Algebraity

FAQ on gen AI use in mathematics research

This article was written by Algebraity (Harrison Lemley) on September 11th, 2026, and is available at both algebraity.moe and tanuki-cd.com. 0% of the prose in this article was written by generative AI, and the views expressed within it are solely those of its author and not of any institution with which he is associated. The entire text of this article is licensed under CC BY 4.0.

Table of Contents

What generative AI tools do you use?

In terms of models, I mainly use OpenAI products, specifically GPT-5.6 Sol and the recently released GPT-6 Astra. I also use Anthropic’s Claude Fable 5 and 5.1, the latter of which was released last week. I use ChatGPT for most research tasks, and I use Claude mostly for code and paper reviews. I have the 20x Pro plan for ChatGPT, which I use very heavily, and I have a premium research plan for Claude, which is a little better than the normal 5x premium plan.

As for the tools I use with these models, I rely heavily on the standard “chat” mode of ChatGPT on the "Pro" and "extra high" reasoning settings. I also use Codex / Claude Code and occasionally ChatGPT Work for longer tasks or those which benefit from subagent parallelism.

How do you use generative AI tools?

My typical use is to send a draft or a question to a chatbot on the high, extra high, or Pro reasoning level and use its response as a springboard for my work. In the early stages of research, this might involve asking a model to do a deep search for papers related to a problem I’m interested in or to spend some time trying to see if there is an easy route to a solution. If it finds something that makes the proof trivial to finish after one or two prompts, I'll then try to strengthen the result or generalize; if not, then I will carefully read its output, keep exploring new ways of attacking the problem, get a better strategy, and prompt again.

Outside of the mathematical process itself, generative AI is also very useful for writing up proofs and organizing papers. I’ve found that having a Pro model generate a full draft of a paper from an outline of the paper and drafts of the proofs usually results in something that is 80-90% of the way to being a finished paper. When I carefully read through all of the results myself, I see parts that should be expanded or contracted and arguments that must be adjusted, but the result is usually quite strong right away.

For code, the process is very similar. I'll explore ideas with the model until I've formed a full plan for implementation, and then I'll usually write the header (“.h”) file in C and explain how I want each function to be implemented. Then I’ll let it generate the code, read through it myself, and test it afterward. The code generated by frontier models for my projects is usually very high quality and requires few adjustments to be complete.

How do generative AI tools benefit your work?

A professor instructing a class of mine once suggested that programming without AI is quickly becoming what programming with punch cards in the late 20th century was like. I agree, but I don’t think this is just a direction we’re moving in: we’re already there. I also think this is true for much more than just programming. Another professor once said to me personally that, what took her 5 years in graduate school to learn, an undergraduate student of hers learned in one weekend using AI by analyzing her papers and asking questions about them. While this may be slightly exaggerated, it's not too far from the kind of speedup that I anticipate from efficient and productive use of these tools in mathematics research.

Many parts of my work would easily take weeks to months without AI, yet I am able to complete them in hours to days. Let’s focus on the example of literature review. With AI, I can ask GPT-6 Pro to find papers for me that I can use as inputs to prove a lemma or theorem that I need, which usually takes the model less than 10 minutes. If you add an hour or more of verification and reading to that, we can round up and say it takes 2 hours in total to find the result, verify it, and integrate it into my paper. How long would it take without AI? Well, it would probably involve asking people I know about the result, searching through the literature and reading random papers for hours until I find something that looks promising, and then eventually finding what I need, perhaps after weeks of searching. Of course, it won’t always be this slow, but this is a common story in academia. With AI, getting stuck for weeks like this is much less likely, and the same amount of searching can be achieved orders of magnitude faster.

What about proving lemmas or intermediate results? That might also take days to weeks without AI. It takes a lot of thinking and careful processing to understand what’s going on well enough to formalize it. With AI, I can just ask GPT-6 Pro if it can prove the result, and it often can, so that I can just read and verify the proof and then move on to the rest of the argument instead of spending days to weeks on a small intermediate point. Even if it can’t produce a full proof right away, it’s usually able to make a lot of significant progress and make the rest of the proof much easier for me to complete.

For code, it makes implementation much faster. Especially in cases where the design of the functions is relatively simple but typing everything up and checking for small mistakes would take a long time to do by hand, AI implementation can save a lot of time and produce significantly better output. Understanding 100% of my code is very important to me, but even at this high standard, it’s still many times faster to produce code with AI according to my specifications, read the result carefully, and internalize the code than it is to methodically write each function by hand. I still write some code by hand, but it’s for practice and for fun, not for production, as I cannot justify doing things at 1/5 the speed to get almost the same result.

Why do you use generative AI tools?

