Narayana Sequence and The Brocard-Ramanujan Equation

2022/06/18 by Mustafa Ismail, Ismail, Mustafa, Salah Rihanaa +3
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2206.09174

openalex publication_date 2022/06/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \lbrace a_n\rbrace_{n≥ 0} be the Narayana Sequence defined by the recurence a_n=a_n-1+a_n-3 for all n≥ 3 with intital values a₀=0 and a₁=a₂=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=u² where u is a Narayana number.

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