Theorem. The prime numbers are inexhaustible.
Let’s cast this as a geometrical problem. We will study the scheme . The points of this scheme are (identified with) the prime ideals of , and it is a fun exercise to prove that they are precisely: the zero ideal ; and principal ideals where is a prime number in (hint: simply Euclid’s Lemma).
Suppose for a contradiction that there are only a finite number of prime numbers. Let be the product of all primes. Then, the integer can be interpreted as a regular function over that vanishes nowhere. Any such function is necessarily invertible, but the only units in are . Hence, either or , two impossibilities.
Discussion. The key fact is that any nowhere vanishing function is invertible. The reason this is true is because the ideal generated by a non-unit element is always contained in a maximal ideal. In our case, working with , this is just saying that every integer has a prime dividing it.