×

Integral points on the elliptic curve \(y^2=x^3-4p^2x\). (English) Zbl 1513.11137

In this paper, the integral points on the elliptic curve \(E\) defined by the equation \(y^2=x^3-4p^2x\), where \(p\) is a prime \(\geq 17\), are determined. The proof is divided in four cases according to the form of \(p\). For instance, if \(p=a^4+b^4\), then \((x,\pm y)=(-4a^2b^2,\pm 4ab|a^4-b^4|)\). The proofs are relied on some properties of quadratic and quartic Diophantine equations. If \(N(p)\) denotes the number of pairs of nontrivial integer points of \(E\) and \(p\equiv\pm 1\ (\bmod\ 8)\), then, it is proved that \(N(p)\leq 4\), if \(p\equiv 1\ (\bmod\ 8)\) and \(N(p)\leq 1\), if \(p\equiv -1\ (\bmod\ 8)\). Note that in case where \(p=17\), the nontrivial integral points of \(E\) are: \((x,\pm y)=(-16,\pm 120),(-2,\pm 48),(162,\pm 2016),(578,\pm 13872)\), and so, the upper bound for \(N(p)\) given above is attainable.

MSC:

11G05 Elliptic curves over global fields
11D25 Cubic and quartic Diophantine equations
11Y50 Computer solution of Diophantine equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bennett, M. A., Integral points on congruent number curves, Int. J. Number Theory 9 (2013), 1619-1640 · Zbl 1318.11047 · doi:10.1142/S1793042113500474
[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 · doi:10.1090/S0002-9939-99-05041-8
[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 · doi:10.1006/jnth.1999.2430
[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 · doi:10.1090/S0025-5718-06-01852-7
[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 · doi:10.1090/S0025-5718-06-01841-2
[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 · doi:10.1016/j.jnt.2008.12.001
[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 · doi:10.3836/tjm/1327931392
[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 · doi:10.4064/aa160-4-3
[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
[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
[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 · doi:10.12988/ija.2007.07026
[12] Tunnell, J. B., A classical Diophantine problem and modular forms of weight \(3/2\), Invent. Math. 72 (1983), 323-334 · Zbl 0515.10013 · doi:10.1007/BF01389327
[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 · doi:10.3336/gm.44.1.04
[14] Walsh, P. G., On the number of large integer points on elliptic curves, Acta Arith. 138 (2009), 317-327 · Zbl 1254.11035 · doi:10.4064/aa138-4-2
[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 · doi:10.1216/RMJ-2011-41-1-311
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.