# The Catalan numbers have no forbidden residue modulo primes

## TL;DR

This paper proves that for any prime p ≥ 5, Catalan numbers modulo p cover all residues infinitely often, and any integer can be expressed as a product of central binomial coefficients modulo p.

## Contribution

It establishes that Catalan numbers modulo primes p ≥ 5 are surjective onto all residues and characterizes the automaton structure of their residues.

## Key findings

- Catalan numbers modulo p attain all residues for p ≥ 5.
- All residues occur infinitely often in the sequence.
- Any positive integer can be expressed as a product of central binomial coefficients modulo p.

## Abstract

Let $C_n$ be the $n$th Catalan number. For any prime $p \geq 5$ 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 $C_n \mod p$ consisting of the constant states and transitions between them has a cycle which visits all vertices.