A modular approach to the generalized Ramanujan-Nagell equation

2021/11/10 by Elif Kızıldere Mutlu, Mutlu, Elif Kızıldere, LE Mao-hua +4
Mathematics · #11D61 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11D61

paper · pdf · doi:10.48550/arxiv.2111.05626

10 pages, accepted for publication in Indagationes Mathematicae

openalex publication_date 2021/11/10 · arxiv created 2022/04/25 · arxiv updated 2022/04/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let k be a positive integer. In this paper, using the modular approach, we prove that if k≡ 0 \pmod4, 30< k<724 and 2k-1 is an odd prime power, then under the GRH, the equation x²+(2k-1)^y=k^z has only one positive integer solution (x,y,z)=(k-1,1,2). The above results solve some difficult cases of Terai's conecture concerning this equation.

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