[seqfan] Proof or counter sample needed

[Robert Israel]

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