A recurrence relation for elliptic divisibility sequences
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866929279577620480 |
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| author | Verzobio, Matteo |
| author_facet | Verzobio, Matteo |
| contents | 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 $\{β_n\}_{n\geq 0}$ 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 $\{β_n\}_{n\geq 0}$, in general does not hold $β_{m+n}β_{m-n}β_{r}^2=β_{m+r}β_{m-r}β_{n}^2-β_{n+r}β_{n-r}β_{m}^2$ for $m\geq n\geq r$. We will prove that the recurrence relation above holds for $\{β_n\}_{n\geq 0}$ under some conditions on the indexes $m$, $n$, and $r$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_07573 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A recurrence relation for elliptic divisibility sequences Verzobio, Matteo Number Theory 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 $\{β_n\}_{n\geq 0}$ 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 $\{β_n\}_{n\geq 0}$, in general does not hold $β_{m+n}β_{m-n}β_{r}^2=β_{m+r}β_{m-r}β_{n}^2-β_{n+r}β_{n-r}β_{m}^2$ for $m\geq n\geq r$. We will prove that the recurrence relation above holds for $\{β_n\}_{n\geq 0}$ under some conditions on the indexes $m$, $n$, and $r$. |
| title | A recurrence relation for elliptic divisibility sequences |
| topic | Number Theory |
| url | https://arxiv.org/abs/2102.07573 |