# A recurrence relation for elliptic divisibility sequences

Verzobio M. A recurrence relation for elliptic divisibility sequences. arXiv, 2102.07573.

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https://doi.org/10.48550/arXiv.2102.07573
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Corresponding author has ISTA affiliation

Abstract

In literature, there are two different definitions of elliptic divisibility
sequences. The first one says that a sequence of integers $\{h_n\}_{n\geq 0}$
is an elliptic divisibility sequence if it verifies the recurrence relation
$h_{m+n}h_{m-n}h_{r}^2=h_{m+r}h_{m-r}h_{n}^2-h_{n+r}h_{n-r}h_{m}^2$ for every
natural number $m\geq n\geq r$. The second definition says that a sequence of
integers $\{\beta_n\}_{n\geq 0}$ 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 $\{\beta_n\}_{n\geq 0}$, in general does not
hold
$\beta_{m+n}\beta_{m-n}\beta_{r}^2=\beta_{m+r}\beta_{m-r}\beta_{n}^2-\beta_{n+r}\beta_{n-r}\beta_{m}^2$
for $m\geq n\geq r$. We will prove that the recurrence relation above holds for
$\{\beta_n\}_{n\geq 0}$ under some conditions on the indexes $m$, $n$, and $r$.

Publishing Year

Date Published

2021-02-15

Journal Title

arXiv

Article Number

2102.07573

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### Cite this

Verzobio M. A recurrence relation for elliptic divisibility sequences.

*arXiv*. doi:10.48550/arXiv.2102.07573Verzobio, M. (n.d.). A recurrence relation for elliptic divisibility sequences.

*arXiv*. https://doi.org/10.48550/arXiv.2102.07573Verzobio, Matteo. “A Recurrence Relation for Elliptic Divisibility Sequences.”

*ArXiv*, n.d. https://doi.org/10.48550/arXiv.2102.07573.M. Verzobio, “A recurrence relation for elliptic divisibility sequences,”

*arXiv*. .Verzobio M. A recurrence relation for elliptic divisibility sequences. arXiv, 2102.07573.

Verzobio, Matteo. “A Recurrence Relation for Elliptic Divisibility Sequences.”

*ArXiv*, 2102.07573, doi:10.48550/arXiv.2102.07573.**All files available under the following license(s):**

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arXiv 2102.07573