My answer is simple: it’s a matter of values. There was a time when I wondered if I should actually use AI as a part of my research, or if it would ultimately be a bad thing for me. I thought about the fact that many mathematicians would refuse to take my work seriously, that I would repeatedly receive criticism for my choice to use AI, and that some would not attribute the intellectual authorship of my work to me.

Nonetheless, I realized that the answer is one of values and priorities. What do I care about? If I have to choose between producing a high-quality paper exhibiting a significant result in a week using AI, or doing so in 3 months on my own, which do I choose? The former gets completed results out to the mathematical community much faster and allows me to learn much more quickly, while attracting criticism and claims of inauthenticity; the latter means others might be more impressed with me and my work, but it also means a lot more time and effort for less scientific impact.

What I decided in the end is that what matters most to me is scientific output. I don’t care if I'm “impressive” to others or if, to the contrary, they see me as a fraud. What I care about is science. When I do mathematics, it’s because I want to bring truths into the world that have never before been imagined, and to disseminate them in a way that is freely accessible and usable to others—that’s why I do research in the first place, and why I release all of my work under open licenses, e.g. GPL and MIT for code and CC BY for papers. Refusing a 20x speedup of my research with no penalty to accuracy and scientific value (and in fact a huge benefit to these things) would be a complete and total rejection of the reason I do my work in the first place and what I want to bring into the world.

In short, I use AI because it makes my work much better and much faster. It means I’m able to identify an open problem, form an approach, develop a proof, write a fully tested and benchmarked implementation, and post a preprint to arXiv all by myself in a single week—indeed, it took me only 8 days to go from an idea to a complete 20-page paper and programmatic implementation in C and Python for 1. If someone doesn’t like that, or if they want to say I’m not a “real mathematician,” that doesn’t bother me. If they take a look at my work, they will see it is “real mathematics,” and at the end of the day, that’s what I care about.

Are your papers really yours?

Yes, they are. I am the sole agent behind any of my papers, except for papers where I have human co-authors. I use AI tools as a part of my work, but that’s it—they don’t do the work for me, as much as a computer doesn't do the work of a mathematician who uses a SageMath or Mathematica script to computationally establish an important claim of their paper. This conclusion is supported by the official stances of prestigious publishers and journals: Springer Nature holds the position that AI is a tool and cannot be credited as the author of a paper (2), and Elsevier permits AI use in the research process but places responsibility for a submission in the hands of its human author (3).

An LLM is not an agent who can take credit or responsibility for a work. I verify every result presented to me by AI manually and work the math out by hand, and I am the only agent who has any final say over my papers. If anything is wrong with my work, I'm the sole individual responsible for it: AI cannot take responsibility for a high-quality paper, nor can it take the blame for a mathematical error.

Do you own your results?

Yes, I do. Richard Stallman and Linus Torvalds do not own my work just because I use the GNU/Linux operating system, nor does Overleaf own any percentage of my work just because I use it to write my papers online. In the same way, no matter how heavily I utilize AI in my work, AI companies do not own my work in any sense (4). LLMs themselves are not persistent entities which can take responsibility or credit for a living breathing scientific result.

Ultimately, no matter how much AI is used in the process of producing my results, I'm still responsible for them. It is I who decides on a problem, originates the search, chooses a proof strategy, verifies all of the results and their novelty, puts the work together into a final verified paper, and takes responsibility for the result. That’s why I feel no conflict between my use of AI and the feeling of creation and discovery that I get from my work: no matter how much this process is enhanced by AI, it is still my process, and the results are my own.

Should I be less impressed with you and your work because you use AI?

Probably. You should also probably be less impressed by someone who produced 5 papers in 2020 than by someone who produced just 1 paper in 1850.

When technology makes work easier, it naturally requires less work to produce the same results. The kind of productivity that would have required a genius 300 years ago can be achieved by a graduate student today. This acceleration is provided by computers as much as it is by AI: the internet has made work much easier and thus less “impressive,” and AI is now in the process of causing a similar change to take place (5; 6).

I say “probably” instead of “yes” because, when the same result takes less work to prove, it’s possible to prove better results. Thus, what one might have been able to do without AI before might be much less valuable than what one can now prove with AI. Nonetheless, the right metric is not how “impressed” someone is by my work: it’s how valuable my work actually is and how much the knowledge I create actually contributes to science.

Should I, the reader of this FAQ, use AI?

If you care about proving stronger results in less time and having a lower chance of publishing an incorrect or incomplete result, then yes, absolutely. Using the strongest frontier models as a part of your research can only help you if you do it in an informed way, and doing research without AI is, as is written on 7, like writing a web app in assembly. Without AI, you will spend a lot more time to get to the same result, and it won’t be better in the end.

This doesn’t mean you have to use AI for every part of your work. At first, you might just want to use it to do a final prose check on your paper and confirm that it cannot find a mathematical flaw in your work. However, the more comfortable you become with AI, the better: you don’t lose any time or risk the accuracy of your work by having LLMs review your work before submission or search for new problems or papers related to your work while you work on something else. The sooner you start using new, efficient tools for your work, the better, and using AI for even just a small fraction of your work is a huge help.

