622: Haiku Proof
Title text: After somewhere around 40 hours, there's no academic reason to go to the class. Only go for the hallucinations.
Euclid's theorem states that there are an infinite number of primes, prime numbers being numbers that are only divisible by themselves and 1. The most notable proof of this theorem, and the one presented in this comic, was first given by Euclid himself in his Elements. A more traditional form of this proof follows:
- If we suppose that there are a finite number of primes, then they must have a product, i.e. p1p2...pn = q. Now consider q + 1. If this number is prime itself, then we have discovered a new prime number, contrary to the assumption that we had listed them all. If it is not prime, it must have a prime divisor. Since all of the pk are a factor of q, they cannot be a divisor of q + 1. So q + 1 is divisible by a prime not on the list, which again is a contradiction. Therefore, there must be infinitely many primes.
The comic essentially takes this proof and states it in the form of a haiku, which is a traditional form of Japanese poetry which is in Japanese broken up into patterns of morae, a unit that measures the length of sound. A haiku consists of three lines with 5, 7 and 5 morae respectively per line. This is less easy to do in English as English doesn't use the concept of syllable length, so we count haikus according to the syllables in each line.
The haiku proof given is slightly off, as the first line talks about the "top prime's divisors," which makes no sense because the top prime doesn't have any divisors besides itself and one. You need to take the product of all primes, not just one. But, hey, it's a hallucination.
The comic and title text conclude that going to class while sleep-deprived is an interesting, but entirely noneducational, experience.
- [Students are sitting at desks.]
- Megan: How do you know there are an infinite number of primes?
- Miss Lenhart: I'll answer in haiku!
- Miss Lenhart: Top prime's divisors'
- [Miss Lenhart floats into the air.]
- Miss Lenhart: Product (plus one)'s factors are...?
- [Miss Lenhart wafts over the students.]
- Miss Lenhart: Q.E.D., bitches!
- Cuebal [in thought bubble]: Wow, after the 48-hour sleep-dep mark, lectures get really interesting.