Chapter 42: Chapter 26: Shura Field
The office was silent.
The only sound was the TICK-TOCK of the wall clock.
Chen Zhuo looked at the problem.
He took the fountain pen.
That familiar, cold, metallic feeling traveled from the nerves in his fingertips to the center of his brain.
He didn’t start writing immediately.
He was deconstructing the problem in his head.
A prime number, p.
An exponent of p-2.
Divisibility.
These keywords, combined, instantly awakened a theorem from deep within his mind.
Fermat’s Little Theorem.
a^(p-1) ≡ 1 (mod p) (when a is not a multiple of p).
It was one of the cornerstones of number theory.
Chen Zhuo pushed up his glasses.
This problem.
For a middle school student, it was definitely beyond the curriculum. It was practically sadistic.
Even for a high school competition, it wouldn’t be considered easy.
That’s because it required you to not only know Fermat’s Little Theorem, but also understand how to flexibly apply modular inverses.
But in Chen Zhuo’s eyes, this was actually a very interesting problem.
What was 2^(p-2)?
According to Fermat’s Little Theorem, 2^(p-1) ≡ 1 (mod p).
Therefore, 2^(p-2) ≡ 2^(-1) (mod p).
In other words, the modular inverse of 2 under modulo p.
Similarly, 3^(p-2) is the inverse of 3.
And 6^(p-2) is the inverse of 6.
The problem then becomes proving that:
2^(-1) + 3^(-1) + 6^(-1) - 1 ≡ 0 (mod p).
This was too easy.
Chen Zhuo almost wanted to laugh.
1/2 + 1/3 + 1/6 = 3/6 + 2/6 + 1/6 = 6/6 = 1.
- 1 = 0.
Proof complete.
This was the beauty of mathematics.
What seemed like a complex exponential operation, when viewed through the lens of number theory, was reduced to the simplest elementary school fraction arithmetic.
The greatest truths are the simplest.
Chen Zhuo uncapped the pen.
He didn’t use any scratch paper.
He began writing directly in the blank space on the test paper.
No need to draw diagrams, no need to assume air resistance.
All it took were a few clean, sharp lines of congruence equations.
∵ p is prime, p > 3
∴ gcd(2, p) = 1, gcd(3, p) = 1, gcd(6, p) = 1
By Fermat’s Little Theorem:
2^(p-1) ≡ 1 (mod p) ⇒ 2^(p-2) * 2 ≡ 1 (mod p)
...
Chen Zhuo wrote quickly.
The nib of the fountain pen made a soft SCRATCHING sound on the paper.
In less than two minutes, Chen Zhuo stopped writing.
The last line read:
∴ Original Expression ≡ 1 - 1 ≡ 0 (mod p)
Q.E.D.
Chen Zhuo capped the pen and pushed the test paper toward Old Zhao.
"Done."
Old Zhao hadn’t said a word, his eyes fixed on Chen Zhuo’s hand the entire time.
The moment Chen Zhuo wrote the first congruence symbol, "≡," Old Zhao’s pupils dilated.
He knew this was in the bag.
The kid not only knew how to solve it, but he was also using the most standard, most elegant language of number theory.
He hadn’t used the clumsy method of binomial expansion; instead, he cut straight to the heart of the problem.
The modular inverse.
Old Zhao picked up the paper.
He looked at the few lines of beautiful equations.
The logical flow, the elegance of the numbers... it was simply perfect.
"Excellent!"
Old Zhao slammed his hand on the desk, making the lid on his teacup jump.
"A brilliant use of Fermat’s Little Theorem!"
"A brilliant use of the modular inverse!"
Old Zhao looked at Chen Zhuo, the fanaticism in his eyes practically threatening to drown the boy.
"I knew it."
"I knew your brain was born for mathematics."
"That damn physics teacher, Old Zhou, what the hell does he know about this kind of beauty?"
Old Zhao stood up, unhooked a large chain of keys from his belt, and fumbled through them with a JINGLING-JANGLING sound for a long time.
Finally, he found a slightly rusted, brass-colored key.
He placed the key in front of Chen Zhuo.
"Take it."
Chen Zhuo looked at the key. ’These teachers really love giving out keys.’
"What’s this?"
"The key to the archives room on the top floor."
Old Zhao’s voice was filled with a magnanimous air.
"Inside, you’ll find every National High School Mathematics League test from 1985 to the present, as well as a lot of materials from the national training team."
"Those training team lecture notes... I had to swallow my pride and beg my old college classmate, who’s the chairman of the Provincial Mathematical Society, for them. I lugged them back box by box."
