Czechoslovak Mathematical Journal, Vol. 69, No. 3, pp. 853-862, 2019


Integral points on the elliptic curve $y^2=x^3-4p^2x$

Hai Yang, Ruiqin Fu

Received November 21, 2017.   Published online March 26, 2019.

Abstract:  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 $E y^2=x^3-4p^2x$. Further, let $N(p)$ denote the number of pairs of integral points $(x,\pm y)$ on $E$ with $y>0$. We prove that if $p\geq17$, then $N(p)\leq4$ or $1$ depending on whether $p\equiv1\pmod8$ or $p\equiv-1\pmod8$.
Keywords:  elliptic curve; integral point; quadratic equation; quartic Diophantine equation
Classification MSC:  11G05, 11D25, 11Y50
DOI:  10.21136/CMJ.2019.0529-17


References:
[1] M. A. Bennett: Integral points on congruent number curves. Int. J. Number Theory 9 (2013), 1619-1640. DOI 10.1142/S1793042113500474 | MR 3103908 | Zbl 1318.11047
[2] M. A. Bennett, G. Walsh: The Diophantine equation $b^2X^4-dY^2=1$. Proc. Am. Math. Soc. 127 (1999), 3481-3491. DOI 10.1090/S0002-9939-99-05041-8 | MR 1625772 | Zbl 0980.11021
[3] A. Bremner, J. H. Silverman, N. Tzanakis: Integral points in arithmetic progression on $y^2=x(x^2-n^2)$. J. Number Theory 80 (2000), 187-208. DOI 10.1006/jnth.1999.2430 | MR 1740510 | Zbl 1009.11035
[4] K. A. Draziotis: Integer points on the curve $Y^2=X^3\pm p^kX$. Math. Comput. 75 (2006), 1493-1505. DOI 10.1090/S0025-5718-06-01852-7 | MR 2219040 | Zbl 1093.11020
[5] K. Draziotis, D. Poulakis: Practical solution of the Diophantine equation $y^2= x\*(x+2^ap^b)\*(x-2^ap^b)$. Math. Comput. 75 (2006), 1585-1593. DOI 10.1090/S0025-5718-06-01841-2 | MR 2219047 | Zbl 1119.11073
[6] K. Draziotis, D. Poulakis: Solving the Diophantine equation $y^2= x(x^2-n^2)$. J. Number Theory 129 (2009), 102-121; corrigendum 129 (2009), 739-740. DOI 10.1016/j.jnt.2008.12.001 | MR 2468473 | Zbl 1238.11038
[7] Y. Fujita, N. Terai: Integer points and independent points on the elliptic curve $y^2=x^3-p^kx$. Tokyo J. Math. 34 (2011), 367-381. DOI 10.3836/tjm/1327931392 | MR 2918912 | Zbl 1253.11043
[8] Y. Fujita, N. Terai: Generators and integer points on the elliptic curve $y^2=x^3-nx$. Acta Arith. 160 (2013), 333-348. DOI 10.4064/aa160-4-3 | MR 3119784 | Zbl 1310.11036
[9] G. H. Hardy, E. M. Wright: An Introduction to the Theory of Numbers. The Clarendon Press, Oxford University Press, New York (1979). MR 0568909 | Zbl 0423.10001
[10] B. K. Spearman: Elliptic curves $y^2=x^3-px$ of rank two. Math. J. Okayama Univ. 49 (2007), 183-184. MR 2377178 | Zbl 1132.11328
[11] B. K. Spearman: On the group structure of elliptic curves $y^2=x^3-2px$. Int. J. Algebra 1 (2007), 247-250. DOI 10.12988/ija.2007.07026 | MR 2342998 | Zbl 1137.11040
[12] J. B. Tunnell: A classical Diophantine problem and modular forms of weight $3/2$. Invent. Math. 72 (1983), 323-334. DOI 10.1007/BF01389327 | MR 0700775 | Zbl 0515.10013
[13] P. G. Walsh: Maximal ranks and integer points on a family of elliptic curves. Glas. Mat., III. Ser. 44 (2009), 83-87. DOI 10.3336/gm.44.1.04 | MR 2525656 | Zbl 1213.11125
[14] P. G. Walsh: On the number of large integer points on elliptic curves. Acta Arith. 138 (2009), 317-327. DOI 10.4064/aa138-4-2 | MR 2534137 | Zbl 1254.11035
[15] P. G. Walsh: Maximal ranks and integer points on a family of elliptic curves II. Rocky Mt. J. Math. 41 (2011), 311-317. DOI 10.1216/RMJ-2011-41-1-311 | MR 2845948 | Zbl 1234.11074

Affiliations:   Hai Yang, School of Science, Xi'an Polytechnic University, Xi'an, Shaanxi, 710048, P. R. China, e-mail: xpuyhai@163.com; Ruiqin Fu, School of Science, Xi'an Shiyou University, Xi'an, Shaanxi, 710065, P. R. China, e-mail: xsyfrq@163.com


 
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