[seqfan] Re: A014206 and computer algebra systems

[M. F. Hasler]

On Thu, Jul 4, 2019 at 8:39 AM Thomas Baruchel <baruchel at gmx.com> wrote:

> 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.
>

Your  P(n) = ((x^2-n)^2-n)^2-n -x = Q(n) * R(n)
with  Q(n) = x^2 - x - n ,
R(n) = x^6 + x^5 + (1-3*n)*x^4 + (1-2*n)*x^3 + (3*n^2 - 3*n + 1)*x^2 + (n^2
- 2*n + 1)*x - n^3 + 2*n^2 - n + 1

The numbers A014206 are those for which R(n) can be factored in two
polynomials of degree 3
(see examples below), so the CAS can give explicit solutions.
(You should be able to work out the remaining details...)

- Maximilian

f(c) = factor( ((x^2-c)^2-c)^2-c -x)
f(2) =( x - 2 ) (  x + 1 ) ( x^3 - 3*x + 1 ) ( x^3 + x^2 - 2*x - 1 )
f(3) =(  x^2 - x - 3 ) ( x^6 + x^5 - 8*x^4 - 5*x^3 + 19*x^2 + 4*x - 11 )
f(4) = ( x^2 - x - 4 ) (  x^3 - x^2 - 6*x + 7 ) ( x^3 + 2*x^2 - 3*x - 5)
f(5) =( x^2 - x - 5 ) ( x^6 + x^5 - 14*x^4 - 9*x^3 + 61*x^2 + 16*x - 79 )
f(6) =( x - 3 )( x + 2 ) ( x^6 + x^5 - 17*x^4 - 11*x^3 + 91*x^2 + 25*x -
149 )
f(7) = ( x^2 - x - 7 ) ( x^6 + x^5 - 20*x^4 - 13*x^3 + 127*x^2 + 36*x - 251
)
f(8) = ( x^2 - x - 8 ) ( x^3 - 2*x^2 - 11*x + 23) ( x^3 + 3*x^2 - 6*x - 17)
f(9) = ( x^2 - x - 9 ) ( x^6 + x^5 - 26*x^4 - 17*x^3 + 217*x^2 + 64*x - 57 )