Prime numbers are more than any assigned multitude of prime numbers.

Let *A, B,* and *C* be the assigned prime numbers.

I say that there are more prime numbers than *A, B,* and *C.*

Take the least number *DE* measured by *A, B,* and *C.* Add the unit *DF* to *DE.*

Then *EF* is either prime or not.

First, let it be prime. Then the prime numbers *A, B, C,* and *EF* have been found which are more than *A, B,* and *C.*

Next, let *EF* not be prime. Therefore it is measured by some prime number. Let it be measured by the prime number *G.*

I say that *G* is not the same with any of the numbers *A, B,* and *C.*

If possible, let it be so. Now *A, B,* and *C* measure *DE,* therefore *G* also measures *DE.* But it also measures *EF.* Therefore *G,* being a number, measures the remainder, the unit *DF,* which is absurd.

Therefore *G* is not the same with any one of the numbers *A, B,* and *C.* And by hypothesis it is prime. Therefore the prime numbers *A, B, C,* and *G* have been found which are more than the assigned multitude of *A, B,* and *C.*

Therefore, *prime numbers are more than any assigned multitude of prime numbers.*

Q.E.D.

This proposition states that there are more than any finite number of prime numbers, that is to say, there are infinitely many primes.

Consider the number *m* + 1. If it’s prime, then there are at least *n* + 1 primes.

So suppose *m* + 1 is not prime. Then according to VII.31, some prime *g* divides it. But *g* cannot be any of the primes *a*_{1}, *a*_{2}, ..., *a _{n}*, since they all divide

This proposition is not used in the rest of the *Elements*.