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Apéry-style Irrationality Proofs
Adapt Apéry's proof of the irrationality of $\zeta(3)$ to other constants.
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About the problem
Attempts by AI
AI prompts
Mathematician survey
About the problem
In 1979, Apéry proved that $\zeta(3)$ is irrational. At its core, the proof uses a particular recurrence relation, shown below. When initialized two different ways, this gives a pair of series that can be configured to converge “quickly” to $\zeta(3)$. This fast convergence suffices for a proof of irrationality.
\[n^3 u_n = (34n^3- 51n^2 + 27n - 5) u_{n-1} - (n-1)^3 u_{n-2}\]The aim of this problem is to find analogous recurrences and initializations that can be used to prove the irrationality of other “famous” constants.
Warm-up: we ask for the $\zeta(3)$ case. The solution here is well-known and models readily recapitulate it.
Attempts by AI
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AI Prompts
At the request of the problem contributor, we hold out the prompt formulation.
Mathematician survey
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