## Missouri Journal of Mathematical Sciences

### $G$-Sets and Linear Recurrences Modulo Primes

#### Abstract

Let $p$ be a prime number with $p\neq 2$. We consider second order linear recurrence relations of the form $S_n=aS_{n-1}+bS_{n-2}$ over the finite field $Z_p$ (we assume $b\neq 0$). Results regarding the period and distribution of elements in the sequence $\{ S_0, S_1, \ldots \}$ are well-known (see works by Kuipers, Niederreiter, Wall, and Webb). We examine these second order recurrences using matrices, groups, and $G$-sets.

#### Article information

Source
Missouri J. Math. Sci., Volume 25, Issue 1 (2013), 27-36.

Dates
First available in Project Euclid: 28 May 2013

https://projecteuclid.org/euclid.mjms/1369746395

Digital Object Identifier
doi:10.35834/mjms/1369746395

Mathematical Reviews number (MathSciNet)
MR3087686

Zentralblatt MATH identifier
1292.11037

Subjects
Primary: 11B50: Sequences (mod $m$)

#### Citation

McKenzie, Thomas; Overbay, Shannon; Ray, Robert. $G$-Sets and Linear Recurrences Modulo Primes. Missouri J. Math. Sci. 25 (2013), no. 1, 27--36. doi:10.35834/mjms/1369746395. https://projecteuclid.org/euclid.mjms/1369746395

#### References

• M. Artin, Algebra, Prentice Hall, New Jersey, 1991.
• L. Kuipers and J. S. Shiue, A distribution property of the sequence of Fibonacci numbers, The Fibonacci Quarterly, 10.4 (1972), 375–392.
• H. Niederreiter and J. S. Shiue, Equidistribution of linear recurring sequences in finite fields, Indag. Math., 80 (1977), 397–405.
• D. D. Wall, Fibonacci series modulo m, The American Mathematical Monthly, 67 (1960), 525–532.
• W. A. Webb and C. T. Long, Distribution modulo $p$ of the general second order recurrence, Atti Accad. Lincei, 58 (1975), 92–100.