Are there infinitely many primes? (Euclid, 300BC)
Are there infinitely many primes starting with 7 in base 10? (follows from prime number theorem)
Are there infinitely many primes ending with 7 in base 10? (follows from Dirichlet’s theorem)
Are there infinitely many primes with 49% of the digits being 7 in base 10? would follow from the Riemann hypothesis
Are there infinitely many pairs of primes differing by 2? (twin prime conjecture)

Key feature: To show that a set (of primes) is infinite, want to estimate the number of elements $\leq x$ .

Definition. Define

π (x) = | {primes \leq x} | = \sum_{p \leq x} 1 .

Euclid showed: $\lim_{x \to \infty} π (x) = \infty$ .

Theorem (Prime number theorem).

\lim_{x \to \infty} \frac{π (x) \log x}{x} = 1 .

$π (x) \sim \frac{x}{\log x}$ . (Conjectured: Legendre, Gauss. Proved: Hadamard, de la Vallée Poussin)