622: Haiku Proof

Explain xkcd: It's 'cause you're dumb.
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Haiku Proof
After somewhere around 40 hours, there's no academic reason to go to the class. Only go for the hallucinations.
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. Since each pk 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 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.

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A prime number must also be a natural number greater that one. -- ‎ (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. -- 14:40, 5 April 2015 (UTC)
"All primes' divisors'" would've been correct (although the "divisors" is still unnecessary). -- 10:16, 16 July 2015 (UTC)

I made a new poem: The product of all- Primes, plus one, divisors are?- Q.E.D., ******s! 23:56, 10 January 2017 (UTC)

I want a t-shirt with that last line on it.Aronurr (talk) 21:01, 3 March 2020 (UTC)