Primitive Divisors of Certain Elliptic Divisibility Sequences

Abstract

Let $E: y^2=x^3+D$ be an elliptic curve, where $D$ is an integer that contains no primes $p$ with $6 \mid {\ord}_pD$. For a nontorsion rational point $P$ on $E$, write $x(nP)=A_n(P)/B_n^2(P)$ in lowest terms. We prove that for the sequence $\{B_{2^m}(P)\}_{m \ge 0}$, the term $B_{2^m}(P)$ has a primitive divisor for all $m \ge 3$. As an application, we give a new method for solving the Diophantine equation $y^2=x^3+d^n$ under certain conditions.