Apéry's theorem

Remarks

The proof constructs two sequences of rational numbers $a_n$ and $b_n$, where the $b_n$ and the $2 d_n^3 a_n$ are integers, with $d_n$ the least common multiple of $1, 2, \dots, n$. The differences $b_n \zeta(3) - a_n$ are nonzero and tend to $0$ so fast that $2 d_n^3 (b_n \zeta(3) - a_n)$ still tends to $0$. If $\zeta(3) = p/q$, then $2 q d_n^3 (b_n \zeta(3) - a_n)$ would be a nonzero integer tending to $0$, which is impossible. A useful series in the proof is $$\zeta(3) = \frac{5}{2} \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^3 \binom{2n}{n}}.$$ For even arguments much more is known: $\zeta(2k)$ is a rational multiple of $\pi^{2k}$, so it is transcendental, as in the Basel problem. It is not known whether $\zeta(5)$ is irrational, and it is not known whether $\zeta(3)$ is transcendental.

Going deeper

For a student

Apéry's sequences satisfy a three-term recurrence with polynomial coefficients, and their growth controls the speed of approximation. Later work gave other proofs, for example with Beukers' integrals over the unit cube. Rivoal proved that infinitely many of the values $\zeta(2k+1)$ are irrational, and Zudilin proved that at least one of $\zeta(5), \zeta(7), \zeta(9), \zeta(11)$ is irrational.