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 \(N(2^r,p)\le 1\).
Generalized Ramanujan–Nagell equation Number of solutions Upper bound estimate
Mathematics Subject Classification
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The authors express their gratitude to the referee for his very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the N. S. F. (11501452) and the P. S. F. (2014JQ2-1005) of P. R. China.