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I came across a result that if $p^n \mid f_m$ for some $n\geq1$ then $p^{n+1} \mid f_{pm}$. I was wondering if this is true.

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    Seems to be a homework problem on page 49 at http://math.ucsd.edu/~erickson/research/pdf/fibonacci3.pdf2011-06-07

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Here is one way, but it uses a pretty strong result. The Fibonacci numbers satisfy a multiplication formula, specifically: $F_{kn}=\sum_{i=1}^{k}\binom{k}{i}F_{i}F_{n}^{i}F_{n-1}^{k-i}.$ Combining this formula with the fact that the Fibonacci number satisfy the division property $F_k|F_{nk}$, you can see why your result follows.