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Let $D$ be a positive integer, and let $p$ be an odd prime with $p\nmid D$. In this paper we use a result on the rational approximation of quadratic irrationals due to M. Bauer, M. A. Bennett: Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation. Ramanujan J. 6 (2002), 209–270, give a better upper bound for $N(D,p)$, and also prove that if the equation $U^2-D{V}^2=-1$ has integer solutions $(U,V)$, the least solution $(u_1,v_1)$ of the equation $u^2-p{v}^2=1$ satisfies $p\nmid v_1$, and $D>C\left(p\right)$, where $C\left(p\right)$ is an effectively computable constant...