The Catalan numbers have no forbidden residue modulo primes

Abstract

Let $C_n$ be the $n$th Catalan number. For any prime $p\ge 5$ we show that the set $\{C_n:n\in \mathbb{N}\}$ contains all residues mod $p$. In addition all residues are attained infinitely often. Any positive integer can be expressed as the product of central binomial coefficients modulo $p$. The directed sub-graph of the automata for $C_n\mathrm{mod}p$ consisting of the constant states and transitions between them has a cycle which visits all vertices.

5 pages

References in corpus (1)

• Structure and asymptotics for Catalan numbers modulo primes using automata