X Link](https://x.com/ErnestRyu/status/1981236897868632367) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-23T05:51Z 7923 followers, 1287 engagements


"From this formulation ChatGPT quickly produced a sequence of arguments that led to a contradiction. Once again this automation easily saved me about XX minutes. The final proof is a streamlined reformulation of this contradiction. 11/N"  
[X Link](https://x.com/ErnestRyu/status/1981237325737972046) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-23T05:52Z 7923 followers, 2892 engagements


"I used ChatGPT to solve an open problem in convex optimization. *Part II* 1/N"  
[X Link](https://x.com/ErnestRyu/status/1981632663623762422) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-24T08:03Z 7924 followers, 24.5K engagements


"For 0r=1 ChatGPT impressively constructs a divergent counterexample in one shot. 3/N"  
[X Link](https://x.com/ErnestRyu/status/1981632994801758634) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-24T08:04Z 7923 followers, 1113 engagements


"So I sifted through the energy functions ChatGPT produced and found X correct ones. In the subsequent queries I explicitly specified these energy functions to prevent ChatGPT from going down incorrect paths. 6/N"  
[X Link](https://x.com/ErnestRyu/status/1981633587100406175) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-24T08:07Z 7924 followers, XXX engagements


"(ii) In the r=3 case the proof had X steps: X(t) is bounded and X(t) converges. ChatGPT easily proved convergence for 1r3 when prompted with the r=3 proof and the correct energy functions E and F. 7/N"  
[X Link](https://x.com/ErnestRyu/status/1981633764901232979) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-24T08:08Z 7923 followers, XXX engagements


"The r = X case contains the key technical insight. I expected the generalization to X r X to be largely straightforward given the r = X proof and ChatGPT was indeed very helpful in carrying out this extension. 10/N"  
[X Link](https://x.com/ErnestRyu/status/1981634352208359915) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-24T08:10Z 7924 followers, 1735 engagements


"In particular Radu Ioan Bo Jalal Fadili Dang-Khoa Nguyen reached out with a preprint of their own capitalizing on the ideas in the continuous-time proof to also establish point convergence of Nesterov 1983 and more @RaduIoanBot 8/N"  
[X Link](https://x.com/ErnestRyu/status/1981978375758684624) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-25T06:57Z 7924 followers, 2350 engagements


"@RaduIoanBot They also argue that the FISTA method Beck Teboulle 2009 exhibits point convergence. Check out their work when it goes public (As far as I know Bot Fadili Nguyen 2025 did not use AI tools.) 10/N"  
[X Link](https://x.com/ErnestRyu/status/1981978704587931840) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-25T06:58Z 7924 followers, 1978 engagements


"@RaduIoanBot Conclusion: Starting from the key idea of the first tweet we extended the convergence result to several related settings and resolved the main 42-year-old open problem with ChatGPT doing most of the heavy lifting along the way. 11/N"  
[X Link](https://x.com/ErnestRyu/status/1981978806509597149) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-25T06:59Z 7924 followers, 1930 engagements


"@RaduIoanBot Overall this entire journey took just a week less than XX hours of my time. ChatGPTs assistance provided a significant speedup and without it I would most likely have given up after three days of slow progress. (As I did in the past.) 12/N"  
[X Link](https://x.com/ErnestRyu/status/1981978967965151273) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-25T06:59Z 7924 followers, 1909 engagements


"@RaduIoanBot Again ChatGPT is now at the level of solving some math research questions but you do need an expert guiding it. I strongly encourage fellow mathematicians to try incorporating AI assistance into their workflow. It takes some getting used to but it can be worth it. 13/N"  
[X Link](https://x.com/ErnestRyu/status/1981979046432100718) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-25T07:00Z 7923 followers, 3063 engagements


"@DocOctagonical The final proof corresponds to a geometric intuition saying that the X(t) must oscillate between z1 and z2 indefinitely and the difference of energy functions yields an ODE mapping this oscillation onto the line through z1 and z2. This ODE shows the oscillation isn't possible"  
[X Link](https://x.com/ErnestRyu/status/1980767724621164671) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-21T22:46Z 7893 followers, 3248 engagements


