Let $a\_n$ is a *binomial sum*, for example
$$
a\_n := \sum\_{k} \binom{n-k}{k} \quad \text{or} \quad \sum\_{k=0}^n\binom{n+k}{n-k}\binom{2k}{k}
\quad \text{or} \quad \sum\_{k=0}^n\sum\_{\ell=0}^k\binom{n}{k}\binom{n+k}{k}\binom{k}{\ell}^3
$$
These are [Fibonacci](https://oeis.org/A000045), [Delannoy](https://oeis.org/A001850) and [Apéry numbers](https://oeis.org/A005259). More generally, we can consider
the sequence $\{a\_n\}$ obtained as a *diagonal* of a rational function over $\Bbb Z$ (so in particular it is *P-recursive*), see e.g. Stanley, "Enumerative Combinatorics", Chapter 6.


**Question:** Given $m\in \Bbb N$ and a binomial sum $\{a\_n\}$, is it decidable whether $m$ divides $a\_n$ for all $n\in \Bbb N$?


This might seem a bit naive, and I supect this should be true, but couldn't find the reference. Note the [Skolem's Problem](https://en.wikipedia.org/wiki/Skolem_problem) which is different but has a similar flavor (and not known to be decidable).



Congruences of binomial sums
I found the reference: Theorem 9.1 in Boris Adamczewski and Jason P. Bell, [Diagonalization and Rationalization of algebraic Laurent series](https://arxiv.org/abs/1205.4090), *Ann. Sci. Éc. Norm. Supér.* **46** (2013), 963–1004.


The authors even discuss the *Apéry sequence* modulo 5, right after the theorem.


It's sort of amazing when the reference request works out so well. I thank Boris Adamczewski and Jason Bell for telling me about this remarkable work.


