After taking the advice, I became a top student.

Others might not be able to keep up with Chen Mo's pace, and because the earlier parts had been erased, they would be completely unable to understand what Chen Mo was writing later on.

But Zheng Mingyang was able to understand it. He had been studying the proof for several days and was an expert in the field, so of course he could understand Chen Mo's proof.

But precisely because he could understand it, when he saw Chen Mo deduce step by step and was about to arrive at the final answer, Zheng Mingyang's heart began to churn.

He initially thought Chen Mo was just lucky and happened to write a form.

Unexpectedly, this first-year high school student gave a complete and rigorous proof on the spot, and the proof was extremely ingenious. He used the Gaussian sum modulo 2q to increase the exponent, and then used parity splitting and duality to connect the unknown sum with the known Gaussian sum.

He had studied analytic number theory for twenty years and was thoroughly familiar with the theory of Gaussian sums, but he had never thought of using this modulus technique to estimate linear exponential sums.

This technique, though simple, requires a strong intuition in number theory and a deep understanding of characteristic structures.

Chen Mo wrote very quickly, muttering to himself.

Vice Principal Zhao Zhenzhong quietly leaned close to Zhou Zhi's ear: "Old Zhou, what characteristics did he mean? Are they personality traits?"

Zhou Zhi's lips twitched slightly: "Professor Zhao, it's a Dirichlet characteristic... a type of number-theoretic function."

"oh."

Zhao Zhenzhong nodded, then asked, "What does 'mod 2q' mean?"

Zhou Zhi paused for two seconds before choosing to tell the truth: "Old Zhao, why don't you stop asking now? I'll explain it to you when we get back."

Zhao Zhenzhong wisely kept his mouth shut.

On the other side, Jiang Fan was frantically searching for Gauss and Gauss on his phone.

However, the search results only confused him more; the formula on Wikipedia was more complicated than he had imagined.

Hu Xin remained relatively calm, because he had long since given up on trying to understand.

He was wondering, is this student still the same Chen Mo from my class?

He thought of the questions Chen Mo asked him after every class, then looked at the whiteboard in front of him, "So, what have I done to deserve answering questions for such a prodigy?"

"Therefore, T = ψ(2)S + ψ(2)S{conjugate}."

"This is a real number because it is conjugate symmetric."

"And |T|=√2q is known, because ψ is a quadratic eigenvalue, and the modulus of the Gaussian sum is the square root modulus."

"Let ψ(2) = ±1, then T = ±(S + S{conjugate}) or T = ±(S − S{conjugate}), depending on the specific sign."

"But in either case, there is always a factor |S| = |T|/2..."

Chen Mo quickly calculated and finally wrote it down on the whiteboard.

|S| = √q!

The office was silent for three seconds.

Then, Zheng Mingyang suddenly stood up.

"This..." His voice trembled slightly.

Although he had some expectations after seeing Chen Mo get into the zone, he still couldn't quite believe his eyes when Chen Mo actually provided complete proof.

After staring blankly at the whiteboard for a long time, Zheng Mingyang finally turned to Chen Mo and asked, "How did you come up with the idea of ​​using 2q?"

Chen Mo never expected that he would actually prove it in the end. Just like when he was explaining the problem to Bai Zhi, he only had a general idea at first, but as he deduced step by step, the conclusion came out as naturally as water flowing into a channel.

Math doesn't seem so difficult after all!

"Since e^{πik/q}=e^{2πik/(2q)}, I wanted to transform the problem into a Gaussian sum modulo 2q, and then I discovered that the quadratic eigenvalue modulo 2q can be decomposed..."

"No, that's not what I'm asking," Zheng Mingyang interrupted him. "I'm asking how you came up with the idea that after parity-even splitting, the odd-numbered terms and even-numbered terms would be conjugates? I've been doing number theory for twenty years, and I've never thought about this symmetry from this perspective before."

Chen Mo thought for a moment and said, "Actually, it's just... mapping the summation interval [1, 2q] to itself, using the transformation k↦2q−k. This transformation turns odd numbers into odd numbers, even numbers into even numbers, and also turns the exponent into a conjugate."

Then I noticed that ψ(2q−k)=ψ(−k)=ψ(−1)ψ(k). As long as ψ(−1)=1, the odd-numbered terms can be perfectly paired with the even-numbered terms.

"Is it that simple?" Zheng Mingyang's voice was a little hoarse.

"It's that simple." Chen Mo nodded. "The only thing is... you need to choose the right construction for ψ so that ψ(−1)=1."

Zheng Mingyang slowly sat back down in his chair, staring blankly at the whiteboard.

He recalled his struggles over the past two days and nights, having read dozens of papers and tried various methods, including perihelion integrals, Poisson summation, and asymptotic expansion of the L function. Each method only sank him deeper into the mire.

This high school student used a symmetry technique from elementary number theory to provide a clean and concise proof in just a few tens of minutes.

It's not because the proof is difficult, but because it's too simple.

It's so simple it's incomprehensible, why didn't I think of it?

Zheng Mingyang closed his eyes, his mind replaying Chen Mo's deduction process over and over again.

He had seen that transformation k↦2q−k countless times.

This transformation is ubiquitous in proving the quadratic reciprocity law and in analyzing the properties of Gaussian sums.

But he never thought of using it for lifting molds.

The problem modulo q is elevated to modulo 2q, utilizing a larger space to accommodate symmetry, and then the unknown and known quantities are linked through parity splitting.

This is a kind of dimensional reduction attack. It doesn't simplify complex problems, but puts simple problems in a larger framework, allowing the symmetry to reveal itself.

"Genius..." Zheng Mingyang murmured to himself.

He opened his eyes, looked at Chen Mo, and said solemnly, "Chen Mo, your proof is enough to publish a short paper."

Chen Mo was taken aback: "Huh?"

He suddenly remembered the suggestion from netizens. At that time, he felt that he was still a long way from publishing a paper. He didn't expect that he would be able to finish it so soon.

"I'm serious." Zheng Mingyang stood up, walked to the whiteboard, and pointed to the line |S|=√q. "This result itself is not new; Gauss and the theory already knew it."

But your proof method, your technique for symmetrical matching in mold raising, is new; it's beautiful, very beautiful!

On the other side of the conference room, everyone was in a state of collective bewilderment.

Zhao Zhenzhong was already looking at the watch.

It wasn't because he was impatient, but because he needed to confirm whether he was dreaming.

The symbols on the whiteboard were like gibberish to him, but he gleaned one thing from Professor Zheng's reaction: this student was very capable.

That's enough!

As the vice principal, all he needed to know was this result.

Zhou Zhi's expression was complex.

As a math competition coach, he had studied some elementary number theory and could understand some of the symbols written by Chen Mo.

Gauss and he knew, and he understood conjugate...

However, he completely missed the conjugate pairing after the odd-even split; he could only pretend to be thinking, while his brain had actually crashed.

Jiang Fan is a physics teacher, and number theory is like a language from another planet to him.

But he has one advantage: he admits what he doesn't understand.

He was holding his phone, quietly sending a WeChat message to Zhou Zhi: "Old Zhou, what's he saying?"

Zhou Zhi glanced at his phone but didn't bother to reply.

Hu Xin's feelings were the most complicated at this moment.

He recalled that two months ago, Chen Mo couldn't even understand functions in class.

Now, this student is explaining cutting-edge techniques in number theory to a university professor.

Who am I? Where am I? What am I doing? These three questions kept playing in his mind.