Infinite Primes

Beauty is the first test: there is no permanent place in the world for ugly mathematics. — G.H. Hardy

I didn’t always like math. I actually sucked at math up until college. After graduating with a math degree, I still suck at math, though, to a lesser degree than before.

Part of my frustrations with math were that people always classified it as beautiful and I never got that[1]. I can tell that Halle Berry is beautiful, or that the way Jordan played basketball is beautiful. But a 3–4–5 right triangle? Please. Its geometric perfection was lost on me for quite a long time.

As I became better at math in college, I discovered it demanded creative rigor similar to writing a short story or directing a movie. In the below script, I show how one of the most classical proofs in math might have come to be.

Primaeus: Hey, Euclid. Do you think there are infinitely many primes?

Euclid: Dude, what kind of question is that? I’m trying to enjoy my lunch.

P: No seriously, what do you think? There’s gotta be a largest one right? I mean it just wouldn’t make sense that they go on forever.

E: Fine, I’ll play your game[2]. Let’s say there is a largest prime[3]. Let’s call it M.

P: Sure thing. M is a fine name for the largest prime.

E: Ok, so we’ll multiply all the primes from 2, 3, 5…all the way to M. We’ll add 1 to that product and call this new number N. So, there’s two possibilities for N. It’s either prime or not.

P: Yeah. It’s either a prime or a product of primes. Right.

E: Well if it’s prime…

P: Then, we just found a prime that wasn’t on our list.

E: Exactly!

P: And, if it isn’t prime?

E: Well if it isn’t that means there’s some prime, we’ll call it P, that divides it. Right?

P: Yeah that makes sense. P times some other number equals N.

E: Cool. So what do we know about P?

P: P can’t be one of the numbers in the set [2, 3, 5,…, M].

E: Right. But why?

P: Because we said so.

E: Haha, we never said that! But here what if P were in the set [2, 3, 5,…, M]. What would that mean?

P: Oh, oh. I see it. Then P would have to divide [2 x 3 x 5 x … x M] (or, N — 1).

E: Yep.

P: But we already know that P divides N and since N = [2 x 3 x 5 x … x M] + 1. It would mean that P would have to divide 1…

E: Which would be ridiculous.

P: Man, I guess there must be an infinite number of primes. Who woulda guessed?

E: Not me. What do you have after lunch period.

P: Modern Greek Philosophy.

E: Ugh, the worst.

—
Notes
[1] This is, of course, due in part to the fuzzy definitions of beauty. Is it in the eye of the beholder? Or, is it an inherent property of the observed quantity?

[2] This proof method is called proof by contradiction and relies on showing that denying the premise would lead to absurd outcomes.

[3] The largest known prime is this number: 257,885,161–1.