Sujeong arrived at her accommodation. But the hotel's name seemed somehow familiar.
"Hilbert's Hotel?"
Her mind was hazy like fog, and she couldn't for the life of her remember where she had seen it.
"Let's just go in."
Dressed in a nineteenth-century European-style gown, she pulled her suitcase and approached the reception desk.
However, a sign reading "No vacancy" sat upon the counter.
"Ah... No rooms."
A hotel employee approached the disappointed Sujeong and said,
"Lady, do not worry. We are an infinite hotel."
"What? Then what is that?"
Sujeong pointed at the "No vacancy" sign on the counter.
"Aha! Though we are indeed fully occupied, we can specially make an empty room for you, Lady."
What on earth was he talking about? Sujeong wore a puzzled expression.
"How?"
"Move the guest in Room 1 to Room 2, the guest in Room 2 to Room 3... If we shift our guests infinitely in this manner, Room 1 will become empty. You may stay there."
While Sujeong stood there blankly, a group of guests arrived. A man who looked like a bus driver approached the counter.
"I say. We are the Infinite Bus Company. Can our passengers stay here?"
"No problem at all."
"Did I not say the Infinite Bus Company? We have infinitely many passengers."
The employee smiled faintly and tapped the ledger with the fountain pen in his hand.
"We are an infinite hotel.
If we move all of the hotel's guests to even-numbered rooms, all odd-numbered rooms become empty. Therefore, no matter how many guests there are, they can stay without issue."
Sujeong could no longer endure this seemingly meaningless conversation.
"If it's infinite anyway, you can just put them in any empty room!"
The employee stared at Sujeong, then spoke with a stiff expression.
"Lady, did I not tell you from the very beginning? This hotel has no vacant rooms. If you enter recklessly..."
He had somehow drawn near and whispered into Sujeong's ear.
"...You never know who you'll meet in that room."
Kyaaaaaaaah—
Sujeong screamed and fled.
"..."
Seoha was eating donuts with Sujeong.
Thanks to her realistic acting, even the cafeteria staff were glancing over and stealing looks at them.
"You had a dream like that?"
Sujeong gulped down a glass of water as if her throat were parched.
"Yeah, I read about Hilbert's Paradox of the Grand Hotel before going to bed, so it showed up in my dream."
If a hotel has infinite rooms, then even if fully occupied, empty rooms can always be prepared depending on how you divide the sets—that was Hilbert's paradox.
"You've been reading something interesting. Why Hilbert all of a sudden?"
Sujeong shrugged as she set down her cup.
"I have mentoring tomorrow. I was asked to explain countable infinity and one-to-one correspondence to a middle school student."
"Aha! That's a good idea. Using Hilbert's paradox, you can show it intuitively."
The set of natural numbers: 1, 2, 3, 4, 5...
The set of rational numbers: 1/2, 2/3, 1/4...
At a glance, since there are infinitely many rational numbers between 1 and 2, they seem far more numerous than natural numbers; however, in mathematics, the size of a set is defined by Cardinality (cardinal number).
The key lies in whether a one-to-one correspondence is possible.
As Hilbert's Hotel demonstrates, in countable infinity, even natural numbers (rooms) and rational numbers (guests) can be placed into one-to-one correspondence with each other.
The paradox of infinity where the part equals the whole is deeply philosophical, and it greatly shakes human intuition.
"Why are you interested in this, too?"
Instead of answering, Seoha pulled a book from his bag.
[The Theory of the Riemann Zeta-Function]
Sujeong looked at the book and was so shocked she nearly fainted.
"Th-the Riemann zeta function? Don't tell me you want to solve the Riemann Hypothesis?"
"Even if not now, I intend to solve it someday. I'm still in the investigation stage, though."
To explain the Riemann Hypothesis in a manner similar to Hilbert's approach, it would go like this.
There is a hotel with infinitely many rooms. But unlike Hilbert's Hotel, not all the rooms are occupied.
Rooms 2, 3, 5, 7, 11, 13, 17, 19...
Guests are only in rooms bearing prime numbers.
At first glance, the positions of the occupied rooms appear scattered at random.
However, the Riemann Hypothesis states that the arrangement of these "prime guests" is not random and follows an invisible mathematical rhythm.
And the key that reveals that rhythm is precisely the Riemann zeta function.
"You're someone who believes the secret of the universe is hidden in prime numbers, too? But the prevailing view is that they're irregular."
Sujeong looked indifferent.
Prime numbers were a topic that had fascinated many mathematicians.
Yet no one had been able to prove their regularity. Even if the Riemann Hypothesis were proven, one could only predict the distribution of primes with precision; it would still be impossible to know where each individual prime is located.
"The majority of mathematicians hold the opinion that there is no exact rule in the arrangement of primes. They say that very randomness is the essence of prime numbers."
That is why primes are sometimes called "the atoms of mathematics."
Existences that cannot be divided further and whose appearance cannot be predicted.
"But if the Riemann Hypothesis is true, doesn't the term 'irregularity' become ambiguous? Could we call it order within chaos?"
"Rather than a law, it would be about predicting density and flow to some degree. But even with just that, humanity can learn truly many things."
Seeing how serious Seoha was, Sujeong straightened her posture.
