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MathGroup Archive 2005

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Re: Solve or Reduce on a monstrosity of an expresssion (and a prize!)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg57390] Re: Solve or Reduce on a monstrosity of an expresssion (and a prize!)
  • From: Andrzej Kozlowski <andrzej at akikoz.net>
  • Date: Thu, 26 May 2005 04:31:41 -0400 (EDT)
  • References: <200505230620.CAA04045@smc.vnet.net> <EEA0FBBD-9C31-419A-8D0B-7C73AE4DA32E@akikoz.net> <Pine.LNX.4.58.0505231817420.9452@boston.eecs.umich.edu> <458D701E-37FA-425F-89C4-52A5628E22CF@akikoz.net> <Pine.LNX.4.58.0505241736010.24188@boston.eecs.umich.edu> <1D91709F-1E4D-4B98-95F5-695F7BD65577@akikoz.net> <Pine.LNX.4.58.0505242049410.3527@boston.eecs.umich.edu> <F016112F-BF6B-4D4E-8A3C-1F38AB3DDC5F@akikoz.net> <Pine.LNX.4.58.0505251303210.3527@boston.eecs.umich.edu>
  • Sender: owner-wri-mathgroup at wolfram.com

It seems to me that if you really want to maximize the chances of  
this problem being solved you should do two things. First, you should  
let everyone know why you want it solved: if people think it is  
something sufficiently important they will be more keen to spend time  
trying to solve it. Second, you should try to popularize it outside  
the Mathematica community. There are free equation solvers used for  
research purposes much more powerful than Mathematica and there are  
researchers who develop them, like Jean-Charles Faugere, who can  
solve anything that can be solved by human being or machine.
Even then I doubt this will ever get solved, but you will have a  
fighting chance.

Andrzej


On 26 May 2005, at 02:20, Daniel Reeves wrote:

>
> yikes.  ok, let's assume that all works.  So it remains to show  
> that both
> of the following expressions of n are zero for all n >= 2.  Strangely,
> FullSimplify can do this for all specific n up to 113 but 114 seems to
> stump it.
>
> here are the expressions:
>
> ((-12*(-Sqrt[3] + 2*Sqrt[3]*n + 18*Sqrt[3]*n^2 + 34*Sqrt[3]*n^3 +
>      19*Sqrt[3]*n^4 + 6*n*(1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2] +
>      6*n^2*(1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2] - 3^(1/3)*(-1 + n)*(1 +  
> 2*n)*
>       Sqrt[-1 + 6*n + 7*n^2]*
>       (n*(1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n +  
> 7*n^2])/n))^
>        (1/3))*(-3^(2/3) + 3^(2/3)*n^2 -
>      3*n^(4/3)*(9 + 18*n + 9*n^2 + ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3  
> + 21*n])/
>          n)^(1/3) - 3^(1/3)*(n*(1 + n)*(9 + 9*n +
>          (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2])/n))^(2/3))*
>     (3^(2/3) - 2*3^(2/3)*n + 2*3^(2/3)*n^3 - 3^(2/3)*n^4 -
>       3*(-1 + n)*n^(7/3)*(9 + 18*n + 9*n^2 + ((1 + n)^(3/2)*(1 + 2*n)*
>            Sqrt[-3 + 21*n])/n)^(1/3) + 3*(-1 + n)*
>        (n*(9 + 18*n + 9*n^2 + ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3 +  
> 21*n])/n))^
>         (1/3) + 3^(1/3)*(-1 + n)^2*
>        (n*(1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n +  
> 7*n^2])/n))^
>         (2/3))^2)/((-1 + n)^4*(1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2]*
>     (n*(1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n +  
> 7*n^2])/n))^
>      (2/3)*(3^(1/3) - 3^(1/3)*n^2 +
>       (n*(1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n +  
> 7*n^2])/n))^
>        (2/3))^3) + (6*(-Sqrt[3] + 19*Sqrt[3]*n^3 +
>      3*n*(Sqrt[3] + 2*Sqrt[-1 + 6*n + 7*n^2]) +
>      3*n^2*(5*Sqrt[3] + 4*Sqrt[-1 + 6*n + 7*n^2]))*
