Coming to Terms with Not Being Remarkable

This isn't meant to be self-deprecating or anything. But it is meant to be sort of a wake-up call. I was on Twitter (X) today and happened upon a tweet:

"the tail end of generation alpha is gonna be cracked btw"

Below that, a link to a Reddit post reading:

"The may be fairly basic, so please bear with me. My son thinks that a prime number squared is only divisible by that number (and itself and 1, of course). For example, 7x7 = 49, is only divisible by 7 (and 1, 49). I think he is right, but I don't know for sure. Can anyone confirm?

... "

Of course, the kid's completely right. The 4 y/o kid. Who is 4. Years old. It took me about 2 minutes to put together a really basic script to test it out. It's not a formal proof, and doesn't really prove anything, but it does check for all the prime numbers up to 10k if it's true, and it is:

for (let i = 0; i < 10000; i++) {
    let isPrime = true;
    for (let j = 0; j < i; j++) {
        if (j !== i && j !== 1 && i % j === 0) {
            isPrime = false;
            continue;
        }
    }
    if (isPrime) {
        const exceptions = [];
        console.log('checking', i);
        const squared = i**2;
        for (let k = 0; k < squared; k++) {
            if (k !== i && k !== 1 && k !== squared && i % k === 0) {
                exceptions.push(i, k);
            }
        }
        console.log('exceptions', exceptions);
    }
}

What's more disheartening than that is the comments I was reading in the thread discussing it. One read:

"Well he's right. Every integer is defined by a single set of primes raised to exponents and multiplied. It's trivial then that squaring the number is equivalent to doubling each exponent. Since a prime is itself to the first power, its square is itself the power two."

And the original poster replies:

"Yes, it's a trivial sub-case of the theorem of arithmetic, but for a 4 year old to discover and convince himself of its veracity with nobody leading him to that conclusion is incredible."

Incredible indeed. Now, this is the internet-- worse, it's Reddit + Twitter-- so it's likely not true that this 4 y/o did this. And it's equally unlikely that the twitter guy and the other twitter guy just know this offhand and are discussing it as casually as it reads. In other words, I fully imagine these guys are sitting at their desktops with ChatGPT open, asking it to explain the arguments, and then casually copying and pasting or paraphrasing to add an heir of nonchalance to their diatribe.

The point is that none of this is particularly trivial, regardless of the tooling be used or anyone's age. Perhaps if you visualize and prime your nerves to reason about the concepts, it can become trivial to imagine. But, from the 4 year old to the poster to the commenter, I am wholly beat. And this is just a random twitter thread. If I assume good faith and veracity, I must humble myself before God, who is apparently all around me, but not me.

--

One quick note is that there are 1231 prime numbers between 0 and 10k, including 0 and 1. This is more than I would have guessed.

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Another quick note on the fundamental theorem of arithmetic: I didn't know this was true. It's also called the "unique factorization theorem"