8/ Takeaways:.I learned a lot about interacting with LLMs — and about Agent Mode. With the right engineering, you can turn an LLM into a genuine research assistant.

7/ CAS tools (Pari/GP, Magma, SageMath) give the right answer instantly… but almost never show what a real proof certificate looks like.

6/ GPT-5 Pro guessed the Galois group correctly as PGL(2, F₇) and justified it — but relied partly on a Chebotarev density heuristic. I wasn’t happy. I pushed it to use the Stauduhar method: heavy group theory + resolvents.

5/ Computing them is hard. The main tool: resolvents (special invariants). I’ve used them before with LLMs/LRMs to study reasoning patterns on this task.

4/ Why bother?.Galois groups are fundamental: they capture polynomial symmetries. They instantly tell you (via Abel’s theorem) if a polynomial is solvable by radicals — the stuff we learned in high school.

3/ GPT-5 Pro acted as the “director” — keeping pieces together, coordinating my workflow, and reviewing correctness step-by-step. ⚠️ Hard maths incoming.

2/ Building verifiable mathematical docs with LLMs is still tough. You must:.• interact with external CAS tools.• rerun code many times (hallucinations are real).• fight the sparse training set problem.

1/ Context is king. I gave the model a detailed PDF, full of formulas and tricky math. That deep context was key to guiding the agent into building correct code.

This entry will be about the interaction with GPT-5 Pro and ChatGPT Agent Mode. My goal was to provide a certificate of correctness for the Galois group computation of a polynomial. x^8 − x^7 + 3x^6 − 3x^5 + 2x^4 − 2x^3 + 5x^2 + 5x + 1.

RT @andrzejdragan: W tym roku mija 25 lat odkąd wykładam szczególną teorię względności, głównie dla najlepszych studentów Wydziału Fizyki i….

I asked Grok 4, Claude Opus 4.1, Gemini 2.5 Pro and GPT 5 Pro the following question:. can a pentagonal prism be called Archimedean polyhedron?. The answer is tricky because there is no obvious consensus here. This is more a judgement question. All 4 models are very different.

The convergence to semicircular distribution was proved by Brian Birch in a paper from 1968. and ChatGPT can help you understand a bit more about this context (text makes sense!).

Visual information is important for understanding. I use a lot Claude Sonnet and Opus for building (vibe coding) applets that enhance my understanding of mathematical formulae. Here is how I use it for understanding vertical Sato-Tate distribution.

@leanprover Done a simple arithmetic formalization in one shot of GPT 5 Pro. My point is to test an absolutely minimalistic setup. I am literally curious how much formalization works already out of the box with the SOTA models.

@leanprover It took a bit of fixing (Lean version mismatch and my ignorance about tactics) but I was trying to get a really vibe coded experience - no intervention on my side. GPT 5 Pro got it slightly faster (but not much). than GPT 5:.

Vibe coding mathematics is fun. I tried to use @leanprover combined with ChatGPT (5 and 5 Pro) to see if I can talk to the model what I want to prove and the model would formalize (code) it for me. Lean is quite complicated so was not expecting much. But it worked!

GPT 5 Pro as interactive research tutor. I have a hard new paper .I prompt the model with a tutoring instruction. The model explains to me in an interactive way the content of the paper. @alexwei_ I only miss a "light" Pro version to run it faster.

Coming soon: A little something I'm writing with Ken Ono.

I tried to develop (via Hopf alebras and differential equations) the core of trigonometry in positive characteristic. Surprisingly I got this similar to existing paper. but it feels like the model hallucinated this idea itself.

Here is a case study in the frontier math research. I know a general answer and have a work in progress preprint where I do more but we spent almost 2 years thinking about this topic. The model got to this level in 15 minutes.