"Some of that material has been out of print for years."
"According to school rules, that’s a restricted room. Only the head of the teaching and research group is allowed in."
Old Zhao lowered his voice as if entrusting him with a state secret.
"You can go whenever you want and look at whatever you want."
"As long as you don’t set the place on fire, you can do whatever you want in there."
Old Zhao paused, then continued.
"Also."
"I’ve already signed you up for next year’s National Junior High School Mathematics Competition."
"Starting today, you’re a core member of the school’s math training team."
"From now on, Monday and Wednesday evenings are for math training. Tuesday and Thursday afternoons, you can go fool around with Old Zhou."
"As for your other classes..."
Old Zhao waved his hand dismissively.
"As long as your midterm and final exam scores are decent enough, not only will I approve your absences, but if anyone has a problem with it, they can come see me!"
Dual privileges.
Dual-core drive.
Chen Zhuo reached out and took the brass key.
It was cool to the touch, and a little heavy.
"Thank you, Teacher Zhao."
Chen Zhuo stood up and bowed to Old Zhao.
It was a genuine, heartfelt gesture.
In this day and age, to meet two teachers who would break the rules for him, and even fight over him, was his good fortune.
Chen Zhuo walked out of the office.
The key to the physics lab was in his left pocket, and the key to the math archives was in his right.
He walked with a new spring in his step.
Just as he rounded the corner of the building, he ran right into someone.
Old Zhou.
Old Zhou was holding a textbook under his arm, on his way to teach a second-year class.
He saw Chen Zhuo coming out of the grade-level office, and then he saw Old Zhao standing at the doorway, looking refreshed and triumphant as he watched Chen Zhuo leave.
What kind of person was Old Zhou?
He was sharp as a tack.
His squinted eyes darted around, and he immediately understood what had happened.
Old Zhou stopped in his tracks, blocking Chen Zhuo’s path.
"Old Zhao called you in?"
Old Zhou’s voice was sour, as if he’d just downed a pint of aged vinegar.
Chen Zhuo stopped and nodded.
"Yeah."
"What kind of nonsense did that old fox feed you?"
Old Zhou glanced at the bulge in Chen Zhuo’s right pocket.
"He said physics is the flesh, and math is the bones," Chen Zhuo answered honestly.
"Without bones, flesh is just a pile of mush."
"Hah!"
Old Zhou was so angry he had to laugh.
"That old coot, he’s still got such a sharp tongue."
Old Zhou shifted the book under his arm to his other hand, a look of disdain on his face.
"Bones?"
"What are you with only bones and no flesh?"
"A skeleton! A specimen! Something dead!"
"Physics is what’s alive! It’s flesh and blood, something that can run and jump!"
Old Zhou poked Chen Zhuo’s chest with his finger.
"Kid, don’t you let him lead you astray."
"Mathematics is a tool. A hammer, a wrench. We physicists are the ones who use tools to build machines. Have you ever seen a craftsman bow down to his hammer?"
Chen Zhuo looked at Old Zhou’s expression—a mixture of exasperation and a fierce, protective instinct—and couldn’t help but want to laugh.
But he held it in.
As a perfectly mature adult, he was quite experienced in handling these kinds of situations.
"Teacher,"
Chen Zhuo pushed up his glasses, his tone sincere.
"I think Teacher Zhao has a point."
"Oh?"
Old Zhou’s eyebrows shot up.
"If the hammer isn’t good, you can’t build a precise machine."
Chen Zhuo patted his right pocket, where the key to the math archives was.
"I want to sharpen my hammer first, then I’ll build your machines."
Old Zhou was stunned for a second.
He stared at Chen Zhuo for a few moments.
Then, he smiled.
A smile that smoothed out the wrinkles on his face.
"Alright, you little rascal."
Old Zhou cursed playfully and ruffled Chen Zhuo’s hair.
"Playing both sides, are we? Wanting to have it all?"
"Fine. You’ve got ambition."
"As long as you don’t find it too tiring, go on and run between both."
Old Zhou waved his hand and stepped aside to let him pass.
"Scram, scram. Don’t make me late for class."
Chen Zhuo nodded and walked away with a light step.
Old Zhou stood there, watching Chen Zhuo’s retreating back, then glanced at Old Zhao, who was standing at the office door some distance away, smirking at him.
The two old men, with a combined age of nearly a hundred, glared at each other from a dozen meters apart down the hallway.
"Old coot."
"Old fossil."
The two of them cursed at almost the exact same time.
Then, they each turned away, looking equally pleased with themselves.
Outside the hallway, the sun was shining brightly.
The cicadas were still chirping.
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