| Title:
|
An elliptic curve having large integral points (English) |
| Author:
|
He, Yanfeng |
| Author:
|
Zhang, Wenpeng |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
60 |
| Issue:
|
4 |
| Year:
|
2010 |
| Pages:
|
1101-1107 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The main purpose of this paper is to prove that the elliptic curve $E\colon y^2=x^3+27x-62$ has only the integral points $(x, y)=(2, 0)$ and $(28844402, \pm 154914585540)$, using elementary number theory methods and some known results on quadratic and quartic Diophantine equations. (English) |
| Keyword:
|
elliptic curve |
| Keyword:
|
integral point |
| Keyword:
|
Diophantine equation |
| MSC:
|
11D25 |
| idZBL:
|
Zbl 1224.11051 |
| idMR:
|
MR2738972 |
| . |
| Date available:
|
2010-11-20T14:01:12Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140809 |
| . |
| Reference:
|
[1] Baker, A.: The Diophantine equation $y^2=ax^3+bx^2+cx+d$.J. Lond. Math. Soc. 43 (1968), 1-9. Zbl 0157.09801, MR 0231783, 10.1112/jlms/s1-43.1.1 |
| Reference:
|
[2] Stroeker, R. J., Tzanakis, N.: On the elliptic logarithm method for elliptic Diophantine equations: reflections and an improvement.Exp. Math. 8 (1999), 135-149. Zbl 0979.11060, MR 1700575, 10.1080/10586458.1999.10504395 |
| Reference:
|
[3] Stroeker, R. J., Tzanakis, N.: Computing all integer solutions of a genus $1$ equation.Math. Comput. 72 (2003), 1917-1933. Zbl 1089.11019, MR 1986812, 10.1090/S0025-5718-03-01497-2 |
| Reference:
|
[4] Zagier, D.: Large integral points on elliptic curves.Math. Comput. 48 (1987), 425-436. Zbl 0611.10008, MR 0866125, 10.1090/S0025-5718-1987-0866125-3 |
| Reference:
|
[5] Walker, D. T.: On the Diophantine equation $mx^2-ny^2=\pm 1$.Am. Math. Mon. 74 (1967), 504-513. MR 0211954, 10.1080/00029890.1967.11999992 |
| Reference:
|
[6] Walsh, G.: A note on a theorem of Ljunggren and the Diophantine equations $x^2-kxy^2+y^4=1,4$.Arch. Math. 73 (1999), 119-125. MR 1703679, 10.1007/s000130050376 |
| . |
