On the Finiteness of Perfect Powers in Elliptic Divisibility Sequences

Alfaraj, Abdulmuhsin

Journal de théorie des nombres de Bordeaux, Tome 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.

Classification : 11B83, 11D61, 11G05
Keywords: Modular methods, Elliptic divisibility sequences, Perfect powers

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