# [seqfan] Proof or counter sample needed

Artur grafix at csl.pl
Sun Oct 26 20:47:54 CET 2008

```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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>

```