>     (-3^(2/3) + 3^(2/3)*n^2 - 3*n^(4/3)*
>       (9 + 18*n + 9*n^2 + ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3 + 21*n])/ 
> n)^
>        (1/3) - 3^(1/3)*(n*(1 + n)*(9 + 9*n +
>          (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2])/n))^(2/3))*
>     (3^(2/3) - 3^(2/3)*n - 3^(2/3)*n^2 + 3^(2/3)*n^3 +
>       3*n^(7/3)*(9 + 18*n + 9*n^2 + ((1 + n)^(3/2)*(1 + 2*n)*Sqrt 
> [-3 + 21*n])/
>           n)^(1/3) - 3*(n*(9 + 18*n + 9*n^2 + ((1 + n)^(3/2)*(1 +  
> 2*n)*
>             Sqrt[-3 + 21*n])/n))^(1/3) +
>       3^(1/3)*(n*(1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*
>             Sqrt[-1 + 6*n + 7*n^2])/n))^(2/3) -
>       3^(1/3)*n*(n*(1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*
>             Sqrt[-1 + 6*n + 7*n^2])/n))^(2/3))^2)/
>    ((-1 + n)^2*n*(1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2]*
>     (n*(9 + 18*n + 9*n^2 + ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3 +  
> 21*n])/n))^
>      (1/3)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2])/n)*
>     (3^(1/3) - 3^(1/3)*n^2 +
>       (n*(1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n +  
> 7*n^2])/n))^
>        (2/3))^2) + ((3*3^(1/3) - 6*3^(1/3)*n^2 + 3*3^(1/3)*n^4 +
>      3*3^(2/3)*n^(7/3)*(9 + 18*n + 9*n^2 + ((1 + n)^(3/2)*(1 + 2*n)*
>           Sqrt[-3 + 21*n])/n)^(1/3) - 9*n^(5/3)*
>       (9 + 18*n + 9*n^2 + ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3 + 21*n])/ 
> n)^
>        (2/3) - 9*n^(8/3)*(9 + 18*n + 9*n^2 + ((1 + n)^(3/2)*(1 + 2*n)*
>           Sqrt[-3 + 21*n])/n)^(2/3) - 3*3^(2/3)*
>       (n*(9 + 18*n + 9*n^2 + ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3 +  
> 21*n])/n))^
>        (1/3) - 6*n^(8/3)*((1 + n)*(9 + 9*n +
>          (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2])/n))^(2/3) +
>      6*(n*(1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n +  
> 7*n^2])/n))^
>        (2/3) + 3^(2/3)*(n*(1 + n)*(9 + 9*n +
>          (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2])/n))^(4/3) -
>      3*3^(1/3)*(1 + n)*(9*n^2 + Sqrt[3]*Sqrt[-1 + 6*n + 7*n^2] +
>        n*(9 + 2*Sqrt[3]*Sqrt[-1 + 6*n + 7*n^2])))*
>     (9 + (3^(1/3)*((Sqrt[3]*(-1 + n)^3*(1 + n)^(3/2))/
>          (n*(1 + 2*n)*Sqrt[-1 + 7*n]) + 2*(9*(1 + n)^2 +
>           ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3 + 21*n])/n))*
>        (-(3^(1/3)*(-1 + n)*(1 + n)) +
>         (n*(1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n +  
> 7*n^2])/n))^
>          (2/3)))/(n^(1/3)*(1 + n)*(9 + 18*n + 9*n^2 +
>          ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3 + 21*n])/n)^(4/3)) -
>      (3*n^(2/3)*((-2*3^(2/3)*(-1 + n)^2*(1 + n))/n +
>         (2*3^(1/3)*(-1 + n^2)^2*(-Sqrt[3] + 3*Sqrt[3]*n + 15*Sqrt[3] 
> *n^2 +
>            19*Sqrt[3]*n^3 + 6*n*(1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2]))/
>          ((-1 + n)*n^(4/3)*(1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2]*
>           ((1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n +  
> 7*n^2])/n))^
>            (1/3))))/((-1 + n^2)*(9 + 18*n + 9*n^2 +
>          ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3 + 21*n])/n)^(1/3))))/
>    (-(3^(1/3)*(-1 + n)*(1 + n)) +
>      (n*(1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n +  
> 7*n^2])/n))^
>       (2/3))^2 + (3*n^(4/3)*(1 + n)*
>     (9 + 18*n + 9*n^2 + ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3 + 21*n])/n) 
> ^(2/3)*
>     (3*n^(1/3) + (-3^(2/3) + 3^(2/3)*n^2 -
>        3^(1/3)*(n*(1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*
