Alexander Bass
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Fibonacci Moduli

For the Fibonacci sequence, defined by F1=1, F2=1,Fn=Fn−1+Fn−2F_{1} = 1,\> F_{2} = 1,\quad F_{n} = F_{n-1} + F_{n-2}

I believe the following is true (Haven’t found any counterexamples)

(Fnmod⁡n)≠0when(nmod⁡6)∈{2,3,4} (F_{n} \operatorname{mod} n) \ne 0\quad \text{when}\quad (n \operatorname{mod} 6) \in \{2,3,4\}
This note is one of many taken during 2025