# 622: Haiku Proof

Haiku Proof |

Title text: After somewhere around 40 hours, there's no academic reason to go to the class. Only go for the hallucinations. |

## Explanation

Euclid's theorem states that there are an infinite number of primes. (In case it's been a while, remember that prime numbers are 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.
*p*_{1}*p*_{2}...*p*_{n}=*q*. Now consider*q*+ 1. Since each*p*_{k}is a factor of*q*, they cannot be a factor of*q*+ 1. So*q*+ 1 has no prime factors, which means it must be prime itself; however, we assumed we had listed every prime number, which means a contradiction has occurred. Then our supposition is false, so there must be an infinite number of primes.

The comic essentially takes this proof and states it in the form of a haiku, which is a traditional form of Japanese poetry where the three lines must have 5, 7, and 5 syllables, respectively. Actually, the 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.

## Transcript

- [Students are sitting at desks.]
- Student #1: How do you know there are an infinite number of primes?
- Professor: I'll answer in haiku!

- Professor: Top prime's divisors'

- [The professor floats into the air.]
- Professor: Product (plus one)'s factors are...?

- [The professor wafts over the students.]
- Professor: Q.E.D., bitches!
- Student #2 [in thought bubble]: Wow, after the 48-hour sleep-dep mark, lectures get
*really*interesting.

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# Discussion

A prime number must also be a natural number greater that one. -- 204.8.8.13 (talk) *(please sign your comments with ~~~~)*

I removed the paragraph about the haiku being off, as it is not "top prime's divisors," but "top prime's divisors' " (notice the second apostrophe). So the question is actually what the (prime) factors of the product of all prime divisors plus one are. KillaBilla (talk) 21:57, 12 June 2014 (UTC)

- I've put it back, since the paragraph is correct - the proof is incorrect. That second apostrophe just means it is the product belonging to the top prime's divisors. The product of the top prime's divisors is just the top prime. --141.101.98.234 14:40, 5 April 2015 (UTC)
- "All primes' divisors'" would've been correct (although the "divisors" is still unnecessary). --108.162.254.185 10:16, 16 July 2015 (UTC)