From child prodigy to chief scientist

Chapter 119 Everyone, I've proven myself!

"Let E: y²=3+a+b be an elliptic curve defined over the field of rational numbers, and p be a prime number, and p be a superodd prime of E. Prove that when p>3, the natural density of the superodd prime p that satisfies ap(E)=0 among all prime numbers is 1/2."

The problem that Antoine came up with was a superodd prime counting problem that combined arithmetic geometry and elliptic curves. It seems that this problem was what had just caused the argument between him and his companion.

After hearing the question, everyone thought about it carefully, and quite a few people even started to calculate.

This problem is clearly not a typical IMO problem; its difficulty far exceeds the scope of IMO.

Although the IMO does test questions on "point counting of elliptic curves with modulus p", the questions are usually very basic.

Antoine's question went much deeper, going beyond simply calculating specific values ​​to considering the natural density of prime numbers.

Based on Xu Xiao's experience, this should already be considered an entry-level research question.

However, after a quick thought, Xu Xiao came up with a way to prove it.

"Using the modulus and quadratic features, it should be possible to do it."

Xu Xiaochao borrowed paper and pen from one of his teammates, sat down and worked out the deductions, and then successfully completed the proof.

At that moment, Xu Xiao saw that the number of inspiration fragments on the system interface had increased by one.

"As expected—studying problems with these math geniuses really does yield unexpected benefits."

If I could spend every day here with them, I might be able to complete the task of finding inspiration entries in no time.

Xu Xiao was hesitating whether to share his proof method with everyone when Schultz suddenly announced, "Everyone, I've proven it!"

Schultz's words immediately drew everyone's attention, and they all crowded around to ask him about his proof method.

"My approach is to first transform the problem into a modular coefficient problem, then introduce quadratic eigenvalue torsion, and finally apply Chebotarev's density theorem —"

After listening to Schultz's line of reasoning, everyone looked as if they had suddenly understood.

The question setter, Antoine, nodded repeatedly, clearly very surprised by Schultz's proof approach.

"As expected of Schultz, he solved the problem that had been bothering me for many days so quickly."

Antoine's previous idea was to use Delin's theorem to prove it, but he had not made any progress.

Although Schultz's method is more basic, it is very suitable for this problem.

Schultz didn't feel particularly proud of being praised like that; he was more enjoying the pleasure of solving mathematical problems themselves.

What Schultz didn't know was that Xu Xiao's proof was basically the same as his, and that he had completed the proof long before Schultz.

Of course, since Schultz had already explained the proof method, Xu Xiao didn't claim that he had also completed the proof.

Putting himself aside, Xu Xiao felt that Schultz's mathematical talent was indeed extraordinary, and he truly deserved to be called one of the greatest mathematicians of the 21st century.

After everyone finished discussing the issue, Sasha, the Soviet team member who had just drawn the knot projection diagram on the paper, then showed everyone his question.

"I also have an unsolved problem here—let K be a prime knot, and Vk(t) be its Jones polynomial."

Proof: The Jones polynomial of every nontrivial prime knot has at least one root located on the unit circle at t=1.

Sasha's research is in a completely different direction, belonging to the problem of low-dimensional topological knot Jones polynomial root distribution.

This is also a problem that goes far beyond the scope of IM0, and in this era, it can be considered a cutting-edge problem in low-dimensional topology.

While it's unrealistic to expect to publish a paper based on this type of problem, its difficulty level is at least one or two levels higher than the final IMO problem.

After listening to Sasha's question, everyone focused on thinking about it again.

Xu Xiao hadn't encountered this type of question much before, but of course she had some understanding of the relevant knowledge points.

With the combined effect of her strong thinking ability and mathematical intuition, Xu Xiao quickly found a solution to the problem once again.

"It can be done using the unit root property of Kaufman brackets; it doesn't seem particularly complicated."

Xu Xiao picked up a pen and first drew a projection of a trefoil knot on the draft paper, then proceeded to deduce the result along this line of thought.

After a short derivation process of just a few dozen lines, Xu Xiao successfully completed the proof and once again gained a valuable fragment of inspiration.

Seeing that the others were still deep in thought, Xu Xiao decided to break the silence herself, so that they could move on to the next question as soon as possible.

"Um—everyone, I've thought of a way."

Upon hearing Xu Xiao's words, everyone subconsciously raised their heads and looked at him curiously.

"Xu Xiao, what method have you come up with?"

At this point, most people felt that Xu Xiao had at most come up with a method that could be tried, and had not actually completed the proof.

The respect people have for Xu Xiao stems largely from the fact that he was able to get the opportunity to participate in IMO at such a young age.

But no one really thinks that Xu Xiao's math skills are among the best here.

After all, even Terence Tao only won a bronze medal when he first participated in IM0, which was only considered average among all the team members at the time.

No matter how strong Xu Xiao's mathematical talent is, it is unlikely that he can surpass the other top team members who are a few years older than him at this age.

Schultz also looked at Xu Xiao with curiosity, unsure of what kind of idea he had come up with.

Even now, he has not truly witnessed Xu Xiao's mathematical abilities, and he is filled with anticipation for it.

Under everyone's watchful eyes, Xu Xiao began to explain her method.

"My idea is to utilize the root-of-unity property of Kaufman brackets. First, define the Jones polynomial using Kaufman brackets. From the symmetry of Kaufman brackets, we can deduce that the Jones polynomial satisfies reciprocity —"

Listening to Xu Xiao's explanation, the other team members present gradually realized that Xu Xiao did not just have an immature idea, but had actually proven it.

Sasha reacted the fastest this time; as the one who thought of the problem, he was naturally the most familiar with it.

"I see. I had been trying to use the quantum group approach, but the calculation of the modulus of quantum invariants became more and more complicated, and I didn't know how to proceed. I never thought that we could use the root symmetry of reciprocal polynomials."

While feeling enlightened, Sasha also felt a little regretful. He should have been able to think of such a method, but his stupid, stubborn brain just couldn't come up with it.