An elliptic sequence is not a sampled linear recurrence sequence

摘要

Let E be an elliptic curve defined over the rationalsand in minimal Weierstrass form, and let P=(x1/z12,y1/z13)be a rational point of infinite order on E, where x1,y1,z1are coprime integers. We show that the integersequence (zn)n≧1 defined by nP=(xn/zn2,yn/zn3) for all n≧ 1does not eventually coincide with (un2)n≧1for any choice of linear recurrence sequence (un)n≧1 with integer values.