But isn’t AI evil? Doesn’t it use water and electricity and ruin communities?

New data centers used to power AI use water and electricity, and they have the potential to disrupt local communities (8; 9). However, these costs are being brought up in discourse surrounding generative AI tools in a way which is unfair to the technology: the relatively low cost of AI is being blown out of proportion, and its value is being severely understated.

I don’t have the opportunity to go through all of these points in one article, but I’m going to address a few of the major issues here to put the minds of potential future AI-empowered mathematicians at ease and answer the most commonly mentioned points that arise in discourse around AI use.

Water. AI data centers use a lot of water; at the same time, they provide immense value in research, programming, and other kinds of work. Is this benefit worth the cost? To answer that question, we need concrete figures. All data centers in the US (not even just for AI) use around 900 billion liters of water in aggregate each year, including both cooling and electricity generation (8). That sounds like a lot—but California almond orchards alone have been using around 6-10 times that amount each year since before generative AI even existed (10; 11). Since this figure includes more than just data centers used for AI, generative AI tools use less than 20% of the amount of water that California almonds do. But is AI really valuable enough to deserve that much water? Based on the huge enhancement to scientific productivity (5; 6), the answer is a definitive “yes.”

Electricity. In terms of national capacity, the electricity consumed by AI is not a problem. Almost two times the annual electricity use of the entire US could be supplied by the sun alone if corn fields used to produce ethanol were replaced with fields of solar panels—that electricity figure includes data centers (12; 13). So, is electricity a problem here? In terms of raw generation capacity, not at all. Additionally, the total electricity used by AI is not as high as other everyday commodities: in 2020, residential air conditioning in the States alone consumed over 140% of the electricity consumed by all US data centers in 2023 (14; 8), while in 2025, data centers represented only 1.5% of global electricity use despite the growing AI industry (15). This shows that, while the net electricity demand of AI is high, it’s the same order of magnitude of major components of everyday household consumption, while its potential positive influence on science is arguably significantly greater (5; 6).

Effects on communities. While having companies change one’s neighbourhood suddenly can be very frustrating, there are many good reasons for these changes. The local effects of data centers used to power generative AI are real and may be quite significant, but they are fortunately quite limited. The effects on local communities are very concentrated, and the vast majority of people will not be affected: nearly half of total data center capacity of the US is in five small geographic clusters (9). Thus, while local effects are very real, they are isolated, and they do not eliminate the immense global value provided by the technology. Finally, it's worth noting that the effects on local communities are not new and are not one-sided: the introduction of data centers to communities is much like the construction of factories which occurred during the industrial revolution, and while there are some negative consequences, this also has the potential to support the local economy, create jobs, and increase the value of the community, thus making the introduction of data centers a trade-off for local communities rather than a guaranteed net negative.

Are there any risks to using AI for my work?

Yes. While using AI can be greatly advantageous to your work, there are certainly some risks you should keep in mind when deciding how much and what kind of AI use is best for your work.

The main risk is intellectual property. Some AI companies will train their models based on what you enter into their tools, which can create questions of provenance and attribution. This is a real question, and AI companies have been responding to it by giving users more control on whether or not their data is used for model training. OpenAI now allows users to opt out of model training (16), while Anthropic disables it by default and promises to only train their models on the data of users who opt in to this practice (17). Thus, while data provenance is a legitimate concern to take note of before using generative AI tools, it doesn’t mean you can't protect your data while using AI, and it doesn’t have to be a deal-breaker.

Another risk is a problem whose association with generative AI is classical: the possibility of generating results with significant but hard-to-spot inaccuracies or “hallucinations.” This can waste human time and energy and even result in the publication of incorrect results if not caught. Fortunately, this is becoming increasingly uncommon, and there are good ways to prevent this. First, you should never present mathematics you haven’t verified by hand to the public—that’s true whether you use AI or not. Also, even if you do rely heavily on AI for your work, hallucinations and incorrect results are becoming increasingly uncommon for frontier models (18), so wasting your time on inaccurate results is much less likely today than it was a year ago. For example, in my own research, while past models would produce incorrect ideas and I was unable to rely on them for serious work, I’ve personally never seen GPT-5.6 Sol or GPT-6 Pro produce an incorrect mathematical result; instead, they let me know when they have failed to make progress on a problem and report what went wrong instead of confidently presenting a bogus result. Ultimately, unlike deterministic computer programs, AI is fallible, and that’s a real disadvantage—nonetheless, it is far from making the tool hazardous or unsafe if it is used responsibly.

How can I get started using AI?

