12 problems

Beukers' conjecture.

Let $A'_n=\sum_{k=0}^n\binom nk^2\binom{n+k}k$ be the Apéry numbers of the second kind. For an odd prime $p$, if $p=x^2+4y^2$ with $x,y\in\mathbb Z$, then…

Beukers's conjecture. If the representation contains $\alpha$ occurrences of the digit $5$, then

Let $n$ be a positive integer, let $r$ be a positive integer, let $q$ be an indeterminate, and write … for the $q$-integer and $q$-binomial coefficient, respectively. Let…

Su's conjecture. The congruence

Let $\{a_n\}_{n\ge 0}$ denote the Apéry numbers. Chowla et al.'s congruence conjecture. For every non-negative integer $n$, … and … The conjecture was proved by Gessel in the equiv…

Chowla–Cowles–Cowles conjecture. The Apéry numbers associated to $\zeta(3)$ are periodic modulo $8$, alternating between the values $1$ and $5$.

The Apéry numbers are the sequences … A sequence is infinitely log-convex when every sequence obtained by iterating the log-convexity operator remains log-convex. Chen–Xia's conjec…

Let $A_k$ be the Apéry numbers, … For a prime $p>5$, let $H_n=\sum_{0<k\leq n}1/k$ be the $n$-th harmonic number, and let $B_0,B_1,B_2,\ldots$ be the Bernoulli numbers. Sun's conje…

Beukers' conjecture. The integer $p^\alpha$ divides $A_2(n)$.

Second-kind Apéry congruence conjecture. The numbers $A^{(3)}_n$ satisfy the following congruences modulo $9$:

Zagier's conjecture. There are no other triples $(a,b,\mathfrak l)$ giving rise to an ordinary differential equation with a solution in ${\mathbb Z}[[t]]$ beyond the cases found by…