Primitive Divisors of Lucas Sequences in Polynomial Rings
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913734154256384 |
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| author | Da Conceição, Joaquim Cera |
| author_facet | Da Conceição, Joaquim Cera |
| contents | It is known that all terms $U_n$ of a classical regular Lucas sequence have a primitive prime divisor if $n>30$. In addition, a complete description of all regular Lucas sequences and their terms $U_n$, $2\leq n\leq 30$, which do not have a primitive divisor is also known. Here, we prove comparable results for Lucas sequences in polynomial rings, correcting some previous theorem on the same subject. The first part of our paper develops some elements of Lucas theory in several abstract settings before proving our main theorem in polynomial rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04957 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Primitive Divisors of Lucas Sequences in Polynomial Rings Da Conceição, Joaquim Cera Number Theory Commutative Algebra 11B39, 11C08, 13A05 It is known that all terms $U_n$ of a classical regular Lucas sequence have a primitive prime divisor if $n>30$. In addition, a complete description of all regular Lucas sequences and their terms $U_n$, $2\leq n\leq 30$, which do not have a primitive divisor is also known. Here, we prove comparable results for Lucas sequences in polynomial rings, correcting some previous theorem on the same subject. The first part of our paper develops some elements of Lucas theory in several abstract settings before proving our main theorem in polynomial rings. |
| title | Primitive Divisors of Lucas Sequences in Polynomial Rings |
| topic | Number Theory Commutative Algebra 11B39, 11C08, 13A05 |
| url | https://arxiv.org/abs/2410.04957 |