After taking the advice, I became a top student.

"Let's go, we'll talk as we walk."

After Chen Mo left the classroom, Zhou Zhi spoke up, taking the lead towards the administration building. He still didn't understand what had happened and asked curiously, "Have you finished all the questions for this provincial competition?"

"It's done."

Chen Mo nodded.

How difficult did you find the questions?

"It's quite difficult!"

Chen Mo answered truthfully.

Zhou Zhi didn't know what to say. Since it was so difficult, why did you hand in your paper early?

"If the information is correct, you should have won the provincial first prize this time."

Time was of the essence, so Zhou Zhi tried to get to the point: "The president of the Chengdu Mathematical Society and two professors from the Department of Mathematics at Yenching University are coming to see you. If you do well, you might have a chance to be recommended to Yenching University."

For the students of No. 1 High School, Yanbei University was a place that only existed in legends; they would never even dream of such a thing.

Therefore, even if there is only a very small chance, Zhou Zhi hopes that Chen Mo can seize the opportunity.

Chen Mo nodded, but didn't care too much. He was confident that getting into Yanbei University would not be difficult, whether through competitions or the college entrance examination.

……

"Not bad, not bad, your foundation is quite solid."

In the conference room, Zhao Yiming nodded.

Bai Zhi just explained her thought process for solving the first sub-question of the third question. Although she didn't solve the second and third sub-questions, it was still quite good for a high school student.

At this moment, Chen Mo and Zhou Zhi walked into the conference room.

After much anticipation, it finally appeared.

Zheng Mingyang looked at the young boy with his tender face, but saw a maturity on his face that was beyond his years.

"I am Zheng Mingyang, a professor in the Department of Mathematics at Yenching University."

Zheng Mingyang stood up and introduced himself, "Chen Mo, in the competition, when you proved the third sub-question of the third major problem, you wrote this constant equation ∑_k=1^{q}χ(k)e^{πik/q}=√q⋅ε(χ). I am very interested in this constant equation. Can you tell me how you came up with it? Is it always true?"

As he spoke, Zheng Mingyang arrived at the whiteboard in the conference room and wrote down the equations with a marker.

Chen Mo looked at it, somewhat embarrassed, and said, "Professor Zheng, actually I was just simplifying the calculation at the time, and I didn't provide a rigorous proof."

Zheng Mingyang was somewhat disappointed, but he hadn't really expected Chen Mo to be able to provide a proof anyway. "So how did you come up with this equation back then?"

"In the process of deriving the third question, I felt that the conventional proof was too cumbersome, so I wondered if there was a simpler way to prove it, and then this equation came to my mind."

Chen Mo truthfully stated that he was in a state of "clear mind" at the time, a state that even he cannot replicate now.

"oh."

Zheng Mingyang had no doubts; mathematics is indeed sometimes like that, requiring that fleeting moment of inspiration.

But inspiration is destined to remain just inspiration; it cannot be grounded in reality as a theorem or formula.

Every theorem and formula requires a rigorous proof.

pity.

Zhao Yiming was also somewhat disappointed. He had originally thought that this trip would be fruitful, but he did not expect it to end like this.

However, he should have realized this long ago. How could a high school student possibly complete such a proof, and solve a problem that even an old professor who has studied this field for decades could not solve?

They were just being too unrealistic from the start.

Seeing this, the leaders of No. 1 Middle School finally realized that these two professors from Yanbei University were not there to discuss the recommendation for admission, but to ask Chen Mo some questions.

Zhang Hu turned to look at Zhou Zhi, his eyes questioning.

Good heavens, he was also a science and engineering graduate, but he couldn't understand the formulas on the blackboard at all.

Zhou Zhi was also somewhat bewildered, but he could roughly understand what Chen Mo and Zheng Mingyang were discussing.

However, what he didn't understand was that a professor from Yanbei University had traveled a long way just to ask a high school student for advice.

Is this reasonable?

"But I feel it should hold true because... well... I verified a few minor cases."

Chen Mo looked at the equation on the whiteboard and suddenly spoke.

For some reason, the last big problem from when he first joined the math competition team flashed through his mind. He had some inspiration back then, and if Bai Zhi hadn't asked him, he probably wouldn't have written out the complete solution, and he might never have known that the method was correct.

"So, perhaps I can try to prove it!"

Chen Mo suddenly developed a strong interest in this equation.

"Which ones were verified?" Zheng Mingyang asked from the side.

陈末来到白板前，拿起马克笔，在白板上写写画画起来，「比如q=4，取χ为模4的非主特徵χ(1)=1，χ(3)=−1，那幺左边等于1⋅e^{πi/4}+(−1)⋅e^{3πi/4}=e^{πi/4}−e^{3πi/4}。

e^{πi/4}=√2/2+(i√2)/2，e^{3\pi i/4}=−√2/2+(i√2)/2，相减得√2，右边√4=2，所以ε(χ)=√2/2，模长确实为1。」

"What if q=5?"

Zheng Mingyang nodded. He had also been trying to prove this equality in the past few days and had naturally conducted similar verifications, so he knew that Chen Mo had not made a mistake.

Chen Mo frowned, thought for a moment, and said, "There are four types of non-principal features modulo 5, but if it is the Legendre symbol χ(k)=(k/5), then the left side is ∑_k=1^{4}(k/5)e^{πik/5}, which I have also calculated and is √5, so ε(χ)=1."

Zheng Mingyang frowned: "So you only verified quadratic features? What about the other features? For example, non-quadratic features modulo 5?"

Chen Mo paused for a moment, then said, "Ah, non-quadratic eigenvalues? I haven't verified it, but I feel that this equality might only hold true for real eigenvalues. During the exam, I happened to use the case with real eigenvalues, because the combination of the Möbius function and the Euler function only leads to real eigenvalues."

Zheng Mingyang's eyes lit up: "Go on."

Chen Mo continued writing on the whiteboard, "Let's re-examine this invariant. Let χ be a real eigenvalue modulo q, i.e., χ(k) = ±1, and χ(−1) = 1 or −1. Then χ is actually a quadratic eigenvalue, which is related to a certain quadratic field..."

"So consider Gauss and..."

Chen Mo kept writing down formulas on the whiteboard, as if this were not a meeting room, but his study room, and the whiteboard in front of him was not a whiteboard, but his draft paper.

He was completely absorbed in this proof.

The whiteboard was limited, and soon it was filled with Chen Mo's formulas. Then he picked up an eraser, erased the formulas he had just written, and began to write new formulas.

Zhang Hu was about to speak when Zheng Mingyang stopped him, shaking his head to indicate that he should not disturb Chen Mo.

He knew that Chen Mo's condition at this moment was something that could not be forced.

In Chinese terms, this can be called a sudden enlightenment!