Involve: A Journal of Mathematics

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

Recursive sequences and polynomial congruences

J. Lehman and Christopher Triola

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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 m[ω]=m[x]f. When m=p is prime, this order can be described in terms of the factorization of f in the polynomial ring p[x]. We use this connection to develop efficient algorithms for determining the factorization types of monic polynomials of degree k5 in p[x].

Article information

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

Received: 29 October 2007
Accepted: 26 January 2010
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11B50: Sequences (mod $m$) 11C08: Polynomials [See also 13F20] 11T06: Polynomials

linear homogeneous recurrence relations polynomial congruences finite rings finite fields


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

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