Non-trivial integer solutions of xr+yr=Dzp

KARA Y., Mocanu D., Özman E.

Indagationes Mathematicae, cilt.37, sa.4, ss.1510-1523, 2026 (SCI-Expanded, Scopus)

  • Yayın Türü: Makale / Derleme
  • Cilt numarası: 37 Sayı: 4
  • Basım Tarihi: 2026
  • Doi Numarası: 10.1016/j.indag.2025.11.001
  • Dergi Adı: Indagationes Mathematicae
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH, DIALNET
  • Sayfa Sayıları: ss.1510-1523
  • Anahtar Kelimeler: Frey–Mazur Conjecture, Generalized Fermat equation, Modular method
  • Boğaziçi Üniversitesi Adresli: Evet

Özet

In this paper, we use the modular method over totally real fields together with some standard conjectures (the Weak Frey–Mazur Conjecture and the Eichler–Shimura Conjecture) to prove that infinitely many equations of the type xr+yr=Dzp do not have any non-trivial primitive integer solutions, where r≥5 is a fixed prime, whenever p is large enough. For r≡3(mod4), we get the same result only assuming the Weak Frey–Mazur Conjecture.