| Title:
|
Integral points on the elliptic curve $y^2=x^3-4p^2x$ (English) |
| Author:
|
Yang, Hai |
| Author:
|
Fu, Ruiqin |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
69 |
| Issue:
|
3 |
| Year:
|
2019 |
| Pages:
|
853-862 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $p$ be a fixed odd prime. We combine some properties of quadratic and quartic Diophantine equations with elementary number theory methods to determine all integral points on the elliptic curve $E\colon y^2=x^3-4p^2x$. Further, let $N(p)$ denote the number of pairs of integral points $(x,\pm y)$ on $E$ with $y>0$. We prove that if $p\geq 17$, then $N(p)\leq 4$ or $1$ depending on whether $p\equiv 1\pmod 8$ or $p\equiv -1\pmod 8$. (English) |
| Keyword:
|
elliptic curve |
| Keyword:
|
integral point |
| Keyword:
|
quadratic equation |
| Keyword:
|
quartic Diophantine equation |
| MSC:
|
11D25 |
| MSC:
|
11G05 |
| MSC:
|
11Y50 |
| idZBL:
|
Zbl 07088820 |
| idMR:
|
MR3989282 |
| DOI:
|
10.21136/CMJ.2019.0529-17 |
| . |
| Date available:
|
2019-07-24T11:20:36Z |
| Last updated:
|
2021-10-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147793 |
| . |
| Reference:
|
[1] Bennett, M. A.: Integral points on congruent number curves.Int. J. Number Theory 9 (2013), 1619-1640. Zbl 1318.11047, MR 3103908, 10.1142/S1793042113500474 |
| Reference:
|
[2] Bennett, M. A., Walsh, G.: The Diophantine equation $b^2X^4-dY^2=1$.Proc. Am. Math. Soc. 127 (1999), 3481-3491. Zbl 0980.11021, MR 1625772, 10.1090/S0002-9939-99-05041-8 |
| Reference:
|
[3] Bremner, A., Silverman, J. H., Tzanakis, N.: Integral points in arithmetic progression on $y^2=x(x^2-n^2)$.J. Number Theory 80 (2000), 187-208. Zbl 1009.11035, MR 1740510, 10.1006/jnth.1999.2430 |
| Reference:
|
[4] Draziotis, K. A.: Integer points on the curve $Y^2=X^3\pm p^kX$.Math. Comput. 75 (2006), 1493-1505. Zbl 1093.11020, MR 2219040, 10.1090/S0025-5718-06-01852-7 |
| Reference:
|
[5] Draziotis, K., Poulakis, D.: Practical solution of the Diophantine equation $y^2= x\*(x+2^ap^b)\*(x-2^ap^b)$.Math. Comput. 75 (2006), 1585-1593. Zbl 1119.11073, MR 2219047, 10.1090/S0025-5718-06-01841-2 |
| Reference:
|
[6] Draziotis, K., Poulakis, D.: Solving the Diophantine equation $y^2= x(x^2-n^2)$.J. Number Theory 129 (2009), 102-121 corrigendum 129 2009 739-740. Zbl 1238.11038, MR 2468473, 10.1016/j.jnt.2008.12.001 |
| Reference:
|
[7] Fujita, Y., Terai, N.: Integer points and independent points on the elliptic curve $y^2=x^3-p^kx$.Tokyo J. Math. 34 (2011), 367-381. Zbl 1253.11043, MR 2918912, 10.3836/tjm/1327931392 |
| Reference:
|
[8] Fujita, Y., Terai, N.: Generators and integer points on the elliptic curve $y^2=x^3-nx$.Acta Arith. 160 (2013), 333-348. Zbl 1310.11036, MR 3119784, 10.4064/aa160-4-3 |
| Reference:
|
[9] Hardy, G. H., Wright, E. M.: An Introduction to the Theory of Numbers.The Clarendon Press, Oxford University Press, New York (1979). Zbl 0423.10001, MR 0568909 |
| Reference:
|
[10] Spearman, B. K.: Elliptic curves $y^2=x^3-px$ of rank two.Math. J. Okayama Univ. 49 (2007), 183-184. Zbl 1132.11328, MR 2377178 |
| Reference:
|
[11] Spearman, B. K.: On the group structure of elliptic curves $y^2=x^3-2px$.Int. J. Algebra 1 (2007), 247-250. Zbl 1137.11040, MR 2342998, 10.12988/ija.2007.07026 |
| Reference:
|
[12] Tunnell, J. B.: A classical Diophantine problem and modular forms of weight $3/2$.Invent. Math. 72 (1983), 323-334. Zbl 0515.10013, MR 0700775, 10.1007/BF01389327 |
| Reference:
|
[13] Walsh, P. G.: Maximal ranks and integer points on a family of elliptic curves.Glas. Mat., III. Ser. 44 (2009), 83-87. Zbl 1213.11125, MR 2525656, 10.3336/gm.44.1.04 |
| Reference:
|
[14] Walsh, P. G.: On the number of large integer points on elliptic curves.Acta Arith. 138 (2009), 317-327. Zbl 1254.11035, MR 2534137, 10.4064/aa138-4-2 |
| Reference:
|
[15] Walsh, P. G.: Maximal ranks and integer points on a family of elliptic curves II.Rocky Mt. J. Math. 41 (2011), 311-317. Zbl 1234.11074, MR 2845948, 10.1216/RMJ-2011-41-1-311 |
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