[seqfan] Proof or counter sample needed

[Artur]

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}

If so than:

1) existed such rational x that 1 + 10 x^3 - 12 x^5 + 5 x^6  is square 
<x belonging to set{0,3/11,1}>
and
2) (2 (-1 - 5 x^3 + 8 x^5 + Sqrt[1 + 10 x^3 - 12 x^5 + 5 x^6]))/(-1 +
 5 x - 20 x^3 + 16 x^5)
or (2 (-1 - 5 x^3 + 8 x^5 - Sqrt[1 + 10 x^3 - 12 x^5 + 5 x^6]))/(-1 +
 5 x - 20 x^3 + 16 x^5)
is integer

Best wishes
Artur


Artur pisze:
> Dear Seqfans,
>
> Who is able to proof or find counter-sample following conjecture
> 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
> is factorizable if and only when k belonging to finite set {2,4,243}
> no more such k up to k=10^7
>
> Who have any idea let me know
>
> Best wishes
> Artur
>
>
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