Preface: Eternity From Long Ago

Sophia: 37, 37!
Sophia: Look at the trail I made in the water!
37: Yes, I was counting, the end result was the number 4.
37 walked over to Sophia, raising four fingers seriously.
37: That means what you did was spot on. "4" signifies justice and strength.
Sophia: Wonderful!
Sophia: Let me try again, I'll try to make more skips this time.
Sophia: One, two, three ...
Excitedly, Sophia picked up a few more smooth pebbles, only to receive a puzzled look from her companion.
37: You got the right answer already. Why throw more?
Sophia: ... ?
The question made Sophia pause, and she matched her puzzled face.
But 37 seemed to be the more confused.
37: Bigger numbers aren’t always better. If you're testing your throwing technique, 4 is the best possible number.
37: Or do you want to test your passion and coordination? 5 would be better for that!
Sophia: No, I’m not trying to test any number ...  It's a game ...
The red-haired girl felt caught in the awkwardness of the moment.
She hadn’t been on the island for very long, and the seriousness with which they took arithmetic still left her feeling alien.
Thankfully, she had a companion who was well-versed in these matters.
37: A game ...  Oh, a game!
37: Like kottabos, where you fling wine dregs from two-handled cups at dinner parties?
Sophia: ... I guess so. I mean it almost sounds like skipping stones ...
She had agreed without really understanding the specifics, but not wanting the conversation to stall.
... or to disappoint her new friend.
37: Ah ... alright.
37: I don't get the point of all this.
Sophia: Huh ... ?
37’s brow furrowed—as if let down by her companion.
37: Throwing wine or pebbles, hitting a target, sinking a plate, or skipping stones on water—
37: What's the point of these actions? What are we trying to achieve?
37: If it's about being the highest-ranked, I get it. Like 210, he's always trying to get the top score in kottabos.
37: So is 6. He's always chasing the end of each series.
37: But sometimes it seems people do things without any purpose at all.
Sophia: Well, they're just playing.
Sophia: That's what most games are about—spending time with friends and having a fun time ... without any greater purpose, I guess.
37: ... Fun?
37: You mean ... these things don't have any transcendent purpose?
37: They don't climb up the hierarchy, seek knowledge, purify their souls, or seek out the truth?
37: So all they're doing is idling in the sensory world.
The young girl's tone grew much more distant, even cynical, surprising Sophia.
Sophia: No, I think ... well, it's not entirely like that ...
Sophia: Sorry, 37 ...  I'm new here, and there's a lot I don't understand. If I'm wrong—
The blue-haired girl was resting her chin in her hands,she had already drifted away in thought.
37: Oh, sorry, what were you saying?
37: Never mind these trivial matters! We have more important things to do!
37: Mother asked me to familiarize you with the island. So, is 4 the number you wanted to validate on the beach?
Sophia: ...
Sophia tried to finish her apology, but 37 dismissed it with a wave, she was already caught in another train of thought.
This kind of thing would happen several times until she got used to it.
But back then, her still-fledgling heart was unpolished by the sea breeze and sand, and she was full of questions.
Sophia: 37, I still don't quite understand ... this"Apeiron" belief in numbers.
Sophia: Before coming here, I dealt with numbers in my studies, but it was just arithmetic.
Sophia: When I solved problems, my dad would reward me with a candy, and I’d save them in a jar to buy flowers for Mom before going to the hospital ...
37: ... ?
Sophia realized she was saying too much. It wasn’t the right context, nor did it concern her new friend.
Sophia: ... Anyway, my impression of numbers is all caught up in those textbook exercises. Far removed from these abstract, transcendent meanings.
Sophia: Like I don't understand, why you believe that numbers are eternal?
37: Because they are eternal!
37’s answer, more a declaration, was yet more puzzling, making Sophia regret asking at all, and risking offense.
Thankfully, prime numbers like 37 are nimble-minded—at least that is what she would say.
37: I don't know how to explain it with words ...  But I have the perfect example!
