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From: "dilettante" <n...@nonono.no>
Newsgroups: sci.math
Subject: Re: Fibonacci sequence
Date: Mon, 8 Oct 2012 14:19:40 -0500
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"AP" <marc.picher...@wanadoo.fr.invalid> wrote in message 
news:jq76785qi8948cunpaqto5icqflbdmkjeq@4ax.com...
> Fibonacci sequence is
> F_0=0  F_1=1
> and F_n=F(n-1)+F(n-2)
>
> F'_n=F_ mod m with m>=2
>
> F'_n is periodic with period T(m)<=m^2
>
> because among the m^2+1 couples
>
> (F'_0,F'_1) ......(F'_m^2,F'_(m^2+1))
> two are egals
>
> so F'_T(m)=0 and m|F_T(m)  ad also m|F_kT(m)  (because F_p|F_(pq) )
>
> But for many examples we can find
> k<T(m) such as k|F_q
>
>
> if m=6, T(6)=24 but 6|F_12=144
>
> if m=7 T(7)=16 but 7|F_8=21
>
> question : one can find always q<T(m) such as m|F_q ?
> Thanks

 What do you mean by always? If you mean for every m, then clearly not: if m 
= 4 the period is 6, and F_6 is the first F_q to be divisible by 4(excluding 
the trivial case q = 0, of course).