Prime ideal divisors of parametric recurrence sequences
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866917254556286976 |
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| author | N, Darsana Rout, Sudhansu Sekhar |
| author_facet | N, Darsana Rout, Sudhansu Sekhar |
| contents | We prove new arithmetic results for parametric linear recurrence sequences specialized at roots of unity, denoted by $(U_n(ζ))_{n\geq 0}$. In particular, we obtain exponential lower bounds for the largest prime ideal divisor and norm of the radical of the principal ideal generated by $U_n(ζ)$. We further derive an effective upper bound for the $S$-part of $U_n(ζ)$, showing that it is strictly smaller than a fixed power of its absolute norm for sufficiently large $n$. Finally, we establish an effective finiteness result for Diophantine equations of the form $U_n(ζ)=w_1+\cdots+w_r,$ with $S$-unit variables. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_06667 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Prime ideal divisors of parametric recurrence sequences N, Darsana Rout, Sudhansu Sekhar Number Theory We prove new arithmetic results for parametric linear recurrence sequences specialized at roots of unity, denoted by $(U_n(ζ))_{n\geq 0}$. In particular, we obtain exponential lower bounds for the largest prime ideal divisor and norm of the radical of the principal ideal generated by $U_n(ζ)$. We further derive an effective upper bound for the $S$-part of $U_n(ζ)$, showing that it is strictly smaller than a fixed power of its absolute norm for sufficiently large $n$. Finally, we establish an effective finiteness result for Diophantine equations of the form $U_n(ζ)=w_1+\cdots+w_r,$ with $S$-unit variables. |
| title | Prime ideal divisors of parametric recurrence sequences |
| topic | Number Theory |
| url | https://arxiv.org/abs/2602.06667 |