Why the Primes Never End

Prime numbers thin out as you climb the number line, yet they never run dry. Euclid's proof is a small marvel: suppose there were only finitely many primes $p_1, p_2, \dots, p_n$. Form

$N = p_1 p_2 \cdots p_n + 1.$

Every prime divides the product but leaves remainder $1$ when dividing $N$, so $N$ has a prime factor outside the list — a contradiction. The primes are infinite.

The deeper question is how they thin out. The Prime Number Theorem says the count of primes below $x$ behaves like $x / \ln x$, a result that took a century and the machinery of complex analysis to pin down.