Primes forever

Euclid's argument, from the Elements (Book IX, Proposition 20), shows the primes never run out. Take any finite list of primes $p_1, \dots, p_k$ and form $N = p_1 p_2 \cdots p_k + 1$. Dividing $N$ by any $p_i$ leaves remainder $1$, so no prime on the list divides $N$. But $N > 1$ has at least one prime factor. That prime is not on the list — so no finite list contains every prime.

How rare are they? There are $25$ primes below $100$, $168$ below $1{,}000$ and $78{,}498$ below a million. Gauss and Legendre guessed, and Hadamard and de la Vallée Poussin proved in 1896, the prime number theorem: the number of primes up to $x$ is approximately $x / \ln x$, in the sense that the ratio tends to $1$.

Primes can be far apart: $n! + 2, n! + 3, \dots, n! + n$ are $n - 1$ consecutive numbers with no prime among them. Are there also infinitely many twin primes like $11, 13$ and $17, 19$? Nobody knows. In 2013 Yitang Zhang proved that infinitely many pairs of primes differ by at most $70$ million, and collaborative work soon lowered the bound to $246$ — but the twin prime conjecture itself remains open, as does the Riemann hypothesis, which would pin down the error in the prime number theorem.