{"doi":"10.48550/arXiv.2102.07573","publication":"arXiv","type":"preprint","article_processing_charge":"No","publication_status":"submitted","main_file_link":[{"open_access":"1","url":" https://doi.org/10.48550/arXiv.2102.07573"}],"status":"public","day":"15","date_updated":"2023-02-21T10:22:57Z","year":"2021","_id":"12314","language":[{"iso":"eng"}],"extern":"1","author":[{"full_name":"Verzobio, Matteo","orcid":"0000-0002-0854-0306","first_name":"Matteo","last_name":"Verzobio","id":"7aa8f170-131e-11ed-88e1-a9efd01027cb"}],"external_id":{"arxiv":["2102.07573"]},"month":"02","oa":1,"citation":{"short":"M. Verzobio, ArXiv (n.d.).","chicago":"Verzobio, Matteo. “A Recurrence Relation for Elliptic Divisibility Sequences.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2102.07573\">https://doi.org/10.48550/arXiv.2102.07573</a>.","ama":"Verzobio M. A recurrence relation for elliptic divisibility sequences. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2102.07573\">10.48550/arXiv.2102.07573</a>","ieee":"M. Verzobio, “A recurrence relation for elliptic divisibility sequences,” <i>arXiv</i>. .","ista":"Verzobio M. A recurrence relation for elliptic divisibility sequences. arXiv, 2102.07573.","mla":"Verzobio, Matteo. “A Recurrence Relation for Elliptic Divisibility Sequences.” <i>ArXiv</i>, 2102.07573, doi:<a href=\"https://doi.org/10.48550/arXiv.2102.07573\">10.48550/arXiv.2102.07573</a>.","apa":"Verzobio, M. (n.d.). A recurrence relation for elliptic divisibility sequences. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2102.07573\">https://doi.org/10.48550/arXiv.2102.07573</a>"},"article_number":"2102.07573","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","abstract":[{"text":"In literature, there are two different definitions of elliptic divisibility\r\nsequences. The first one says that a sequence of integers $\\{h_n\\}_{n\\geq 0}$\r\nis an elliptic divisibility sequence if it verifies the recurrence relation\r\n$h_{m+n}h_{m-n}h_{r}^2=h_{m+r}h_{m-r}h_{n}^2-h_{n+r}h_{n-r}h_{m}^2$ for every\r\nnatural number $m\\geq n\\geq r$. The second definition says that a sequence of\r\nintegers $\\{\\beta_n\\}_{n\\geq 0}$ is an elliptic divisibility sequence if it is\r\nthe sequence of the square roots (chosen with an appropriate sign) of the\r\ndenominators of the abscissas of the iterates of a point on a rational elliptic\r\ncurve. It is well-known that the two sequences are not equivalent. Hence, given\r\na sequence of the denominators $\\{\\beta_n\\}_{n\\geq 0}$, in general does not\r\nhold\r\n$\\beta_{m+n}\\beta_{m-n}\\beta_{r}^2=\\beta_{m+r}\\beta_{m-r}\\beta_{n}^2-\\beta_{n+r}\\beta_{n-r}\\beta_{m}^2$\r\nfor $m\\geq n\\geq r$. We will prove that the recurrence relation above holds for\r\n$\\{\\beta_n\\}_{n\\geq 0}$ under some conditions on the indexes $m$, $n$, and $r$.","lang":"eng"}],"date_created":"2023-01-16T11:46:36Z","title":"A recurrence relation for elliptic divisibility sequences","oa_version":"Preprint","date_published":"2021-02-15T00:00:00Z"}