The first thing to do when getting started with generative AI is to pick the tools you want to use. While open-source models can be sufficient for simple tasks and I'm a strong supporter of open source software, what they're capable of is a fraction of what frontier models developed by leading AI companies can do. For this reason, I recommend getting a premium subscription to ChatGPT or Claude to start. It's possible to start with the $20/month ChatGPT Plus or Claude Pro plans, but I recommend starting at $100/month to get access to the strongest frontier models for math. I have the $200/month plan for ChatGPT and the equivalent of the $100/month Claude plan, and these products are worth every cent and more to me each month for learning and for my research.

Once you've chosen a model, the best thing to do is to try it out and see how it functions on tasks you'd want it to perform. Generative AI is highly customizable, and you have the ability to control how the models respond using a system prompt—thus, after asking a few different questions and testing a model's capabilities, you may wish to tweak its settings to control its presentation, reasoning effort, and priorities for its responses. You should also "stress test" your chosen model(s) on different reasoning settings, giving them problems ranging from graduate-level exercises to open problems in your field and seeing how well each of them performs and which levels are too low or too high for a given kind of task.

Finally, once you are acquainted with what your chosen tool(s) can do, I recommend you put it to use—by asking it what it can do for you. If you describe your research process to the model, explain what part of the process you want it to play, and describe what you've already decided it's good at, it will be able to provide you with ideas on how to use its capabilities for your chosen purpose. I think this is a good way to develop a plan for how to use AI for your work.

Works Cited

[1] Lemley, H. (2026). Quasilinear multiplication in the real Cayley--Dickson tower [Preprint]. arXiv. https://arxiv.org/abs/2609.11588

[2] Springer Nature. (n.d.). AI for our communities. Retrieved September 11, 2026, from https://group.springernature.com/gp/group/ai/ai-guidance-for-our-researchers-and-communities

[3] Elsevier. (2026, June). Generative AI policies for journals. https://www.elsevier.com/about/policies-and-standards/generative-ai-policies-for-journals

[4] OpenAI. (2026, January 1). Terms of use. https://openai.com/policies/row-terms-of-use/

[5] Agrawal, A. K., McHale, J., & Oettl, A. (2026). AI in science (NBER Working Paper No. 34953). National Bureau of Economic Research. https://doi.org/10.3386/w34953

[6] Toner-Rodgers, A. (2024). Artificial intelligence, scientific discovery, and product innovation [Preprint]. arXiv. https://doi.org/10.48550/arXiv.2412.17866

[7] TANUKI-CD. (n.d.). About. https://tanuki-cd.com/about.html

[8] Shehabi, A., Smith, S. J., Hubbard, A., Newkirk, A., Lei, N., Siddik, M. A., Holecek, B., Koomey, J. G., Masanet, E. R., & Sartor, D. A. (2024). 2024 United States Data Center Energy Usage Report (LBNL-2001637). Lawrence Berkeley National Laboratory. https://doi.org/10.71468/P1WC7Q

[9] International Energy Agency. (2025). Energy and AI. https://www.iea.org/reports/energy-and-ai

[10] U.S. Department of Agriculture, National Agricultural Statistics Service. (2025, May 12). 2025 California almond forecast. https://www.nass.usda.gov/Statistics_by_State/California/Publications/Specialty_and_Other_Releases/Almond/Forecast/202505almpd.pdf

[11] Sanden, B. (2009, January 28). Irrigation 101: When, how much & how often to irrigate [Presentation]. University of California Cooperative Extension, Kern County. https://cestanislaus.ucanr.edu/files/111748.pdf

[12] National Renewable Energy Laboratory. (2019). 2019 Annual Technology Baseline: Utility-scale PV. https://atb-archive.nrel.gov/electricity/2019/index.html?m=1&t=insusdschpcb

[13] Technology Connections. (2026, January 30). You are being misled about renewable energy technology [Video]. YouTube. https://www.youtube.com/watch?v=KtQ9nt2ZeGM

[14] U.S. Energy Information Administration. (2024, March 15). How much electricity is used for air conditioning in the United States? https://www.eia.gov/tools/faqs/faq.php?id=1174

[15] International Energy Agency. (n.d.). Artificial intelligence. Retrieved September 11, 2026, from https://www.iea.org/topics/artificial-intelligence

[16] OpenAI. (n.d.). What if I want to keep my history on but disable model training? OpenAI Help Center. Retrieved September 11, 2026, from https://help.openai.com/en/articles/8983130-what-if-i-want-to-keep-my-history-on-but-disable-model-training

[17] Anthropic. (2026, March 16). Is my data used for model training? Anthropic Privacy Center. https://privacy.claude.com/en/articles/10023580-is-my-data-used-for-model-training

[18] OpenAI. (2026, September 3). GPT-6 Astra: A new generation of intelligence. https://openai.com/index/gpt-6-astra/