Perfect powers in Catalan and Narayana numbers
Sara Checcoli, Michele D'Adderio

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.
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Taxonomy
Topicsgraph theory and CDMA systems · Advanced Combinatorial Mathematics · Graph Labeling and Dimension Problems
