# [seqfan] Proof or counter sample needed

Artur grafix at csl.pl
Sun Oct 26 21:54:36 CET 2008

```Dear Robert,
Yes k is integer.
Artur

Robert Israel pisze:
> On Sun, 26 Oct 2008, Artur wrote:
>
>
>> P.S.
>> Easest to proof should be conjecture that:
>> Quintic polynomial
>>
>> 4 k - k^2 + 5 k^2 x + (20 k - 20 k^2) x^3 + (16 - 32 k + 16 k^2) x^5
>> have one rational root if and only when k belonging to finite set {2,4,243}
>>
>
> I suppose you're talking about integers k.
>
> Don't forget k = 0 (where your polynomial is 16 x^5) and
> k = 1 (where it is 3 + 5 x)
>
> If you allow rational k, then there are lots of solutions, e.g.
> k = 243/242 or 6250/6241 for x = -15
> k = 128/121 or 4/3 for x = -2
> k = 128/125 or 972/961 for x = 6
>
> Robert Israel                                israel at math.ubc.ca
> Department of Mathematics        http://www.math.ubc.ca/~israel
> University of British Columbia            Vancouver, BC, Canada
>
>
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```