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

Title:

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

Creator:

Zhao, Yuang-E and Wang, Tingting

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:c524ba81-0f57-47be-9e9c-a15c0a2175c3
uuid:c524ba81-0f57-47be-9e9c-a15c0a2175c3

Subject:

generalized Ramanujan-Nagell equation, number of solution, and upper bound

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

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 $C(3)=10^7$.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

381-389

Source:

Czechoslovak Mathematical Journal | 2012 Volume:62 | Number:2

Harvested from:

CDK

Metadata only:

false