Chapter 326 - 284: Mathematics, There Is Only One Mathematics

Chapter 326: Chapter 284: Mathematics, There Is Only One Mathematics

2:00 PM.

Fudan University, Guanghua Building East Main Building, a 500-seat lecture hall.

The lecture hall was already packed.

The front row of seats was empty, reserved for several old professors from the School of Mathematics and the School of Physics.

The middle rows were filled with graduate and doctoral students from Fudan University’s School of Mathematics and School of Physics.

The vast sea of heads in the back was a motley crew.

Undergraduates from Fudan, mathematics graduate students from Jiao Da, applied math students from Tongji, doctoral students from East China Normal University...

Even the nearby Fudan High School and Shangzhong had sent a few of their top high school math competition students to attend.

A buzz of discussion filled the room.

"How old is East God this year, again?"

"I saw his resume. Nineteen."

"Holy shit, he’s so young."

"That’s normal. Just look at the history of mathematics."

"Gauss proved the law of quadratic reciprocity at eighteen."

"Galois wrote down his theory on permutation groups at seventeen."

"Abel worked on the insolvability of the general quintic equation at twenty."

"You shouldn’t judge these geniuses by their age."

"Right. You just look at their work."

Amidst this buzz of discussion,

The front door of the lecture hall clicked open.

Academician Shen Wei walked in first, followed by Li Dong.

More than five hundred pairs of eyes snapped toward the stage.

Li Dong was dressed simply in a dark hoodie and jeans.

He nodded to Academician Shen, then walked onto the stage alone.

He wasn’t carrying anything.

He simply stood before the lectern.

He glanced at the students below.

Then, he smiled.

"Good afternoon, everyone."

"I’m Li Dong."

"I’m here today to give a little lesson."

His tone was very calm.

As calm as an ordinary university professor giving a regular specialized course.

"You all thought I was going to talk about the Langlands Program?"

His tone shifted.

"Well, I’m not going to."

A ripple of laughter went through the classroom.

"To be honest, I don’t really want to talk about that stuff anyway."

"Because... your professors will teach it to you in the future. It’s in the textbooks now, after all."

The students below laughed again.

Li Dong paid them no mind and continued.

"Today, I want to play a little game with you all."

"A game?"

Someone in the audience couldn’t help but mutter under their breath.

Li Dong turned around and, stroke by stroke, wrote a line on the blackboard behind him.

[For each positive integer n, how many pairs of integers (a, b) satisfy a² + b² = n?]

After he finished writing, he turned back around.

"A very simple question, isn’t it?"

"An elementary school student could do it."

"So, let’s start at the elementary school level."

He began to list examples on the board.

"n = 1."

"a² + b² = 1."

"The solutions are (±1, 0) and (0, ±1). Four pairs."

"n = 2."

"a² + b² = 2."

"(±1, ±1). Four pairs."

"What about n = 3?"

Low murmurs spread through the audience.

Li Dong smiled.

"Zero pairs."

"Why? A square number modulo 4 can only be 0 or 1. Their sum can only be 0, 1, or 2."

"So 3 is impossible."

"n = 4, four pairs. n = 5, eight pairs. n = 6, zero pairs. n = 7, zero pairs. n = 10, eight pairs."

With each number he announced, the students in the audience scribbled notes faster and faster.

After listing this series, Li Dong turned around.

"Find the pattern."

"Who wants to stand up and tell me what it is?"

The audience fell silent.

Then, a boy wearing glasses sitting in the back raised his hand.

Li Dong pointed.

"You, student. Please stand."

The boy stood up.

Several undergraduates from the Fudan University School of Mathematics sitting in the back rows immediately turned their heads to look at him.

"Damn, Zhang Heng is here too?"

Zhang Heng.

A third-year Ph.D. student at the Fudan University School of Mathematics.

He did his undergraduate studies at Yan University and won a silver medal at the 2018 IMO.

He was currently studying analytic number theory under a Changjiang Scholar at Fudan University.

Within the School of Mathematics, he was known as the type of reliable grad student whose advisor would list him as second author on their next paper without a second thought.

Zhang Heng stood up and cleared his throat.

"This sequence of numbers, I feel..."

"It should be related to the prime factorization of n itself."

"If a prime of the form 4k+3 appears to an odd power in the factorization of n, then there are no solutions."

"If all such primes appear to an even power, then a solution exists."

"As for the specific number of solutions, I’d have to calculate it."

He didn’t answer slowly.

It was the standard textbook answer for someone studying analytic number theory.

Li Dong smiled, without saying whether he was right or wrong.

"A very standard answer."

"Let’s hear from someone else."

Another hand went up in the audience.

This time, it was from the section further back, near the door.

A round-faced, lively-looking young man of about twenty-one or twenty-two.

"Qiu Jiayuan."

Someone nearby immediately muttered.

"The one from Shuimu?"

"Yeah, the 2022 IMO gold medalist. A junior."

"A few months ago, Terence Tao even retweeted a short note he’d casually written."

Qiu Jiayuan stood up.

"I won’t approach it from prime factorization."

"I’ll approach it from a geometric perspective."

"Your question is essentially asking how many integer lattice points lie on the circle centered at the origin with a radius of √n."

"If you sum the solutions for all n up to N, you get the total number of integer lattice points within a distance of √N from the origin."

"By estimating the area, it’s approximately πN."

"Gauss gave this estimate when working on the circle problem."

Li Dong nodded.

"Mm, a geometric perspective. Correct."

"Anyone else?"

Another hand slowly rose from the audience.

This time, it was a girl in a baseball cap, over by the wall.

Li Dong gestured to her.

"Go ahead, student."

She stood up, her voice quiet.

"I’ll approach it using generating functions."

"Consider the theta function θ(q) = Σ q^(n²)."

"The number of solutions, r₂(n), is the coefficient of the qⁿ term in θ(q)²."

"So the problem is, in essence, about studying the object θ²."