On the solutions of some Lebesgue-Ramanujan-Nagell type equations


Mutlu E. K., SOYDAN G.

International Journal of Number Theory, 2024 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Publication Date: 2024
  • Doi Number: 10.1142/s1793042124500593
  • Journal Name: International Journal of Number Theory
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, MathSciNet, zbMATH
  • Keywords: elliptic curve, Exponential Diophantine equation, Galois representation, modular form, S-integral point, Thue equation, Thue-Mahler equation
  • Bursa Uludag University Affiliated: Yes

Abstract

Denote by h = h(-p) the class number of the imaginary quadratic field âš(-p) with p prime. It is well known that h = 1 for p {3, 7, 11, 19, 43, 67, 163}. Recently, all the solutions of the Diophantine equation x2 + ps = 4yn with h = 1 were given by Chakraborty et al. in [Complete solutions of certain Lebesgue-Ramanujan-Nagell type equations, Publ. Math. Debrecen 97(3-4) (2020) 339-352]. In this paper, we study the Diophantine equation x2 + ps = 2ryn in unknown integers (x,y,s,r,n), where s ≥ 0, r ≥ 3, n ≥ 3, h {1, 2, 3} and gcd(x,y) = 1. To do this, we use the known results from the modularity of Galois representations associated with Frey-Hellegoaurch elliptic curves, the symplectic method and elementary methods of classical algebraic number theory. The aim of this paper is to extend the above results of Chakraborty et al.