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

. 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 ∤ 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}

MR Zbl

DOI : 10.1007/s10587-012-0036-3

Classification : 11D61
Mots-clés : generalized Ramanujan-Nagell equation; number of solution; upper bound

@article{10_1007_s10587_012_0036_3,
     author = {Zhao, Yuan-e and Wang, Tingting},
     title = {A note on the number of solutions of the generalized {Ramanujan-Nagell} equation $x^2-D=p^n$},
     journal = {Czechoslovak Mathematical Journal},
     pages = {381--389},
     publisher = {mathdoc},
     volume = {62},
     number = {2},
     year = {2012},
     doi = {10.1007/s10587-012-0036-3},
     mrnumber = {2990183},
     zbl = {1265.11066},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/articles/10.1007/s10587-012-0036-3/}
}

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
SP  - 381
EP  - 389
VL  - 62
IS  - 2
PB  - mathdoc
UR  - https://geodesic-test.mathdoc.fr/articles/10.1007/s10587-012-0036-3/
DO  - 10.1007/s10587-012-0036-3
LA  - en
ID  - 10_1007_s10587_012_0036_3
ER  -

%0 Journal Article
%A Zhao, Yuan-e
%A Wang, Tingting
%T A note on the number of solutions of the generalized Ramanujan-Nagell equation $x^2-D=p^n$
%J Czechoslovak Mathematical Journal
%D 2012
%P 381-389
%V 62
%N 2
%I mathdoc
%U https://geodesic-test.mathdoc.fr/articles/10.1007/s10587-012-0036-3/
%R 10.1007/s10587-012-0036-3
%G en
%F 10_1007_s10587_012_0036_3

Zhao, Yuan-e; Wang, Tingting. A note on the number of solutions of the generalized Ramanujan-Nagell equation $x^2-D=p^n$. Czechoslovak Mathematical Journal, Tome 62 (2012) no. 2, pp. 381-389. doi : 10.1007/s10587-012-0036-3. https://geodesic-test.mathdoc.fr/articles/10.1007/s10587-012-0036-3/

Cité par Sources :