37: It was Euclid who once proved that there are an infinite number of prime numbers.
Sophia: Prime numbers ...  Like how 37 is a prime number?
37: Yes, 13 is prime, 23 is prime, 37 is prime, and 337 is also prime! Their factors are only 1 and themselves. They cannot be divided by any other natural number.
37: Prime numbers can go on infinitely. But how can we prove this?
37 enthusiastically wrote a series of numbers in the sand, finishing with a circle at the end.
37: Euclid's method is to imagine the existence of the largest prime number—this is ○.
37: So, we make a complete series of prime numbers: starting from the smallest 2, all the way to the largest ○.
37: Based on this understanding, let's imagine there's a larger number, △.
37: △ represents the multiplication of all prime numbers in the queue, plus 1.
37: Clearly, it cannot be divided by any of the listed prime numbers. The numbers 2, 3, 5 ...  or any number represented by ○ will result in a remainder of 1.
37: If △ is not a prime number, it should be divisible by a prime number larger than ○.
37: If △ is a prime number, then naturally it's larger than ○.
37: Both scenarios contradict the assumption that ○ is the largest prime number.
37: Therefore, the conclusion is—there's no such thing as the largest prime number ○. Prime numbers are infinite.
37 finished writing, clapped her hands, shook off the sand, and stood up beaming.
Sophia: I ...  heard about this theorem at school. But I still don't understand what it means, really ...
She couldn't grasp the transcendental meaning behind such specific mathematical theorems.
To her, they were strict, correct, but nothing more than chalk particles on a blackboard, monochromatic, sharply defined, and barely soluble in water.
—And that was all.
37: ... Mother told me Euclid wrote down this proof over two thousand years ago.
37: She said, "The person who wrote this proof will die, the paper that recorded this proof will decompose, and we, who read this proof, will also one day disappear."
37: Everything will be washed away like sand on the seashore.
37: But this proof will endure, always correct, beautiful, and intricate, to be admired by every future generation.
37: —I suppose that's eternal.
Sophia: ... !
37: Everyone on the island tells me I'm special.
37: Prime numbers are like the foundation stones of arithmetic. All integers are derived from prime numbers.
37: But I'm one of a set of infinite prime numbers. There will always be more, newer numbers; so I'm not truly special.
37: But even if I am just 37 ...  37 grains of sand on the beach ...  I'm still encompassed within this infinite queue.
37: —And there are an infinite number of such sequences! Sequences of integers, sequences of fractions, sequences of irrational numbers ...  Each and every one infinite!
37 turned around, and now her eyes sparkled with the grains of sand in the sunlight.
37: We're all a part of infinity, and I take pride in that.
37: —I think, Sophia, that is what eternity is.
Under the bright sunlight, the grains of sand of the beach gleamed like gold. As she locked eyes with 37, it was blinding yet poignant.
37: "Don't close your eyes", as the scripture says, "Do not shy away from anything beautiful."
Sophia: Oh ...
Sophia: As time passed, I often remembered the theorem 37 told me.
Sophia: I never quite understood her world— this fascinating, abstract kingdom of mathematics where every number is her friend.
Sophia: But in that moment, I caught a fleeting glimpse of the beautiful brilliance of "eternity" in her eyes.
Sophia: It was enchanting ...  inspiring and admirable.
Sophia: It became the seed that led me to embrace the school.
Sophia: The beach changed in the days that followed.
Sophia: The ship approached the shore, the "Emanation" unfolded before our eyes, and cheers turned into sorrowful cries.
Sophia: Eventually, all that remained of what we lost were cold, lifeless geometric shapes.
Sophia: Yet, through it all 37 stood unwavering.
Sophia: She poured incredible focus into her mother's research, never displaying any sadness or hesitation.
Sophia: Just like the theorem she spoke of that day on the beach—unchanging, resolute, immune to the passage of time.
Sophia: —How could 37 stay so uniquely unaffected by the world around her?
Sophia: ...
Sophia: I couldn't find an answer to that question; all I could do was offer her my help, meager as it was.