I was a top student in America.

Benjamin was slightly disappointed to find that he wasn't the one called to the blackboard.

He already had an idea for this problem.

However, he quickly turned to look at Li Ao on the podium, wanting to see how Li Ao would handle the problem.

In his seat, Michael leaned closer to Kevin, glanced at the half-finished derivation on Kevin's draft paper, and asked in a low voice, "So, have you got any ideas yet?"

"He's made a little progress, but he's still working on it." Kevin shook his head, turning his gaze to the podium. "However, given Li Ao's level of number theory, this problem shouldn't be too difficult for him."

In front of the blackboard, Li Ao held the chalk and scanned the problem conditions from beginning to end.

In just a few seconds, several solutions had already passed through my mind.

He ignored the whispers behind his back and immediately began to write.

"Let u = a^(p-1). The problem statement is equivalent to u = 1 + p^2c. Taking the logarithm of the p-input number, we get v_p(log u) ≥ 2..."

The chalk tapped softly on the blackboard. He skipped the complex congruence expansions and wrote down the conclusion:

「因此，u^(p^(n-1))-1至少含有因子 p^(n+1)。也就是说，a^(p^(n-1)(p-1))≡1 mod p^(n+1)。」

"If we shrink p^(n-1) in the exponent and take the case where c is not divisible by p, the assignment will decrease by one order, and the conclusion will no longer hold."

This is the p-advanced number theory in number theory—using standard tools from advanced mathematics to reduce the dimensionality of competition problems.

After writing the last line, Li Ao turned around:

"Finished writing."

The classroom was quiet for a moment.

Apart from Professor Carlson, the rest of the team members only understood the first half of the congruence condition; they couldn't understand the lines about the p-logarithm.

Benjamin stared at the derivation on the blackboard, his brow furrowing slightly.

Although he didn't quite understand it, his mathematical intuition told him that this approach bypassed a long and complicated congruence expansion and was much simpler than the method he had come up with himself.

"What kind of method is this... It bypasses the tedious congruence expansion, saving so many steps compared to my approach."

As he was pondering this, Professor Carlson, who was standing next to the podium, spoke up.

Carlson was actually somewhat surprised.

He originally thought this question would stump Li Ao, giving this student who had been daydreaming in class a good talking-to. He didn't expect that the other student would not only prove it easily, but also use p-advanced number theory.

After all, this doesn't seem like the kind of knowledge a high school competition student should possess.

"Has this child studied algebraic number theory systematically before?" Carlson wondered to himself.

He coughed twice to cover his surprise, then composed himself and emphasized the limitations of IMO problem-solving tools to everyone:

"Leo, your proof is logically sound. However, using p-advanced number theory to solve the problem does not comply with the IMO rules. In the main competition, advanced calculus tools like p-advanced numbers and group theory are prohibited. If you write it like this in the exam, you won't get a single point."

For students who have been exposed to advanced mathematics, their first reaction to a difficult problem is often to directly call upon the appropriate tools, rather than to think about techniques.

Just like middle school students doing elementary school math olympiad problems, they will subconsciously write out equations.

However, the IMO is not a university final exam; participants must explain the problems thoroughly using elementary mathematics methods.

After hearing this, Li Ao stopped in his tracks as he was about to step down from the podium and nodded: "I see."

After saying that, he didn't hesitate for long and immediately turned around and wrote the elementary solution on the other side of the blackboard.

「设 a^(p-1)=1+p^2t。那幺 a^(p^(n-1)(p-1))就等于(1+p^2t)^(p^(n-1))……」

The chalk writing was done very quickly.

After performing a binomial expansion, the first term after the constant term is exactly 1 + p^(n+1)t, and the subsequent terms all contain p of higher degree, so the remainder is still 1 modulo p^(n+1).

Similarly, if the exponent is missing a p, and t itself is not divisible by p, then the first non-trivial term can only go up to p^n, and the conclusion may not hold.

