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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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.