Perfect powers in Catalan and Narayana numbers

TL;DR

This paper investigates when Catalan and Narayana numbers are perfect powers, proving Catalan numbers are never perfect powers and characterizing perfect squares in Narayana numbers, while conjecturing about higher powers.

Contribution

It proves Catalan numbers are never perfect powers and characterizes all Narayana numbers that are perfect squares, proposing a conjecture for higher powers.

Findings

Catalan numbers are never perfect powers

Narayana numbers are perfect squares infinitely often

Partial results supporting the conjecture for higher powers

Abstract

When a Catalan number or a Narayana number is a (non-trivial) perfect power? For Catalan numbers, we show that the answer is "never". However, we prove that for every b, the Narayana number N(a,b) is a (non-trivial) perfect square for infinitely many values of a, and we show how to compute all of them. We also conjecture that N(a,b) is never a (non-trivial) perfect k-th power for k greater than 2 and we prove some cases of this conjecture.