On the period and the order of appearance of the sequence of Fibonacci polynomials modulo m

Abstract

For a positive integer m, it is well known that the Fibonacci sequence modulo m, {F_n (mod m)}, is periodic and F_r is a multiple of m for some . The smallest possible value of r is called the order of appearance of m, denoted by r(m), in the Fibonacci sequence, and the smallest period of the sequence is denoted by k(m). Let F_n(x)^(m) denote the polynomial obtained by reducing coefficients and exponents of the nth Fibonacci polynomial F_n(x) modulo m. It was proved in the earlier work that the sequence of Fibonacci polynomials modulo m, {F_n(x)^(m)}, is periodic. In this article, we give a proof of this fact which yields the property that F_r(x)^(m) = 0 for some . We call the smallest possible value of r, denoted by r_m, the order of appearance of this sequence, and let k_m denote the smallest period of the sequence. Moreover, we verify some fundamental results and establish the basic result relating k_m and r_m. Some relations among k_p, r_p, k(p), and r(p), where p is a prime number, are also verified.

Subject Classification: (2010):

• 11B39
• 11B50

Keywords:

• Fibonacci number
• Fibonacci polynomial
• Period
• Order of appearance