On the number of positive integer solutions (x, n) of the generalized Ramanujan–Nagell equation $$x^2-2^r=p^n$$ x 2 - 2 r = p n

Abstract

Let p be a fixed odd prime, and let r be a fixed positive integer. Further let \(N(2^r,p)\) denote the number of positive integer solutions (x, n) of the generalized Ramanujan–Nagell equation \(x^2-2^r=p^n\). In this paper, we use the elementary method and properties of Pell’s equation to give a sharp upper bound estimate for \(N(2^r,p)\). That is, we prove that \(N(2^r,p)\le 1\).