Shulou(Shulou.com)11/24 Report--

Goose Meizi Boing, talented mathematician Tao Zhe Xuan engaged in mathematical research, has been inseparable from ordinary people's hands of the "math chicken" GPT!

Under one of his latest math puzzles, Tao Zhe-Xuan made it clear that he "used GPT-4", which offered him a feasible solution.

With the help of GPT-4, he not only successfully broke through this problem, but also shared the answer on MathOverflow:

It provides me with the final idea of solving the problem, and then I just need to keep doing the calculation.

In order to share the convenience of working with GPT-4 with more mathematicians, Tao Zhe Xuan also po his chat notes, which fully record his conversations with GPT-4.

As you can see, in this chat transcript, he called GPT-4 a "professional math collaborator", not just an ordinary math assistant.

This identity is really unusual. I wonder if Tao Zhe-Xuan will list GPT-4 as a co-author (manual dog head) when he writes his paper later.

Not only that, poking the "continue this conversation" button at the bottom of the conversation record, you can also import the conversation record into your own ChatGPT with one click, suddenly breaking into the conversation between the human master and AI.

What new problems did Tao Zhe Xuan solve? The question Tao Zhe Xuan studied was updated on the math website MathOverflow two days ago.

This problem is called "elegant Recursive A301897", which looks like this:

Among them, A301897 is a series included by the integer series website OEIS:

There are a series of numbers from the permutation b of length n in this series. One thing they have in common is that they satisfy the Diaconis-Graham inequality and the equal sign holds.

Diaconis-Graham inequality was jointly proposed by Persi Diaconis and GE Liheng (yes, the father of "maximum number" Ronald Graham) in 1977. it belongs to the study of probability and statistics.

Obviously, this long question is not only mixed with a large number of professional mathematical formulas, but also involves web pages that need to be redirected.

As a "math chicken" who is not very good at mathematics, can GPT-4 really read formulas and give professional advice to solve problems as quickly as a professional mathematician?

In fact, instead of copying and pasting the question directly to GPT-4, Tao Zhe Xuan did three things:

First of all, do some prompt projects to clarify the "identity" and "goal" of GPT-4. The identity is "a cooperative math expert who is good at giving technical advice", with the goal of "providing some advice" rather than solving problems.

Then, instead of copying large pieces of formula directly into the GPT-4 question bar (which he can do), Tao Zhe-Xuan chose to convert the problem description to Markdown format.

Finally, when it comes to the content that needs to be found by GPT-4 himself, Tao Zhe Xuan converts it into a quote comment and gives the URL at the end.

The complete questioning process looks like this:

In other words, instead of asking GPT-4 to directly incarnate as a mathematician to "solve problems", Tao Zhe-Xuan only regarded it as an "expert" who provided inspiration and suggested some research directions for himself.

GPT-4 quickly came up with eight specific suggestions:

1. Understand the problem: before you prove it in depth, make sure you understand every item in the question. If you are not familiar with the Diaconis-Graham inequality, study it to see what each term in the expressions a (n) and R (n, Q) stands for.

2. Simplified expression: sometimes, complex expressions can be simplified. For example, you might want to see if the binomial coefficients of a (n) in the expression can be simplified, or if the sum can be calculated in a simpler way.

3. Induction: because R (n, Q) is recursively defined, a natural way to prove conjecture is to use mathematical induction. You may want to prove that if this conjecture holds for n, then it also holds for nasty 1.

4. Generating function: another method is to use generating function. Try to find the generating functions of a (n) and R (n, 0), and then prove that they are equal. This can sometimes turn a complex combinatorial problem into a simpler algebraic problem.

5. Asymptotic analysis: as you mentioned, analyzing the asymptotic behavior of a (n) and R (n, 0) may have some in-depth understanding of their relationship. This can sometimes be achieved by Stirling's factorial approximation or by analyzing the dominant term in the expression.

6. Numerical evidence: although it is not proof, collecting numerical evidence can sometimes let us know how to prove a conjecture. Because you have PARI / GP code, you can check the conjecture with numerical values, and you may want to read the first few items of each sequence to see if you can find any patterns.

7. Look for similar problems: sometimes, a problem or conjecture is a generalization or variation of a known problem. Look for similar problems or theorems in the combinatorial literature to see if their proofs can be applied to this problem.

8. Seek feedback and cooperation: don't hesitate to ask for feedback from colleagues, or post some of your results on a forum like MathOverflow. Collaboration and peer review are usually the key to solving complex mathematical problems.

In the end, Tao Zhe Xuan solved the problem successfully using recommendation 4, that is, the method of generating functions.

Of course, the process of solving the problem does not need to use GPT-4, using MAPLE worksheet + paper and pen manual calculation can be completed. Tao Zhe-Xuan also tried the method of progressive analysis (recommendation 5), but it turns out that generating functions are more effective.

