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
No AI attempts have been recorded for this problem yet.
AI prompts
At the request of the problem contributor, we hold out the prompt formulation.
Mathematician survey
The author assessed the problem as follows.
Number of mathematicians highly familiar with the problem:
a majority of those working on a specialized topic (≈10)
Number of mathematicians who have made a serious attempt to solve the problem:
5–10
Rough guess of how long it would take an expert human to solve the problem:
1–3 years
Notability of a solution:
a major advance in a broader field; one of the best results of the year
A solution would be published:
in a leading general journal
Likelihood of a solution generating more interesting math:
fairly likely: the problem is rich enough that most solutions should open new avenues
Probability that the problem is solvable as stated:
40-60%