# [seqfan] Proof or counter sample needed

Artur grafix at csl.pl
Sun Oct 26 21:29:55 CET 2008

```P.S.
Because polynomial 1 + 10 x^3 - 12 x^5 + 5 x^6 is factorizable
than square free factor : 1 - 2 x + 5 x^2  have to be square
Weeker conjecture which I  was  proof is
No existed such x <= A081016(n)> another than x=0 that
(2 (-1 - 5 x^3 + 8 x^5 + (-1 - x + x^2) Sqrt[1 - 2 x + 5 x^2]))/(-1 +5 x
- 20 x^3 + 16 x^5)
or
(2 (-1 - 5 x^3 + 8 x^5 - (-1 - x + x^2) Sqrt[1 - 2 x + 5 x^2]))/(-1 +5
x - 20 x^3 + 16 x^5)
is integer
for x=1 we have 0/0 case

because no existed such integer x that 1 + 5 x - 10 x^3 is divisable by
(-1 - x + x^2) Sqrt[1 - 2 x + 5 x^2]) because polynomial 1 + 5 x - 10
x^3 isn't divisable by (-1 - x + x^2)

That x can be only rational fraction (and no integer)

Artur

Artur pisze:
>
> P.S.
> 1 + 10 x^3 - 12 x^5 + 5 x^6 is square if and only when x belonging to
> A081016
> Artur
>
> Artur pisze:
> > 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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> >>
> >>
> >
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```