>              Sqrt[-1 + 6*n + 7*n^2])/n))^(2/3))/
>       ((1 + n)*(9 + 18*n + 9*n^2 + ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3  
> + 21*n])/
>           n)^(1/3)))*(9 - (6*3^(1/3)*(-1 + n)*(-Sqrt[3] + 2*Sqrt[3] 
> *n +
>         18*Sqrt[3]*n^2 + 34*Sqrt[3]*n^3 + 19*Sqrt[3]*n^4 +
>         6*n*(1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2] + 6*n^2*(1 + 2*n)*
>          Sqrt[-1 + 6*n + 7*n^2] - 3^(1/3)*(-1 + n)*(1 + 2*n)*
>          Sqrt[-1 + 6*n + 7*n^2]*(n*(1 + n)*(9 + 9*n +
>             (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2])/n))^(1/3)))/
>       (n*(1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2]*
>        (n*(1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n +  
> 7*n^2])/n))^
>         (2/3)) + (3*3^(1/3)*(-Sqrt[3] + 2*Sqrt[3]*n + 18*Sqrt[3]*n^2 +
>         34*Sqrt[3]*n^3 + 19*Sqrt[3]*n^4 + 6*n*(1 + n)^(3/2)*(1 + 2*n)*
>          Sqrt[-1 + 7*n])*(-3^(1/3) + 3^(1/3)*n^2 -
>         (n*(1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n +  
> 7*n^2])/n))^
>          (2/3)))/((1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2]*
>        (n*(9 + 18*n + 9*n^2 + ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3 +  
> 21*n])/n))^
>         (4/3)) - (2*3^(2/3)*(-1 + n)*(-Sqrt[3] + 2*Sqrt[3]*n +
>         18*Sqrt[3]*n^2 + 34*Sqrt[3]*n^3 + 19*Sqrt[3]*n^4 +
>         6*n*(1 + n)^(3/2)*(1 + 2*n)*Sqrt[-1 + 7*n])*
>        (3^(1/3) - 3^(1/3)*n^2 + (n*(1 + n)*(9 + 9*n +
>             (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2])/n))^(2/3))^2)/
>       (n^(8/3)*(1 + n)^(3/2)*(1 + 2*n)*Sqrt[-1 + 7*n]*
>        (9 + 18*n + 9*n^2 + ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3 +  
> 21*n])/n)^
>         (5/3)) + (3^(1/3)*(-1 + n)^2*((Sqrt[3]*(-1 + n)^3*(1 + n)^ 
> (3/2))/
>          (n*(1 + 2*n)*Sqrt[-1 + 7*n]) + 2*(9*(1 + n)^2 +
>           ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3 + 21*n])/n))*
>        (-(3^(1/3)*(-1 + n)*(1 + n)) +
>         (n*(1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n +  
> 7*n^2])/n))^
>          (2/3)))/((-1 + n^2)*(n*(9 + 18*n + 9*n^2 +
>           ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3 + 21*n])/n))^(4/3)) +
>      (3*(-1 + n)*(-3^(2/3) + 3^(2/3)*n^2 -
>         3^(1/3)*(n*(1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*Sqrt[-1 +  
> 6*n +
>                 7*n^2])/n))^(2/3)))/(n^(4/3)*(1 + n)*
>        (9 + 18*n + 9*n^2 + ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3 +  
> 21*n])/n)^
>         (1/3)) + (3*n^(2/3)*((-2*3^(2/3)*(-1 + n)^2*(1 + n))/n +
>         (2*3^(1/3)*(-1 + n^2)^2*(-Sqrt[3] + 3*Sqrt[3]*n + 15*Sqrt[3] 
> *n^2 +
>            19*Sqrt[3]*n^3 + 6*n*(1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2]))/
>          ((-1 + n)*n^(4/3)*(1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2]*
>           ((1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n +  
> 7*n^2])/n))^
>            (1/3))))/((-1 + n^2)*(9 + 18*n + 9*n^2 +
>          ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3 + 21*n])/n)^(1/3)) +
>      (2*3^(1/3)*(-1 + n)^2*(-(3^(1/3)*(-1 + n)*(1 + n)) +
>         (n*(1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n +  
> 7*n^2])/n))^
>          (2/3))*((-2*3^(2/3)*(-1 + n)^2*(1 + n))/n +
>         (2*3^(1/3)*(-1 + n^2)^2*(-Sqrt[3] + 3*Sqrt[3]*n + 15*Sqrt[3] 
> *n^2 +
>            19*Sqrt[3]*n^3 + 6*n*(1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2]))/
>          ((-1 + n)*n^(4/3)*(1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2]*
