| Title:
|
Linear recurrence sequences without zeros (English) |
| Author:
|
Dubickas, Artūras |
| Author:
|
Novikas, Aivaras |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
64 |
| Issue:
|
3 |
| Year:
|
2014 |
| Pages:
|
857-865 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $a_{d-1},\dots ,a_0 \in \mathbb Z$, where $d \in \mathbb N$ and $a_0 \neq 0$, and let $X=(x_n)_{n=1}^{\infty }$ be a sequence of integers given by the linear recurrence $x_{n+d}=a_{d-1}x_{n+d-1}+\dots +a_0x_{n}$ for $n=1,2,3,\dots $. We show that there are a prime number $p$ and $d$ integers $x_1,\dots ,x_d$ such that no element of the sequence $X=(x_n)_{n=1}^{\infty }$ defined by the above linear recurrence is divisible by $p$. Furthermore, for any nonnegative integer $s$ there is a prime number $p \geq 3$ and $d$ integers $x_1,\dots ,x_d$ such that every element of the sequence $X=(x_n)_{n=1}^{\infty }$ defined as above modulo $p$ belongs to the set $\{s+1,s+2,\dots ,p-s-1\}$. (English) |
| Keyword:
|
linear recurrence sequence |
| Keyword:
|
period modulo $p$ |
| Keyword:
|
polynomial splitting in $\mathbb F_p[z]$ |
| MSC:
|
11B37 |
| MSC:
|
11B50 |
| MSC:
|
11T06 |
| idZBL:
|
Zbl 06391531 |
| idMR:
|
MR3298566 |
| DOI:
|
10.1007/s10587-014-0138-1 |
| . |
| Date available:
|
2014-12-19T16:20:53Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144064 |
| . |
| Reference:
|
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