On the Finiteness of Perfect Powers in Elliptic Divisibility Sequences
Abdulmuhsin Alfaraj1
1 Department of Mathematical Sciences University of Bath Claverton Down Bath, BA2 7AY UK.
Journal de théorie des nombres de Bordeaux, Volume 35 (2023) no. 1, pp. 247-258.

We prove that there are finitely many perfect powers in elliptic divisibility sequences generated by a non-integral point on elliptic curves of the form y 2 =x(x 2 +b), where b is any positive integer. We achieve this by using the modularity of elliptic curves over real quadratic number fields.

Nous prouvons qu’il n’existe qu’un nombre fini de puissances parfaites dans les suites de divisibilité elliptiques générées par un point non entier sur une courbe elliptique de la forme y 2 =x(x 2 +b), où b est un entier positif non nul. Nous y parvenons en utilisant la modularité des courbes elliptiques sur les corps quadratiques réels.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/jtnb.1244
Classification: 11B83, 11D61, 11G05
Keywords: Modular methods, Elliptic divisibility sequences, Perfect powers
Abdulmuhsin Alfaraj 1

1 Department of Mathematical Sciences University of Bath Claverton Down Bath, BA2 7AY UK.
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Abdulmuhsin Alfaraj. On the Finiteness of Perfect Powers in Elliptic Divisibility Sequences. Journal de théorie des nombres de Bordeaux, Volume 35 (2023) no. 1, pp. 247-258. doi : 10.5802/jtnb.1244. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1244/

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