Riv. Mat. Univ. Parma, Vol. 13, No. 1, 2022

Matteo Verzobio [a]

A recurrence relation for elliptic divisibility sequences

Pages: 223-242
Received: 15 February 2021
Accepted in revised form: 8 April 2021
Mathematics Subject Classification: Primary 11G05; Secondary 11B37, 11B39.
Keywords: Elliptic divisibility sequences, Recurrence sequences, Elliptic curves.
Authors address:
[a]: Università di Pisa, Dipartimento di Matematica, Largo Bruno Pontecorvo 5, Pisa, Italy.

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Abstract: In literature, there are two different definitions of elliptic divisibility sequences. The first one says that a sequence of integers \(\{h_n\}_{n\in \mathbb{N}}\) is an elliptic divisibility sequence if it satisfies the recurrence relation \(\,h_{m+n}h_{n-m}h_{r}^2=h_{n+r}h_{n-r}h_{m}^2-h_{m+r}h_{m-r}h_{n}^2\,\) for all natural numbers \(n\geq m\geq r\). The second definition says that a sequence of integers \(\{\beta_n\}_{n\in \mathbb{N}}\) is an elliptic divisibility sequence if it is the sequence of the square roots (chosen with an appropriate sign) of the denominators of the abscissas of the iterates of a point on a rational elliptic curve. It is well-known that the two sequences are not equivalent. Hence, given a sequence of the denominators \(\{\beta_n\}_{n\in \mathbb{N}}\), in general the relation \(\,\beta_{m+n}\beta_{n-m}\beta_{r}^2=\beta_{n+r}\beta_{n-r}\beta_{m}^2-\beta_{m+r}\beta_{m-r}\beta_{n}^2\,\) does not hold for all \(n\geq m\geq r\). We will prove that the recurrence relation above holds for \(\,\{\beta_n\}_{n\in \mathbb{N}}\) under some conditions on the indexes \(m\), \(n\), and \(r\).

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