[seqfan] Re: A014206 and computer algebra systems

[israel at math.ubc.ca]

The polynomial ((x^2-n)^2-)^2-n-x factors over the integers as a quadratic 
times a sextic. If n=m^2+m+2, the sextic factors as the product of two 
cubics: (m^3 - m^2*x - m*x^2 + x^3 + 2*m^2 - 2*m*x + 3*m - 3*x + 1)*(m^3 + 
m^2*x - m*x^2 - x^3 + m^2 - x^2 + 2*m + 2*x + 1) It appears that if n is 
not of that form, the sextic is irreducible. The Galois group of that 
sextic does seem to be soluble when it is irreducible, so in principle 
explicit solutions in radicals should exist, but apparently the CAS's don't 
provide those explicit solutions in radicals.

Cheers,
Robert

On Jul 4 2019, Thomas Baruchel wrote:

>Dear fellow seqfans,
>
> just noticed that A014206 are the integers for which several tried CAS 
> are able to give explicit solutions for the equation ((x^2-n)^2-n)^2-n =x
>
>Here is what I mean with Maxima:
>
>(%i15) f(c) := solve( ((x^2-c)^2-c)^2-c =x, x)$
>(%i16) for i:1 thru 128 do block([c:f(i)], if length(c) > 3 then print(i));
>2
>4
>8
>14
>22
>32
>44
>58
>74
>92
>112
>
>Maybe someone will understand why.
>
>Best regards,
>
>--
>Thomas Baruchel
>
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>Seqfan Mailing list - http://list.seqfan.eu/
>
>