A modular approach to the generalized Ramanujan-Nagell equation


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Mutlu E. K. , Le M., SOYDAN G.

INDAGATIONES MATHEMATICAE-NEW SERIES, vol.33, no.5, pp.992-1000, 2022 (Peer-Reviewed Journal) identifier identifier

  • Publication Type: Article / Article
  • Volume: 33 Issue: 5
  • Publication Date: 2022
  • Doi Number: 10.1016/j.indag.2022.04.005
  • Journal Name: INDAGATIONES MATHEMATICAE-NEW SERIES
  • Journal Indexes: Science Citation Index Expanded, Scopus, MathSciNet, zbMATH, DIALNET
  • Page Numbers: pp.992-1000
  • Keywords: Polynomial-exponential Diophantine equation, Elliptic curve, S-integral point, Modular approach, DIOPHANTINE EQUATIONS, POINTS

Abstract

Let k be a positive integer. In this paper, using the modular approach, we prove that if k & EQUIV; 0 (mod 4), 30 < k < 724 and 2k -1 is an odd prime power, then under the GRH, the equation x2 + (2k -1)y = kz has only one positive integer solution (x, y, z) = (k - 1, 1, 2). The above results solve some difficult cases of Terai's conjecture concerning this equation.(c) 2022 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.