On the solutions of some Lebesgue–Ramanujan–Nagell type equations II

Mutlu E. K., SOYDAN G.

International Journal of Number Theory, 2025 (SCI-Expanded, Scopus) identifier identifier

  • Publication Type: Article / Article
  • Publication Date: 2025
  • Doi Number: 10.1142/s1793042126500235
  • Journal Name: International Journal of Number Theory
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
  • Keywords: elliptic curve, Exponential diophantine equation, Galois representation, modular form, Thue equation, Thue–Mahler equation
  • Bursa Uludag University Affiliated: Yes

Abstract

In this paper, we continue the study of Mutlu and Soydan (Int. J. Number Theory 20(5) (2024) 1195–1218), regarding the Diophantine equation x2+ps = 2ryn for certain primes p. Theorem 1.1 of that paper missed the elimination of some non-rational newforms, and this is corrected in the corresponding Corollary 1.2 of this paper. Further, as new results, we solve some seventh degree Thue–Mahler equations allowing us to extend and improve our results by solving the case n = 7. Our main tools include the known results from the modularity of Galois representations associated with Frey–Hellegouarch elliptic curves (the strategy of Bennett–Skinner) and a Thue–Mahler solver which was improved by Gherga and Siksek.