>           ((1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n +  
> 7*n^2])/n))^
>            (1/3))))/((-1 + n^2)^2*
>        (n*(9 + 18*n + 9*n^2 + ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3 +  
> 21*n])/n))^
>         (2/3)) + (3^(1/3)*n^(2/3)*((-1 + n)/n -
>         ((-1 + n)*(-3^(2/3) + 3^(2/3)*n^2 - 3^(1/3)*
>             (n*(1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n  
> + 7*n^2])/
>                 n))^(2/3)))/(3*n^(4/3)*(1 + n)*(9 + 18*n + 9*n^2 +
>             ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3 + 21*n])/n)^(1/3)))*
>        (-(((Sqrt[3]*(-1 + n)^3*(1 + n)^(3/2))/(n*(1 + 2*n)*Sqrt[-1  
> + 7*n]) +
>            2*(9*(1 + n)^2 + ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3 +  
> 21*n])/n))*
>           (-(3^(1/3)*(-1 + n)*(1 + n)) + (n*(1 + n)*(9 + 9*n +
>               (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2])/n))^(2/3))) +
>         2*(-1 + n^2)*(9*(1 + n)^2 + ((1 + n)^(3/2)*(1 + 2*n)*Sqrt 
> [-3 + 21*n])/
>            n)*(-3*3^(1/3) + (3*(1 + n)^(2/3)*(-Sqrt[3] + 3*Sqrt[3]*n +
>              15*Sqrt[3]*n^2 + 19*Sqrt[3]*n^3 + 6*n*(1 + 2*n)*Sqrt 
> [-1 + 6*n +
>                 7*n^2]))/((-1 + n)*(1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2]*
>             (n*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n +  
> 7*n^2])/n))^
>              (1/3)))))/((1 + n)*(9 + 18*n + 9*n^2 +
>          ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3 + 21*n])/n)^(4/3)) +
>      3*n*(-1 - ((-1 + n)*(-3^(2/3) + 3^(2/3)*n^2 -
>           3^(1/3)*(n*(1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*Sqrt[-1  
> + 6*n +
>                   7*n^2])/n))^(2/3)))/(3*n^(4/3)*(1 + n)*
>          (9 + 18*n + 9*n^2 + ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3 +  
> 21*n])/n)^
>           (1/3)))*(3 + (3^(1/3)*(-Sqrt[3] + 2*Sqrt[3]*n + 18*Sqrt[3] 
> *n^2 +
>           34*Sqrt[3]*n^3 + 19*Sqrt[3]*n^4 + 6*n*(1 + n)^(3/2)*(1 +  
> 2*n)*
>            Sqrt[-1 + 7*n])*(-3^(1/3) + 3^(1/3)*n^2 -
>           (n*(1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n +  
> 7*n^2])/
>               n))^(2/3)))/((1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2]*
>          (n*(9 + 18*n + 9*n^2 + ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3 +  
> 21*n])/n))^
>           (4/3)) + (n^(2/3)*((-2*3^(2/3)*(-1 + n)^2*(1 + n))/n +
>           (2*3^(1/3)*(-1 + n^2)^2*(-Sqrt[3] + 3*Sqrt[3]*n + 15*Sqrt 
> [3]*n^2 +
>              19*Sqrt[3]*n^3 + 6*n*(1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2]))/
>            ((-1 + n)*n^(4/3)*(1 + 2*n)*Sqrt[-1 + 6*n + 7*n^2]*
>             ((1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n +  
> 7*n^2])/
>                 n))^(1/3))))/((-1 + n^2)*(9 + 18*n + 9*n^2 +
>            ((1 + n)^(3/2)*(1 + 2*n)*Sqrt[-3 + 21*n])/n)^(1/3)))))/
>    ((-1 + n)*(-(3^(1/3)*(-1 + n)*(1 + n)) +
>       (n*(1 + n)*(9 + 9*n + (Sqrt[3]*(1 + 2*n)*Sqrt[-1 + 6*n +  
> 7*n^2])/n))^
>        (2/3))^2))/(18*3^(2/3))
>
> and:
>
> (-2*(1 + n)*
>    (((1 + n)^4*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
>            (1 + n)^4/((-1 + n)*n^2))]))/((-1 + n)*n^2))^(2/3)*
>    ((3*3^(1/3)*(1 + n)^3)/n - (3*(1 + n)^(7/3)*(Sqrt[3] + 3*Sqrt[3] 
> *n -
>        15*Sqrt[3]*n^2 + 19*Sqrt[3]*n^3 - 6*n*Sqrt[(-1 + n)*(1 +  
> 7*n)] +
>        12*n^2*Sqrt[(-1 + n)*(1 + 7*n)]))/((-1 + n)^(1/6)*n^(4/3)* 
> (-1 + 2*n)*
>       Sqrt[1 + 7*n]*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 + n)]*
