Primitive divisors of elliptic divisibility sequences for elliptic curves with $j=1728$
Volume 198 / 2021
Acta Arithmetica 198 (2021), 129-168
MSC: Primary 11G05, 11B39; Secondary 11A41, 11D59, 11G07, 11G50.
DOI: 10.4064/aa191016-30-7
Published online: 4 January 2021
Abstract
Take a rational elliptic curve defined by the equation $y^2=x^3+ax$ in minimal form and consider the sequence $B_n$ of the denominators of the abscissas of the iterate of a non-torsion point. We show that $B_{5m}$ has a primitive divisor for every $m$. Then, we show how to generalize this method to the terms of the form $B_{mp}$ with $p$ a prime congruent to $1$ modulo $4$.
