Paul Erdős had this notion that God kept a book of the best proofs. Erdős called God’s book simply “the book.” Springer recently published Proofs from THE BOOK, a collection of elegant proofs that the authors suppose might be in God’s book.
For many mathematicians, the first proof that comes to mind as a candidate for a proof in the book is Euclid’s proof that there are infinitely many primes. The proof is so short and simple that it can almost be written within the 140-character limit of Twitter. I give the proof in my AlgebraFact Twitter account as follows.
There are infinitely many primes. Proof: If not, multiply all primes together and add 1. Now you’ve got a new prime.
Several people pointed out that I was a little too terse. If N is the supposed product of all primes + 1, N itself doesn’t have to be a new prime. It could be that N has a factor that is a new prime. The smallest prime factor of N must be a prime not on the list. Either way “you’ve got a new prime,” but the proof above leaves out some important details.
Here’s an example. Suppose we believed the only primes were 2, 3, 5, 7, 11, and 13. Let N = 2*3*5*7*11*13 + 1 = 30031. Now N cannot be divisible by 2, 3, 5, 7, 11, or 13 because there is a remainder of 1 when dividing N by any of these numbers. Maybe N is a new prime, or maybe N is composite but N has a prime factor not on our list. The latter turns out to be the case: 30031 = 59*509.
If you can find a better summary of Euclid’s proof in 140 characters or less, please leave it in the comments. Please include the character count with your proposal.