What Should I Do If I Can't Spend All the Money? Waiting Online

Sorry, yesterday's chapter was blocked...

——From the headline: Qubit Compared with directly copying and pasting the question to GPT-4, Tao Zhexuan did three things: First, do some prompt projects to clarify the "identity" and "goal" of GPT-4. The identity is "a mathematical co-expert who is good at giving technical advice", and the goal is to "provide some advice" rather than solve the problem. Then, rather than directly copying large paragraphs of formulas into the GPT-4 question column (which is indeed possible), Tao Zhexuan chose to convert the question description into Markdown format. Finally, when it comes to content that requires GPT-4 to find information by itself, Tao Zhexuan converted it into a reference note and gave the URL at the end. The complete questioning process looks like this: GPT-4 quickly gave 8 specific suggestions: 1. Understand the problem: Before in-depth proof, make sure you understand each item in the problem. If you are unfamiliar with the Diaconis-Graham inequality, study it and understand what each term in the expressions a(n) and R(n, q) represents. 2. Simplify expressions: Sometimes, complex expressions can be simplified. For example, you might want to see if the binomial coefficient of a(n) in the expression can be simplified, or if the sum can be calculated in a simpler way. 3. Induction: Since R(n, q) is defined recursively, a natural way to prove the conjecture is to use mathematical induction. You might want to show that if this conjecture holds for n, then it also holds for n+1. 4. Generating functions: Another method is to use generating functions. 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 provide some in-depth understanding of their relationship. This can sometimes be achieved using Stirling's factorial approximation, or by analyzing the dominant term in the expression. 6. Numerical evidence: Although it is not a proof, collecting numerical evidence can sometimes let us understand how to prove a conjecture. Since you have PARI/GP code that can check the conjecture numerically, you may want to read the first few items of each sequence to see if you can spot any patterns. 7. Look for similar problems: Sometimes, a problem or conjecture is a generalization or variant of a known problem. Look for similar problems or theorems in the combinatorial literature and see if their proofs can be applied to this problem. 8. Ask for feedback and collaborate: 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 often key to solving complex mathematical problems. In the end, Tao Zhexuan used suggestion 4, which is the method of generating functions, to successfully solve the problem. Tao Zhexuan also tried the method of asymptotic analysis (suggestion 5), but it turns out that using generating functions is more effective. To put it simply, Tao Zhexuan only uses GPT-4 to provide himself with different problem-solving ideas, which is "problem-solving inspiration", but he will not leave it to GPT-4 to solve the specific problem, which is a more mathematical method. As for the application of GPT-4 in "searching information", Tao Zhexuan has also taken it to the extreme. For example, when thinking about a problem, he would take out his "eureka" and discuss it with GPT-4. For example, he felt that this problem was similar to Catalan numbers and hoped that GPT-4 would help him find the corresponding information. GPT-4 quickly gave a corresponding answer, which also prompted Terence Tao to generate new inspiration for another question. To put it simply, in just two conversations with GPT-4, Terence Tao demonstrated the correct way for mathematicians to use GPT-4 - to find inspiration and look up information.