Narayana Sequence and The Brocard-Ramanujan Equation

TL;DR

This paper investigates the properties of the Narayana sequence, characterizes its 3-adic valuation, and proves that the Brocard-Ramanujan equation has no solutions where the number is a Narayana number.

Contribution

It fully characterizes the 3-adic valuation of Narayana sequence terms and proves the non-existence of solutions to the Brocard-Ramanujan equation involving Narayana numbers.

Findings

No integer solutions to $m!+1=u^2$ with $u$ a Narayana number.

Complete characterization of the 3-adic valuation of $a_n \\pm 1$.

Establishment of properties linking Narayana sequence to classical Diophantine equations.

Abstract

Let $\left\lbrace a_{n}\right\rbrace_{n\geq 0}$ be the Narayana Sequence defined by the recurence $a_{n}=a_{n-1}+a_{n-3}$ for all $n\geq 3$ with intital values $a_{0}=0$ and $a_{1}=a_{2}=1$. In This paper, we fully characterize the $3-$adic valuation of $a_{n}+1$ and $a_{n}-1$ and then we prove that there are no integer solutions $(u,m)$ to the Brocard-Ramanujan Equation $m!+1=u^2$ where $u$ is a Narayana number.