# [seqfan] Re: Consecutive composite Fibonacci numbers

Alonso Del Arte alonso.delarte at gmail.com
Fri Nov 19 16:09:04 CET 2010

```Neil, the first place that I looked was A090206, the non-prime Fibonacci
numbers, where I hoped to find a comment to this effect. But there is a
sequence, presently absent from the OEIS, that I've been pondering in
connection to this question.

Tony, Robert, it was silly of me to forget about F(kn) being divisible by
F(k). However, the lengths of runs of consecutive composite numbers are odd:
the run is bounded by two odd primes, e.g., 7, *8, 9, 10*, 11; 23, *24, 25,
26, 27, 28*, 29. If there is always a run of exactly an odd number of
consecutive composite Fibonacci numbers, the proof might follow from
F(k)|F(kn), but not immediately. Though the statements I've seen on the Web
don't have the word "exactly."

Al

On Thu, Nov 18, 2010 at 10:34 PM, Robert Israel <israel at math.ubc.ca> wrote:

> This is obvious: F_{kn} is divisible by F_k, so whenever integers
> j,j+1,...,j+n are composite (and > 4), F_j, ..., F_{j+n} are
> also composite.
>
> Robert Israel                                israel at math.ubc.ca
> Department of Mathematics        http://www.math.ubc.ca/~israel
> University of British Columbia            Vancouver, BC, Canada
>
> On Thu, 18 Nov 2010, Alonso Del Arte wrote:
>
> > Who proved that there is always a run of n consecutive composite
> Fibonacci
> > numbers?
> >
> > (I'm sure it's either in Fib. Quart. or in Koshy's book, but I have no
> idea
> > what search terms to use to zero in on this particular result).
> >
> > Any pointers would be appreciated.
> >
> > Al
> >
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```