A note on the number of solutions of the generalized Ramanujan-Nagell equation $x^2-D=p^n$

Zhao, Yuan-e, and Wang, Tingting. "A note on the number of solutions of the generalized Ramanujan-Nagell equation $x^2-D=p^n$." Czechoslovak Mathematical Journal 62.2 (2012): 381-389. <http://eudml.org/doc/246966>.

@article{Zhao2012,
abstract = {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 $C(3)=10^7$.},
author = {Zhao, Yuan-e, Wang, Tingting},
journal = {Czechoslovak Mathematical Journal},
keywords = {generalized Ramanujan-Nagell equation; number of solution; upper bound; generalized Ramanujan-Nagell equation; number of solutions; upper bound},
language = {eng},
number = {2},
pages = {381-389},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on the number of solutions of the generalized Ramanujan-Nagell equation $x^2-D=p^n$},
url = {http://eudml.org/doc/246966},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Zhao, Yuan-e
AU - Wang, Tingting
TI - A note on the number of solutions of the generalized Ramanujan-Nagell equation $x^2-D=p^n$
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 2
SP - 381
EP - 389
AB - 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 $C(3)=10^7$.
LA - eng
KW - generalized Ramanujan-Nagell equation; number of solution; upper bound; generalized Ramanujan-Nagell equation; number of solutions; upper bound
UR - http://eudml.org/doc/246966
ER -