An upper bound for the smallest period of the Fibonacci module sequence is obtained, which is 6N, and more accurate and convenient than the current results. Since the smallest period of the Fibonacci module sequence is twice of that of two-dimensional Arnold transformation, the upper bound for the smallest module period of Arnold transformation is naturally 3N. It is a great advance compared with the existing best upper bound N2/2 in the literatures. Moreover, many important properties are gained, which provide new thought for study on the period of Arnold transformation and the necessary mathematical reference of image coding and processing.
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