Anonymous Board,
[Duck Lord, are you going far away?]
-Scary people are suddenly roaming around the school. It looks like our Duck Lord has done something amazing, so what are we supposed to do now?
You’re not going to suddenly leave for America, or toddle off to university saying, “You guys aren’t on my level, so I can’t hang out with you anymore!” are you?
I’m so worried. I’ve become a body that can’t live without Duck Lord anymore, you know?
└N-no way!
└But thinking that our Duck Lord is being recognized makes me feel proud too.
└That’s true. If Duck Lord gets his hands on it, a stupid coloring problem is nothing lol
└What if Elijah Kronen locks Duck Lord in a cage and takes him away?
└Seeing our school on the news felt so weird. Still, I thought, “Of course!” It just feels like Duck Lord could do anything, you know?
└Maybe later he’ll solve all the Millennium Problems too.
└When that happens, I’m bragging to everyone lololol, saying I went to the same school as Duck Lord.
└If that happens, won’t he rise to the ranks of the saints like Pythagoras?
└Don’t you think that’s a fitting place for Duck Lord? lol
└Duck Lord, are you watching? If you are, wag your tail so we can see!
* * *
[If I were to awaken after a thousand years, this would be the first thing I would ask people.]
“Has the Riemann Hypothesis been proven?”
-David Hilbert, at the International Congress of Mathematicians in Paris, 1900.
I had a strange dream.
The bald old man I had only ever seen in photographs appeared in my dream, grabbed me by the collar, and interrogated me about what on earth had happened to the Riemann Hypothesis.
David Hilbert, the German mathematician, presented twenty-three problems that twentieth-century mathematicians ought to challenge themselves with at the congress held in Paris.
Now, more than a hundred years later, thirteen of them have been solved, seven have only been partially resolved, and three still remain unsolved.
‘Why did Grandpa Hilbert grab me like that? Is there some vengeful spirit attached to the Riemann Hypothesis?’
Probably the most famous difficult problem in human history.
Nearly all the greatest minds of the twentieth century threw themselves at the Riemann Hypothesis.
But no one managed to bring down that impregnable fortress.
Wasn’t that why Hilbert had grabbed me by the collar in my dream?
“What have you people been doing for a hundred years? Why still haven’t you proven it!”
The old man’s booming shout seemed to ring right beside me.
Brrr.
A chill ran through my body. It seemed I really should take the herbal medicine Mom had given me.
I was discharged after a week in the hospital.
Before I returned to school, Mom held on to me and spoke with a stern expression.
“If something like this happens again, we’re going straight back to Okcheon together. I’m not just saying this, so listen carefully. I’ll take you back even if I have to force you.”
I had no excuse. I hadn’t wanted to worry her…
I promised my mother that nothing like this would ever happen again.
After making me swear several more times, Mother returned home with my little sister.
My heart ached when I saw Seoeun with tears brimming in her eyes, saying we had to be apart again.
“Let’s play a lot together during vacation!”
We linked pinkies and promised.
When I returned to the dormitory, the building was empty.
‘Ah, it’s class time!’
Today was Thursday. I had decided to rest for two more days and attend classes starting next week.
‘Hm?’
When I opened the door to my room, cards were packed beneath the gap.
I opened them one by one, and they were all from people I knew.
[Seha, hurry back so we can play soccer together!]
[It feels strange that the window seat on the third floor of the library is empty. It’s like an NPC disappeared.]
[This is the newspaper club. Are you still not interested in an interview?]
[Hey! Don’t go too far away. We still need to be able to see your back if we’re going to follow you!]
[When you come back, let’s meet in the music room.]
My chest tightened.
I looked at them for a while, then carefully tucked them between the pages of a book.
‘Classes still won’t be over yet.’
With nothing to do, I trudged down the narrow path toward the music room.
At the hospital, no one had given me a pen or notebook. I didn’t know what kind of agreement the adults had reached among themselves, but they hadn’t even brought me books to read, so I had to keep going in and out of the hospital’s children’s library.
Then what I thought of was the piano.
I asked for a sketchbook and drew keys on it the same size as real ones.
All that remained was practice.
After passing along the path, I opened the door to the music room, and cool air mixed with the smell of chalk rushed over me.
Click.
When I turned on the light, the piano gleamed like black waves.
I opened my bag and took out the sketchbook keyboard I had brought from the hospital.
Sitting on the hospital bed, I had practiced piano all day long on this paper keyboard.
Following the image in my head, I tried striking the real keys.
I pressed “C” very lightly with my thumb.
