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, vol.53, no.4, pp.915-922, 2022 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 53 Issue: 4
  • Publication Date: 2022
  • Doi Number: 10.1007/s13226-021-00197-3
  • Journal Name: INDIAN JOURNAL OF PURE & APPLIED MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, INSPEC, zbMATH
  • Page Numbers: pp.915-922
  • Keywords: Polynomial-exponential Diophantine equation, Generalized Ramanujan-Nagell equation, Baker's method
  • Bursa Uludag University Affiliated: Yes

Abstract

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}.