| Title:
|
A note on the number of solutions of the generalized Ramanujan-Nagell equation $x^2-D=p^n$ (English) |
| Author:
|
Zhao, Yuan-e |
| Author:
|
Wang, Tingting |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
2 |
| Year:
|
2012 |
| Pages:
|
381-389 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $C(3)=10^7$. (English) |
| Keyword:
|
generalized Ramanujan-Nagell equation |
| Keyword:
|
number of solution |
| Keyword:
|
upper bound |
| MSC:
|
11D61 |
| idZBL:
|
Zbl 1265.11066 |
| idMR:
|
MR2990183 |
| DOI:
|
10.1007/s10587-012-0036-3 |
| . |
| Date available:
|
2012-06-08T09:40:57Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142835 |
| . |
| Reference:
|
[1] Bauer, M., Bennett, M. A.: Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation.Ramanujan J. 6 (2002), 209-270. Zbl 1010.11020, MR 1908198, 10.1023/A:1015779301077 |
| Reference:
|
[2] Beukers, F.: On the generalized Ramanujan-Nagell equation II.Acta Arith. 39 (1981), 113-123. Zbl 0377.10012, MR 0639621, 10.4064/aa-39-2-113-123 |
| Reference:
|
[3] Le, M. H.: On the generalized Ramanujan-Nagell equation $x^2-D=p^n$.Acta Arith. 58 (1991), 289-298. MR 1121088, 10.4064/aa-58-3-289-298 |
| Reference:
|
[4] Le, M. H.: On the number of solutions of the generalized Ramanujan-Nagell equation $x^2-D=p^n$.Publ. Math. Debrecen. 45 (1994), 239-254. MR 1315938 |
| Reference:
|
[5] Le, M. H.: Upper bounds for class numbers of real quadratic fields.Acta Arith. 68 (1994), 141-144. Zbl 0816.11055, MR 1305196, 10.4064/aa-68-2-141-144 |
| Reference:
|
[6] Mordell, L. J.: Diophantine Equations.London, Academic Press. (1969). Zbl 0188.34503, MR 0249355 |
| Reference:
|
[7] Siegel, C. L.: Approximation algebraischer Zahlen.Diss. Göttingen, Math. Zeitschr. 10 (1921), 173-213 German. MR 1544471, 10.1007/BF01211608 |
| . |
