Elliptic Divisibility Sequence - Periodicity of EDS

Periodicity of EDS

A sequence (A_n)_{n ≥ 1} is said to be periodic if there is a number N ≥ 1 so that A_n+N = A_n for every n ≥ 1. If a nondegenerate EDS (W_n)_{n ≥ 1} is periodic, then one of its terms vanishes. The smallest r ≥ 1 with W_r = 0 is called the rank of apparition of the EDS. A deep theorem of Mazur implies that if the rank of apparition of an EDS is finite, then it satisfies r ≤ 10 or r = 12.