# Fibonacci numbers moduli

I have made some observation on very interesting material on Fibonacci series. I need some help in proving them mathematically.

We can observe that the periodicity of Fibonacci numbers modulo m, which non-trivial. Can we prove this mathematically? Also, how we can prove this period is always an even and the period itself modulo m is less than or equal to m2−1. How we prove this one. Also, justify mathematically the period modulo m is less than or equal to 6m, and other interesting facts about the period modulo a prime number.

Thank you so much for this wonderful platform.

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Each term depends only on the two preceding. Modulo $m$, there are only $m^2$ possibilities for the two preceding, so after at most $m^2$ terms there must be a repeat of two consecutive terms, and from there on it must repeat forever.
We can sharpen $m^2$ to $m^2-1$ by noting that if two consecutive terms were 0 modulo $m$ then all terms would be zero modulo $m$.