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In my study I understand that the Fibonacci sequence mod $k$ is periodic, with period less than $k^2$. Can any one generalize this with good proof?

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    Thank you for all mathematicians, who given the comments and suggestions. I got and understand this question by JYRKI LAHTONEN. Thank you so much for all.2011-08-31

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HINT $ $ If you consider an n'th order linear recurrence in "system form", i.e. as a shift map on n-tuples then the periodicity is obvious, being cycles of a permutation on a finite set. See my comments about reinventing the wheel (cycle) in this answer and this one.

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    yes sir. I got it! thank you.2011-08-31