To put it simply, Tao Zhe-Xuan only uses GPT-4 to provide himself with different ideas for solving problems, that is, "problem-solving inspiration", but how to solve them, that is, a more mathematical method, he will not give it to GPT-4 to complete.

As for the application of GPT-4 such as "check data", Tao Zhe Xuan has also played to the extreme.

For example, when thinking about a problem, he will also take out his "flash of inspiration" and discuss it with GPT-4. For example, he feels that this problem is similar to Catalan numbers and hopes that GPT-4 will help him find the corresponding information.

GPT-4 quickly gave a corresponding answer, which prompted Tao Zhe Xuan to have a new inspiration for another question.

To put it simply, in just two conversations with GPT-4, Tao Zhe Xuan showed the correct posture for mathematicians to use GPT-4-looking for inspiration and looking for information.

In this way, even a "math chicken" such as GPT-4 can become a mathematician's AI assistant.

In addition to sharing chat notes between Human Master and AI on GPT, Tao Zhe-Xuan's mastodon blog post is accompanied by an intimate guide to his experience of using ChatGPT and GPT-4.

According to his past practical experience, the first most important point is:

Don't try to get AI to answer questions directly, as it will almost certainly get some professional-looking nonsense.

To prevent GPT from becoming the king of nonsense literature, the effective solutions are as follows:

Let AI play the role of collaborator, and then let it provide strategic advice.

Like this:

Besides, what is the use of "math chicken" GPT in the hands of great mathematicians?

Tao Zhe-Xuan probably means Aunt Sauce:

Although ChatGPT's mathematical ability is not good, it is a good tool for people who do academic research to diverge their thinking.

(it's a little unprofessional for ordinary people, but just right for mathematicians.)

How to explain the phrase "divergent thinking"?

Tao Zhe-Xuan expressed the view that since the answers given by ChatGPT on specific mathematical problems are not entirely correct, it is better to give full play to its characteristics of generating partially correct answers.

In short, let it help you find inspiration balabalabla:

When dealing with mathematical problems, you can have large language models like ChatGPT do some semi-finished semantic search.

In other words, ChatGPT doesn't have to provide exact answers, just generate some possible prompts.

In this way, according to the prompts generated by GPT + traditional search engine search, it is easy to get the answer.

And he also revealed that he had obtained access qualifications from Microsoft before the release of GPT-4.

This is the same version of Microsoft's 154-page "Spark of AGI" paper, which is full of blood without security training but with stronger ability.

From Tao Zhe Xuan's feedback, we can see that GPT-4 is very good at doing some cosplay when talking to human beings, such as acting as a compassionate listener, an enthusiastic responder, a creative source of inspiration, a translator or teacher, or a spokesman for the devil.

At the same time, Tao Zhe-Xuan made bold but rigorous predictions about the performance of AI in mathematical research:

When integrated with tools such as formal proof validators, Internet search, and mathematical symbol packages, AI in 2026, if used properly, will become a trusted co-author in mathematical research, as well as in many other fields.

In addition to mathematical research, GPT-4 has been an omni-directional assistant in Tao Zhe Xuan's life.

He often uses GPT-4 to answer random, vaguely worded questions that previously required careful keyword tuning in search engines.

There was also a colleague who was unhappy because his relatives had received a critical diagnosis. To this end, Tao Zhe Xuan waved his hand and let GPT-4 write a letter of condolences.

How did it turn out? The colleague was moved to cry with tears in his eyes.

Finally, let's talk about Tao Zhe-Xuan using GPT-4 to solve math problems.

Under MathOverflow, some netizens feel that he should not use GPT to answer math questions, which feels like a very sensitive topic.

However, there are still some people who express that they think it is really Taikuri.

Tao Zhe-Xuan was not shy about coming forward to make his position clear, and he didn't think there was anything wrong with it:

The concern now is no different from the focus of discussion in the early days of Wikipedia's popularity.

It is common for everyone to get the initial clue on Wikipedia and attach a link to the argument to show that it is part of my argument.

And Tao Zhe-Xuan's view is quite firm, that is, "I believe that in the future, everyone will think that there is nothing wrong with using GPT to support research."

Join the conversation between Tao Zhe Xuan and GPT-4:

Https://chat.openai.com/share/53aab67e-6974-413c-9e60-6366e41d8414

Reference link:

[1] https://mathoverflow.net/questions/449361/elegant-recursion-for-a301897

[2] https://mathstodon.xyz/@tao/110601051375142142

[3] https://finmath.stanford.edu/~cgates/PERSI/papers/77_04_spearmans.pdf

This article comes from the official account of Wechat: quantum bit (ID:QbitAI), author: Hengyu Xiao Xiao

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