Ding—
Left hand, C-G. Right hand, E-D-C.
I repeated the movements I had practiced thousands of times on paper. It wasn’t easy at first, but I gradually grew used to it. Having read the lesson book helped a great deal too.
Ducky had been quiet since that day.
But I knew he had not disappeared. The hunger of his that grated on my nerves like fingernails scraping metal had merely been satisfied for a while.
For the time being, I thought I would be able to live without any particular difficulty.
Ducky’s Theme was a gloomy piece.
Ducky especially liked piano sonatas. In the slow movements, the left hand laid down the same figure low beneath everything while the right hand’s melody came alive little by little. He liked that structure.
While the bass descended note by note, making a path, the melody belatedly placed notes over it.
‘A safe descent.’
That was the feeling it gave me.
Was this what he had been thinking as he sank within my consciousness?
Then, finding light, it rose violently.
“You play pretty well.”
When I turned my head, Gyeoul was there.
Unlike usual, her hair was tied back into one ponytail, and she was wearing sneakers.
Her face was flushed, as if she was out of breath.
Gyeoul came in with an unconcerned expression and set her bag down on a chair.
“I was passing by and heard the piano.”
She gave the reason first, as if making an excuse.
Seha decided to thank her first.
“The CD you gave me was a huge help. Thank you for so many things. Is there any way I can repay you?”
Gyeoul glanced at Seha from the corner of her eye.
“That problem you said took seven years—you solved it in the end, didn’t you?
I’ll save my request for now. I think I’ll have a use for it later. More importantly, keep playing the piano.”
Gyeoul lightly pressed Seha’s wrist and corrected the angle.
Seha steadied his breathing and began playing “Ducky’s Theme” again.
“It’s a piano sonata? Did you compose it yourself?”
“Yeah.”
“I like it. You told me before that if someone is good at music, they probably have talent in math too, right? I think the opposite is also true. For one thing, your left and right hands are separating really well.”
To use both hands separately, the brain’s left and right hemispheres had to exchange signals back and forth.
For Seha, it was as natural as everyday life.
“Is that so?”
“Yeah. Do you know how many kids can’t do that? I don’t know much about math, but I like the piece you made too.”
“I didn’t expect that.”
Seha had thought that the piece Ducky composed was excessively dark and static.
“It has a clear musical theme. The left hand strikes steadily like a clock, while the melody on top of it keeps changing little by little. The way you place variations over this ostinato—a repeating bass—belongs to the line of Baroque passacaglias or chaconnes. It feels similar to Bach too.”
Seha stared blankly at Gyeoul. That was exactly the point Ducky had referenced.
“It’s amazing that you could tell just by listening.”
Seha was sincerely impressed by her. Gyeoul looked satisfied with Seha’s expression.
“But the harmony also has a Romantic color to it. It’s interesting in a lot of ways.”
When Seha turned his head toward her, Gyeoul hurriedly straightened her back. Since she had been helping him with the keys from behind, the two of them had ended up far too close.
“All right! Let’s try again. It isn’t finished yet, right? When the piece is complete, make sure to tell me. I want to try playing it.”
Gyeoul glanced at the clock.
The minute hand had already passed the time she had reserved the practice room for.
But what she was doing now was no different from practice anyway, so she thought it didn’t matter.
* * *
Institute for Advanced Mathematics, third-floor conference room,
Seha received a call early in the morning. It was a polite request for him to come in, as they had some questions.
‘Do they have something to ask about the verification?’
After sitting in a chair and waiting for a short while, Elijah Kronen personally opened the door and came in. In his hands were two large mugs with steam rising from them.
Thud.
“Cocoa is fine, right?”
“Yes!”
Facing a person he had only ever seen in papers, Seha couldn’t help feeling nervous.
He had heard that Kronen was an incredibly frightening person, but throughout the whole meeting, he wore a gentle expression and asked about trivial things.
Questions like what subjects Seha studied at school, what he had used to learn mathematics when he was young, and whether he had friends.
In the end, unable to endure it any longer, Seha spoke first.
“Um… Didn’t you have something to ask me about the proof?”
Kronen burst into laughter.
“Hahaha!”
Seha had no idea what expression he was supposed to make in front of him.
“You know that proof is complete, and I know it too. So is there any need to waste more time?”
“Pardon?”
“The verification will be handled by capable colleagues. Today, I simply wanted to meet a promising mathematician.”
Kronen’s expression changed as he looked at Seha.
“By the way, you read my paper. You suppressed the oscillation of the zeta function with a Fourier transform.”
