A recurrence relation for elliptic divisibility sequences

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

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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 $\{h_n\}_{n\geq 0}$ 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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Verzobio M. A recurrence relation for elliptic divisibility sequences. arXiv. doi:10.48550/arXiv.2102.07573
Verzobio, M. (n.d.). A recurrence relation for elliptic divisibility sequences. arXiv. https://doi.org/10.48550/arXiv.2102.07573
Verzobio, 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.
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