"This is really exciting and impressive and this stuff is in my area of mathematics research (convex optimization). I have a nuanced take. 🧵 (1/9)"  
[X Link](https://x.com/ErnestRyu/status/1958408925864403068) [@ErnestRyu](/creator/x/ErnestRyu) 2025-08-21T06:00Z 7922 followers, 612K engagements


"I used ChatGPT to solve an open problem in convex optimization. *Part I (b)* 1/N"  
[X Link](https://x.com/ErnestRyu/status/1981235878354980981) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-23T05:46Z 7923 followers, 24.4K engagements


"First and foremost some folks at OpenAI kindly reached out and granted my request for additional GPT-5 Pro queries which was very nice Using the Pro queries Ill continue the AI-accelerated math research. 4/N"  
[X Link](https://x.com/ErnestRyu/status/1981236152767279141) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-23T05:48Z 7917 followers, 1346 engagements


"(iii) However we were unable to prove boundedness. ChatGPT could not find a proof so I spent the time to work through the problem myself. After a reasonable amount of effort I am convinced that ChatGPT is correct that the technique used for r = X does not extend to r X. 8/N"  
[X Link](https://x.com/ErnestRyu/status/1981634016068686152) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-24T08:09Z 7922 followers, XXX engagements


"Funnily enough ChatGPT acted like a bit of a hater. Yesterday I typed up the proof and carefully verified the arguments myself. Then just to further double-check I asked ChatGPT to check the proof. 5/N"  
[X Link](https://x.com/ErnestRyu/status/1981236301115641975) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-23T05:48Z 7923 followers, 1497 engagements


"Well thanks for the encouragement Chat 😒 Ill look into extending the argument to Hilbert spaces. Challenge accepted. 7/N"  
[X Link](https://x.com/ErnestRyu/status/1981236629189906618) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-23T05:49Z 7923 followers, 1381 engagements


"How did ChatGPT accelerate the search Example #1. At a point I derived the following differential relationship: Q= tA+tB/2 Q = A-B where Q is bounded and AB0. Are these conditions sufficient to guarantee Q(t) converges to a limit If so that would complete the proof. 8/N"  
[X Link](https://x.com/ErnestRyu/status/1981236738103398775) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-23T05:50Z 7923 followers, 1307 engagements


"Example #2. At one point I arrived at the ODE 2t a(t)+4a(t)+o(1)=0 where a(t) oscillates infinitely often between two cluster points. However this first-order ODE doesnt seem oscillatory. If I can draw a contradiction from this I would complete the proof. 10/N"  
[X Link](https://x.com/ErnestRyu/status/1981237019037819297) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-23T05:51Z 7923 followers, 1154 engagements


"Next we extend the analysis to the generalized Nesterov ODE. Prior work establishes convergence for r3. We established convergence for r=3. What about 0r3 2/N"  
[X Link](https://x.com/ErnestRyu/status/1981632905349890153) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-24T08:04Z 7923 followers, 1139 engagements


"This lemma statement is the best that we could prove in place of boundedness. When prompted with the lemma statement and the r=3 proof ChatGPT easily produced a correct proof the lemma. 9/N"  
[X Link](https://x.com/ErnestRyu/status/1981634155776802909) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-24T08:09Z 7923 followers, XXX engagements


"NAG was introduced by Nesterov in 1983 as an accelerated improvement upon plain gradient descent yet its point convergence remained unresolved until today. Nesterov ODE was studied as a simplified proxy aimed at gaining insight into the behavior of its discrete counterpart. 3/N"  
[X Link](https://x.com/ErnestRyu/status/1981977293892592060) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-25T06:53Z 7924 followers, 2797 engagements


"@RaduIoanBot The arXiv preprint of Bot Fadili Nguyen 2025 will go public in a few days. They also prove weak convergence in the infinite-dimensional Hilbert space setting. (ChatGPT will now be happy.) 9/N"  
[X Link](https://x.com/ErnestRyu/status/1981978490716172484) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-25T06:57Z 7924 followers, 2977 engagements


"@bootcoff I am an expert in this area and I know the relevant prior works (and the authors) but I also asked ChatGPT to double-check. The prior work doesn't already solve this question"  
[X Link](https://x.com/ErnestRyu/status/1982279109180055668) [@ErnestRyu](/creator/x/ErnestRyu) 2025-10-26T02:52Z 7923 followers, XX engagements