Open Access
2009 Primitive Divisors of Certain Elliptic Divisibility Sequences
Minoru Yabuta
Experiment. Math. 18(3): 303-310 (2009).


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.


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Minoru Yabuta. "Primitive Divisors of Certain Elliptic Divisibility Sequences." Experiment. Math. 18 (3) 303 - 310, 2009.


Published: 2009
First available in Project Euclid: 25 November 2009

zbMATH: 1241.11037
MathSciNet: MR2555700

Primary: 11D25 , 11D45 , 11D61 , 11G05

Keywords: Diophantine equation , elliptic divisibility sequence , primitive divisor

Rights: Copyright © 2009 A K Peters, Ltd.

Vol.18 • No. 3 • 2009
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