Solvability of certain exponential Lebesgue-Nagell equations 𝑥 2 + 𝑝 𝑚 = 𝑦

Abstract

In this article, we first investigate the exponential Lebesgue-Nagell equation as shown in the title. Eventually, we can establish a necessary and sufficient criterion for having an integer solution to such an equation under the conditions that 𝑝 ≡ 3 (mod 4) and 𝑛 ≡ 1 (mod 4). The unique factorization in the ring of Gaussian integers, the existence of primitive divisors of the Lehmer sequences, and also the MAGMA program are the essentials applied in this work.

Keywords

• exponential lebesgue-nagell equations
• gaussian integers
• unique factorization
• lehmer sequences
• primitive divisors