A note on the Diophantine equation x(2)=4p(n)-4p(m) + l(2)

Abu Muriefah F. S., Le M., SOYDAN G.

INDIAN JOURNAL OF PURE & APPLIED MATHEMATICS, cilt.53, sa.4, ss.915-922, 2022 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 53 Sayı: 4
  • Basım Tarihi: 2022
  • Doi Numarası: 10.1007/s13226-021-00197-3
  • Dergi Adı: INDIAN JOURNAL OF PURE & APPLIED MATHEMATICS
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, INSPEC, zbMATH
  • Sayfa Sayıları: ss.915-922
  • Anahtar Kelimeler: Polynomial-exponential Diophantine equation, Generalized Ramanujan-Nagell equation, Baker's method
  • Bursa Uludağ Üniversitesi Adresli: Evet

Özet

Let l be a fixed odd positive integer. In this paper, using some classical results on the generalized Ramanujan-Nagell equation, we completely derive all solutions (p, x, m, n) of the equation x(2) = 4p(n)-4p(m)+l(2) with l(2) < 4p(m) for any l > 1, where p is a prime, x, m, n are positive integers satisfying gcd(x, l) = 1 and m < n. Meanwhile we give a method to solve the equation with l(2) > 4p(m). As an example of using this method, we find all solutions (p, x, m, n) of the equation for l is an element of {5, 7}.