## Involve: A Journal of Mathematics

• Involve
• Volume 3, Number 2 (2010), 129-148.

### Recursive sequences and polynomial congruences

#### Abstract

We consider the periodicity of recursive sequences defined by linear homogeneous recurrence relations of arbitrary order, when they are reduced modulo a positive integer $m$. We show that the period of such a sequence with characteristic polynomial $f$ can be expressed in terms of the order of $ω=x+〈f〉$ as a unit in the quotient ring $ℤm[ω]=ℤm[x]∕〈f〉$. When $m=p$ is prime, this order can be described in terms of the factorization of $f$ in the polynomial ring $ℤp[x]$. We use this connection to develop efficient algorithms for determining the factorization types of monic polynomials of degree $k≤5$ in $ℤp[x]$.

#### Article information

Source
Involve, Volume 3, Number 2 (2010), 129-148.

Dates
Accepted: 26 January 2010
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.involve/1513733291

Digital Object Identifier
doi:10.2140/involve.2010.3.129

Mathematical Reviews number (MathSciNet)
MR2718873

Zentralblatt MATH identifier
1269.11015

#### Citation

Lehman, J.; Triola, Christopher. Recursive sequences and polynomial congruences. Involve 3 (2010), no. 2, 129--148. doi:10.2140/involve.2010.3.129. https://projecteuclid.org/euclid.involve/1513733291

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