Seha’s eyes went round.
“Yes! That’s right. Originally, I had no idea what to do with the tail that kept diverging in the limiting interval, but once I applied that transform, all the noise cleared away. It was a truly incredible idea!”
As Seha spoke about mathematics, his eyes sparkled like a child’s.
The corners of Kronen’s mouth lifted.
“You think so too?”
Several theories had been employed in Seha’s proof of the Four Color Theorem, and among them, Kronen’s “oscillation suppression transform” had played a key role.
Kronen spoke at length with Seha about how he had arrived at that theory. He also told him about several research topics he was currently working on, almost as if showing them off.
It was an enjoyable conversation.
For Seha, this was the first time in his life he had met someone he could communicate with so well.
“So, what do you plan to do from now on?”
Kronen had been worried that, like Perelman, Seha might lose his motivation for research after proving a difficult problem. That was why he had arranged this meeting, but fortunately, his fears had been needless.
He could see it in Seha’s eyes.
The face of a mathematician impatient to begin another line of research at once.
“I still have a lot to learn. Challenging difficult problems like the Millennium Problems right now would be arrogance.”
Kronen heard Seha’s words differently.
‘Though if I were given enough time, I could solve them anytime.’
“Still, there must be some problem you want to solve.”
He was curious.
What problem this boy had set his heart on. If it happened to overlap with his own, he would have to give him advice.
“If I had to say, I’m interested in the Riemann Hypothesis.”
“Hmmm.”
A low sound emerged from Kronen’s mouth.
‘Is he the type who cares about honor more than he appears?’
The Riemann Hypothesis was, quite literally, a mathematician grinder.
But if one solved it, honor incomparable to anyone else in the world would follow. Kronen considered it a perfectly understandable choice.
“The reason?”
Seha hesitated, then answered in a tiny voice.
“The truth is, I’ve been interested in prime numbers for a long time. So I tried solving Goldbach’s Conjecture on my own. But I concluded that in its current state, there’s absolutely no answer.”
“What?”
Kronen was startled.
[2+2=4, 3+3=6, 3+5=8…]
In this way, Goldbach’s Conjecture, which states that “every even integer greater than 2 can be expressed as the sum of two prime numbers,” is simple, yet one of the most difficult problems to prove.
“To prove that, you would have to rewrite number theory from the ground up.”
It would require a new system, one that would practically change every mathematics textbook.
“So…”
“Ah!”
Kronen let out an exclamation as if he finally understood.
“You impudent little brat!
You’re thinking that the Riemann Hypothesis is much easier than Goldbach’s Conjecture! You intend to get a hint from the Riemann Hypothesis and eventually prove Goldbach’s Conjecture.”
The Riemann Hypothesis is deeply related to the distribution of prime numbers.
Therefore, if one could solve it, one would obtain an extremely powerful clue for proving Goldbach’s Conjecture.
This child was treating the Riemann Hypothesis, which had frustrated more mathematicians than any other problem in history, as little more than a bridge toward his goal.
Seha’s face turned red.
“I-it’s just that the Riemann Hypothesis has had a lot of research done on it already…”
As many mathematicians had attempted it, there was also a great deal of accumulated material.
There were tens of thousands of papers related to the Riemann Hypothesis alone, and Seha believed that there had to be a clue somewhere among them.
“And?”
“Somehow, I felt like I could solve it.”
“A feeling.”
The intuition known only to those born with talent.
Tap, tap.
Kronen tapped the table with his fingers.
“It is the inspiration most necessary in mathematics. But it is also dangerous.”
“I heard that you’re challenging one of the Millennium Problems too, Professor.”
“Why, are you curious?”
Kronen grinned at Seha.
“A little.”
“If you promise to come to Princeton, I’ll tell you. Everything, even my progress. I’ll say this in advance: I’m quite close.”
“Then I’m fine.”
Seha shook his head with a disappointed expression.
For the first time in his life, Kronen failed to scout a student.
“Listen, boy!”
He stopped Seha as he stood to leave.
“Yes?”
“Listen carefully. If you seriously intend to challenge a Millennium Problem, even if it’s later, you’d better form a team.
Solving a difficult problem is similar to taking a boat into the vast open sea and searching for a single island. But the captain can’t set the sails and row the oars all by himself, can he?”
Seha stopped for a moment and thought over his words. Then he nodded.
“Thank you.”
After bowing politely, Seha left the conference room.
‘Teammates…’
This was already the second time he had heard that.
When he thought about who would be good, one name came to mind. Though it was a story for the distant future.