[seqfan] Re: Can a repunit be a Fibonacci number?

Jack Brennen jfb at brennen.net
Fri May 13 19:23:58 CEST 2016

Even if you go to generalized base representations, there are only four
examples of Fibonacci numbers less than 10^15 that can be written using
three or more digits from [0,1] in any base:

13 == 111 base 3
21 == 111 base 4
144 == 100 base 12
4181 == 1001111 base 4

(Of course any number >= 3 can be written as 10 or 11 in an appropriate
base, which is why we add the restriction of three or more digits.)

So you have two very small GRU (generalized repunit) Fibonacci numbers,
a square Fibonacci number, and a single unclassified outlier.

On 5/13/2016 9:45 AM, Alonso Del Arte wrote:
> Sorting the base 10 digits of the Fibonacci numbers (A272918) does not
> really seem to go against what we know about Benford's law and the
> Fibonacci numbers. But if we sort the digits in descending order, it seems
> that, aside from the trivial initial exceptions, no term will start with
> the digit 1.
> Of course a base 10 repunit is not the only kind of number that would allow
> a Fibonacci number to be in this analogue to A272918. A Fibonacci number
> that has only 1s and 0s would then become something like 111111111000 in
> this sequence, but that seems unlikely as well.
> Al

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