# Article

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Keywords:
generalized Ramanujan-Nagell equation; number of solution; upper bound
Summary:
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-DV^2=-1\$ has integer solutions \$(U, V)\$, the least solution \$(u_1, v_1)\$ of the equation \$u^2-pv^2=1\$ satisfies \$p\nmid v_1\$, and \$D>C(p)\$, where \$C(p)\$ is an effectively computable constant only depending on \$p\$, then the equation \$x^2-D=p^n\$ has at most two positive integer solutions \$(x, n)\$. In particular, we have \$C(3)=10^7\$.
References:
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