Apéry's proof of the irrationality of $\zeta(3)$ astounded contemporary mathematicians for its wealth of new ideas and techniques in proving the irrationality of a known constant. It is often the case that such new Theorems, whose substance contains completely novel approaches to an old problem, inspire solutions to other, similar problems by providing the tools necessary to tackle them.

However, in the case of Apéry's Theorem, it seems like the new techniques presented in the proof are not easily used within other contexts. This has led me to ask the question:

*Has Apéry's proof of the irrationality of $\zeta(3)$ ever been used to prove the irrationality of other constants?*

Since the proof contains numerous new ideas, I also put forward the following corollary question:

*Which techniques employed in Apéry's proof of the irrationality of $\zeta(3)$ have been used within other proofs, whether in the field of irrationality/transcendence theory or other fields?*

Irrationality proofs using modular formsinAsterisque(1987), for two additional examples (Theorems 3 and 4) besides the old $\zeta(2) = \pi^2/6$, and for an illumination of what goes on behind Apéry's proof. $\endgroup$ – Vesselin Dimitrov Sep 26 '18 at 11:58Irrationality of certain $p$-adic periods for small $p$, IMRN 2005) made use of an overconvergence to prove the irrationality of the $2$-adic and $3$-adic zeta values $\zeta_2(3)$ and $\zeta_3(3)$. $\endgroup$ – Vesselin Dimitrov Sep 26 '18 at 12:09notbased on Apéry's methods. $\endgroup$ – Timothy Chow Sep 27 '18 at 14:35