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Elliptic curves containing sequences of consecutive cubes. (English) Zbl 1405.14081

Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\) and, for any \(P\in E\), denote its coordinates by \((x_P,y_P)\). The paper deals with curves containing sequences of points \(P_i\) whose \(x\)-coordinates are consecutive cubes in \(\mathbb{Q}\), i.e. such that \(x_{P_0}=c^3\) and \(x_{P_i}=(c+i)^3\) for \(i\geqslant 1\). It is easy to see that such sequences have to be finite and the authors use explicit computations to write down parametric families of elliptic curves containing sequences of 5 points with \(x\)-coordinate \((c+i)^3\) with \(-2\leqslant i\leqslant 2\). Moreover, using MAGMA (and Silverman’s specialization theorem), they also show that such points are independent and of infinite order so that all curves in the family have rank \(\geqslant 5\).

MSC:

14H52 Elliptic curves
11B83 Special sequences and polynomials
11D25 Cubic and quartic Diophantine equations
11G05 Elliptic curves over global fields
14G05 Rational points

Software:

ecdata; Magma
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References:

[1] A. Alvarado, An arithmetic progression on quintic curves, J. Integer Seq. 12 (2009). · Zbl 1201.11065
[2] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system, I, The user language, J. Symbol. Comp. 24 (1997), 235–265. · Zbl 0898.68039
[3] A. Bremner, On arithmetic progressions on elliptic curves, Exper. Math. 8 (1999), 409–413. · Zbl 0951.11021
[4] A. Bremner and M. Ulas, Rational points in geometric progressions on certain hyperelliptic curves, Publ. Math. Debr. 82 (2013), 669–683. · Zbl 1274.11106
[5] G. Campbell, A note on arithmetic progressions on elliptic curves, J. Int. Seq. 6 (2003). · Zbl 1022.11026
[6] J.E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, Cambridge, 1997. · Zbl 0872.14041
[7] P.K. Dey and B. Maji, Arithmetic progressions on \(y^2=x^3+k\), J. Int. Seq. 19 (2016). · Zbl 1383.11078
[8] G. Faltings, Endlichkeitsätze fur abelsche varietäten uber Zahlkörpern, Invent. Math. 73 (1983), 349–366.
[9] M. Kamel and M. Sadek, On sequences of consecutive squares on elliptic curves, Glasnik Mat. 52 (2017), 45–52. · Zbl 1386.14089
[10] J.B. Lee and W.Y. Vélez, Integral solutions in arithmetic progression for \(y^{2}=x^{3}+k\), Period. Math. Hungar. 25 (1992), 31–49. · Zbl 0757.11009
[11] A.J. Macleod, \(14\)-term arithmetic progressions on quartic elliptic curves, J. Int. Seq. 9 (2006). · Zbl 1101.11018
[12] L.J. Mordell, Diophantine equations, Academic Press, New York, 1969. · Zbl 0188.34503
[13] J.H. Silverman, Advanced topics in the arithmetic of elliptic curves, Springer-Verlag, New York, 1994. · Zbl 0911.14015
[14] M. Ulas, A note on arithmetic progressions on quartic elliptic curves, J. Int. Seq. 8 (2005). · Zbl 1068.11039
[15] ——–, On arithmetic progressions on genus two curves, Rocky Mountain J. Math. 39 (2009), 971–980. · Zbl 1243.11069
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