"Really? I think the Riemann Hypothesis is a vastly overestimated theory. Solving it won't bring down modern cryptographic systems. The rules of primes will still remain unknowable; while number theory may see some progress, it's not as if any grand mathematical advancement is expected.
Isn't it regarded as legendary simply because it has destroyed so many mathematicians?"
Sujeong wanted to pull Seoha away from the Riemann Hypothesis.
Seoha nodded.
Some mathematicians would agree with her. But Seoha had a reason he absolutely had to solve it.
Seoha's dream, which he had told no one.
It was to discover the so-called "Theory of Everything," a theory beyond the theory of relativity.
A theory that explains everything—that had been the domain of Newtonian mechanics for the past three hundred years.
But Einstein discovered that Newton had not considered the relativity of time and published a theory proving it.
And modern scholars say that the theory of relativity cannot explain the movement of quanta.
The time had come when the next stage of theory was needed.
(String theory had been one of the candidates, but proving it is impossible.)
Seoha believed that the secret of primes was not merely a problem of elementary number theory. Because the order hidden in the zeros of the Riemann zeta function resembled the chaos of the quantum world—that is, the rhythm of the universe.
"If there is a theory beyond Newton and Einstein, the beginning of that stage will inevitably be connected to the Riemann Hypothesis."
This is what Seoha thought.
But it was research he hadn't even begun; he couldn't tell Sujeong everything.
"If we prove the Riemann Hypothesis, couldn't we solve Goldbach's conjecture?"
Sujeong looked at Seoha silently for a long time. Then she opened her mouth with difficulty.
"Seoha. I'm asking you, please don't."
Sujeong's voice was earnest.
"I know you're a genius without equal in this world. But touching infinity is not merely a problem of mathematics. There is definitely something there that destroys human intuition.
Think about it again. Is it truly worth it?"
Falling into the Riemann Hypothesis meant the end of one's career (at least, until now).
The world romanticizes it as mathematicians' tenacity, failure, and endless spirit of inquiry, but what has become of those who actually threw themselves into it?
An endless labyrinth with no known end, a question where one cannot even know if an answer exists.
Mathematicians captivated by it merely lose their way and exhaust their lives in infinite darkness.
All that remains is probably emptiness and frustration.
Sujeong was terrified that Seoha would walk into the endless abyss of mathematics.
"It is well worth it. I cannot reveal everything now, but I intend to approach the secret of primes someday without fail."
Sujeong closed and opened her eyes, letting out a deep sigh.
She knew Seoha. He would never stop until he found the answer.
"What if there is nothing at the end of that path?"
No counterexample had emerged yet, but the Riemann Hypothesis might not be true.
"I will become the person who confirms there is nothing. That is enough. Someone has to look over that cliff directly."
Seoha's voice was calm, yet it contained a strong conviction. As if he were saying it was his destiny to walk this path.
At that moment, Sujeong felt as if she knew what she had to do.
Perhaps meeting Seoha had been inevitable.
To help him, or to stop him before it was too late...
Because even fate would not wish for such talent to vanish in vain.
"Then is there anything I can do? I don't know if someone like me could help."
Her first friend, the benefactor who had made her realize the beauty of mathematics.
To her, Seoha was a special existence.
"It's not yet at a stage where I can say anything for certain. But if I become confident that I can solve it, I will definitely call you, noona. Please help me then."
Sujeong nodded.
She had been thinking of studying more mathematics at university but had not yet decided on a specific theme.
"I should study Number Theory."
Until today, that is.
* * *
"Oppa!"
Seeing Seoha walking from afar, Seoeun ran out of the house. She ran hard and leaped into Seoha's arms.
Seoha smiled brightly and lifted Seoeun's small body high into the air.
Kyaa—
Seoeun rejoiced as if she were flying through the sky.
"Were you waiting?"
"Yeah! You said you'd come today. I was watching the whole time!"
Holding her oppa's hand tightly, Seoeun ran inside as if bursting through the door.
"Mom, Dad! Oppa's here!"
The house was soon filled with bright laughter.
Seoha helped his younger sister with her studies and harvested pumpkins with her in the garden.
After eating meat heartily, Seoeun began nodding her head drowsily and soon collapsed into sleep.
Cheolho carefully picked Seoeun up.
"She was so excited her oppa came, and she's already asleep."
Seoeun's sleeping face, perhaps deep in some dream, was full of smiles.
"Has Seoeun been well lately?"
Miyeong thought for a moment, then smiled gently.
"She used to say she'd become a mathematician like you, but these days she says she wants to be a doctor?"
"All of a sudden?"
"Not exactly. It's been since she visited you at the hospital. It seems she was quite shocked when her oppa collapsed."
Thinking back to that time, Seoha still felt ashamed before his family.
"I'm sorry."
"As long as you know. Just don't do it again next time."
Seoha thought carefully about Seoeun's dream.
Grades wouldn't be a problem. Seoeun's aptitude didn't fall short even when compared to gifted high school students. Even if she lacked something, he could teach her by her side.
What remained was the practical problem.
Her parents would say it was fine, but six years of medical school tuition on a civil servant's income would be no light burden.
The secrets of the universe were well and good, but the peace of the family was also important.
Seoha contemplated ways to raise the tuition before his younger sister went to college.
Fortunately, it did not seem too difficult.