New York Journal of Mathematics
Volume 22 (2016) 1319-1338


F. Luca and T. Ward

An elliptic sequence is not a sampled linear recurrence sequence

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Published: November 8, 2016
Keywords: Elliptic divisibility sequence; nontorsion point; linear recurrence sequence
Subject: 11B37; 11G05

Let E be an elliptic curve defined over the rationals and in minimal Weierstrass form, and let P=(x1/z12,y1/z13) be a rational point of infinite order on E, where x1,y1,z1 are coprime integers. We show that the integer sequence (zn)n≧1 defined by nP=(xn/zn2,yn/zn3) for all n≧ 1 does not eventually coincide with (un2)n≧1 for any choice of linear recurrence sequence (un)n≧1 with integer values.

Author information

F. Luca:
School of Mathematics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa

T. Ward:
Ziff Building, University of Leeds, Leeds LS2 9JT, UK