[seqfan] Re: Mod[Fibonacci[n],n^2]<=Mod[Fibonacci[n],n]

Charles Greathouse charles.greathouse at case.edu
Fri Apr 30 06:22:16 CEST 2010


For your first question, this is the same as numbers n such that
fibonacci(n) % n^2 < n.  Unfortunately I never finished my code for
calculation of Fibonacci numbers mod small values, so I used an
inefficient Pari script to calculate the next terms as
288176, 290422, 565181, 760763

For your second question, I found no other examples.  I'm not sure
what to expect: if Fibonacci numbers behaved randomly mod n^2 we'd
expect finitely many since zeta(2) is finite, but they don't.  Such
numbers, I believe, are the intersection of the first sequence with
A023172.

Charles Greathouse
Analyst/Programmer
Case Western Reserve University

On Thu, Apr 29, 2010 at 11:08 PM, zak seidov <zakseidov at yahoo.com> wrote:
> 1.
> Numbers n such that
> Mod[Fibonacci[n],n^2]<=Mod[Fibonacci[n],n]
> {1,2,3,4,12,45,177,1932,2189,4712,
> 5771,5778,43220}
> More terms?
>
> 2.
> Q.Is Mod[Fibonacci[n],n^2]=0 only for n=1,12?
>
> Thanks, Zak
>
>
>
>
>
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