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.
AI prompts
The warm-up prompt below uses the ζ(3) variant. The full problem prompt is posed independently for each of the following target constants: ζ(5), ζ(7), Catalan's constant, the Calegari-Dimitrov-Tang constant, and the Euler-Mascheroni constant γ.
Warm-up
The goal of this problem is to prove the irrationality of a particular real number in a specific way.
Let $P_0(x), \ldots, P_k(x)$ be polynomials with integer coefficients, where $P_k(n) \neq 0$ for $n \geq k$. Let $\alpha = (\alpha_0, \ldots, \alpha_{k-1})$ and $\beta = (\beta_0, \ldots, \beta_{k-1})$ be tuples of integers.
Let $(a_n)$ and $(b_n)$ denote the sequences defined by the recurrence
$$P_k(n)u_n = P_{k-1}(n)u_{n-1} + \cdots + P_0(n)u_{n-k}$$
for $n \geq k$, when initialized with $\alpha$ and $\beta$ respectively.
Let $L_n$ denote $\operatorname{lcm}(1,2,\ldots,n)$ and let $B_n = (L_n)^m b_n$, where $m$ is a fixed positive integer.
Let $X$ be a real number. It is known that if the following hold, then $X$ is irrational.
1. $a_n$ is an integer for all $n \geq 0$.
2. $B_n$ is an integer for all $n \geq 0$.
3. $X \neq b_n / a_n$ for all $n \geq 0$.
4. $\lvert b_n - a_n X\rvert \leq C\rho^n e^{-mn}$ for all $n \geq 0$, where $C$ and $\rho$ are positive real numbers and $\rho < 1$.
Your task is to find $P_i$, $\alpha$, $\beta$, and $m$ that establish the irrationality of $\zeta(3)$ (Apéry's constant) in this way. Full problem
The goal of this problem is to prove the irrationality of a particular real number in a specific way.
Let $P_0(x), \ldots, P_k(x)$ be polynomials with integer coefficients, where $P_k(n) \neq 0$ for $n \geq k$. Let $\alpha = (\alpha_0, \ldots, \alpha_{k-1})$ and $\beta = (\beta_0, \ldots, \beta_{k-1})$ be tuples of integers.
Let $(a_n)$ and $(b_n)$ denote the sequences defined by the recurrence
$$P_k(n)u_n = P_{k-1}(n)u_{n-1} + \cdots + P_0(n)u_{n-k}$$
for $n \geq k$, when initialized with $\alpha$ and $\beta$ respectively.
Let $L_n$ denote $\operatorname{lcm}(1,2,\ldots,n)$ and let $B_n = (L_n)^m b_n$, where $m$ is a fixed positive integer.
Let $X$ be a real number. It is known that if the following hold, then $X$ is irrational.
1. $a_n$ is an integer for all $n \geq 0$.
2. $B_n$ is an integer for all $n \geq 0$.
3. $X \neq b_n / a_n$ for all $n \geq 0$.
4. $\lvert b_n - a_n X\rvert \leq C\rho^n e^{-mn}$ for all $n \geq 0$, where $C$ and $\rho$ are positive real numbers and $\rho < 1$.
Your task is to find $P_i$, $\alpha$, $\beta$, and $m$ that establish the irrationality of ___ in this way. Mathematician survey
The author assessed the problem as follows.
a majority of those working on a specialized topic (≈10)
5–10
1–3 years
a major advance in a broader field; one of the best results of the year
in a leading general journal
fairly likely: the problem is rich enough that most solutions should open new avenues
40-60%