Fibonacci数列的模数列的周期性

The Periodicity of Modular Sequence of Fibonacci Sequence

对于Fibonacci数列{Fn}以及给定的正整数m,由Fn关于模m的最小非负剩余an,构成一个新的数列{an},称为Fibonacci数列的模数列.本文利用初等数论的知识和数学归纳法,证明了Fibonacci数列的模数列是周期数列,并且是纯周期数列.

For the Fibonacci sequence {Fn} and positive integer m, the new sequence {an} which composed of least non-negative residue an of module m on Fn is called modular sequence of Fibonacci sequence. Based on the elementary number theory and mathematical induction, this paper proves that the modular sequence of Fibonacci sequence is both a periodic sequence and a simple periodic sequence.