Writing out this elementary solution won't take much longer than the previous one.

Professor Carlson quickly checked it over.

After confirming that there were no vulnerabilities, he paused for a moment.

He never expected that Li Ao would not only use algebraic number theory methods, but also write so neatly and efficiently after switching back to elementary methods.

Being able to solve the same problem using two completely different approaches in a short period of time demonstrates that this student has a very solid understanding of number theory.

As far as Carlson knows, even some senior undergraduates find it difficult to achieve this quickly.

"No wonder he doesn't pay much attention in class."

He felt more confident, then suddenly remembered something and casually asked Li Ao, "Have you systematically studied algebraic number theory?"

It's not surprising that he guessed that way.

A high school student who can write a proof of a p-adduct number proficiently has most likely taken advanced number theory courses in advance.

After all, there are quite a few students in the national team who have extra time and energy to get in touch with advanced mathematics competitions every year.

Li Ao put down the chalk and answered truthfully, "I haven't studied it systematically, I only have a basic understanding of the theory of p-adjacency numbers."

After he finished speaking, he stepped down from the podium and returned to his seat.

It's normal for someone who's made it into the US IMO national team to have more skills.

If someone can even use tools like p-matrix, which are typically encountered in upper-level university or even graduate school, then their talent is somewhat outrageous.

Even studying for ten or twenty hours a day might not be enough to accumulate that much knowledge.

This was the first time Michael and Daniel had seen Li Ao demonstrate the entire problem-solving process.

The two looked at each other, both seeing surprise in each other's eyes.

"That's incredible, no wonder he's Leo."

Kevin was even more direct; as soon as Li Ao sat down, he gave him a thumbs up.

Benjamin, standing next to him, couldn't help but ask, "Leo, you're not learning ordinary differential equations while also teaching yourself algebraic number theory, are you?"

"No, not at all." Li Ao shook his head.

Before Benjamin could even catch his breath, Li Ao casually added, "I heard about p-advanced number theory by chance at an academic conference a few days ago. I found it interesting, so I looked up some literature on it."

Upon hearing this answer, Benjamin was speechless.

He originally thought he was the most talented one in this national team, but now, looking at Li Ao, he suddenly felt that his previous pride was somewhat untenable.

When he first arrived at summer camp, he was known as a math genius.

But now, compared to Li Ao, why do I seem so mediocre?

Michael had witnessed Li Ao's skill level yesterday, so he found it much easier to accept now.

He looked at Li Ao and exclaimed, "You're really amazing. I'll have to ask you for help with any questions I don't understand in the future."

There are only six people on the national team, and they train together every day.

Having a player of this caliber on the team will benefit him greatly, and it will definitely be a good thing for the team to strive for first place in the IMO overall score.

Carlson, standing on the podium, was also somewhat stunned after hearing Li Ao's words.

He didn't come to his senses until Li Ao returned to his seat.

Finally, I made up my mind in secret:

"No, we have to bring this kid to Princeton."

He could almost imagine that if other coaches knew there was such a talented high school student in the national team, they would try every means to poach him.

While Princeton's mathematics department is renowned, it doesn't have much of an advantage compared to top universities like MIT and Stanford.

Strike first to gain the upper hand; that's the right thing to do.

Benjamin remained silent for a long time, staring blankly at the words on the blackboard.

From the summer camp placement test onwards, he has always regarded Li Ao as his most important competitor, hoping to regain the advantage that belongs to a gold medalist in the formal training camp and the IMO main competition.

But now, no matter how you look at it, the gap between the two seems to be getting more and more obvious.

Meanwhile, Natalia, who was sitting on the other side, would occasionally glance at Li Ao's seat during the second half of the class.

An undisguised curiosity surfaced in her deep blue eyes.

She couldn't quite understand it.

When we first entered the summer camp, we didn't think Li Ao was that outstanding. How come he's become so strong in less than a summer?

……