>           (-1 + n + 2*n^2))/n)^(1/3)))*(-((1 + n)^2/((-1 + n)*n)) +
>     (n^(2/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*
>           Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n)*n^2))]))^ 
> (1/3)*
>       ((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2) -
>        3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 +  
> n)]*
>               (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3)))/
>      (3*(1 + n)^(7/3)))*((-9*(1 + n)^2)/((-1 + n)*n) +
>     (n^(4/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*
>           Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n)*n^2))]))^ 
> (2/3)*
>       ((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2) -
>         3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1  
> + n)]*(
>                 -1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3))^2)/
>      (1 + n)^(14/3) +
>     (3*n^(2/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*
>           Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n)*n^2))]))^ 
> (1/3)*
>       (-((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2)) +
>        3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 +  
> n)]*
>               (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3)))/(1 + n)^ 
> (7/3) +
>     (3*(-((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2)) +
>        3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 +  
> n)]*
>               (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3)))/
>      ((-1 + n)^(2/3)*(n*(1 + n)*(9*(1 + n)^2 +
>          Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n) 
> *n^2))]))^
>        (1/3)) +
>     9*n*(-1 + (n^(2/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*
>             Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n) 
> *n^2))]))^(1/3)*
>         (-((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2)) +
>          3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1  
> + n)]*
>                 (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3)))/
>        (3*(1 + n)^(7/3)))*(-((1 + n)^2/((-1 + n)*n)) +
>       (n^(2/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*
>               (27*(1 + n) + (1 + n)^4/((-1 + n)*n^2))]))^(1/3)*
>         (-((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2)) +
>          3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1  
> + n)]*
>                 (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3)))/
>        (3*(1 + n)^(7/3)))))/
>   (3^(2/3)*(-((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2)) +
>      3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 + n)]*
>             (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3))^3) -
>  ((1 + n)^(5/3)*
>    ((-1 + n)/(n*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
>            (1 + n)^4/((-1 + n)*n^2))])))^(1/3)*
>    ((Sqrt[3]*(1 + n)^5*Sqrt[(-1 + n)/(1 + 7*n)])/(n*(1 - 3*n +  
> 2*n^2)) +
>     2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
>           (1 + n)^4/((-1 + n)*n^2))]))*(-((1 + n)^2/((-1 + n)*n)) +
>     (n^(2/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*
>           Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n)*n^2))]))^ 
> (1/3)*
>       ((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2) -
>        3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 +  
> n)]*
>               (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3)))/
>      (3*(1 + n)^(7/3)))*((-9*(1 + n)^2)/((-1 + n)*n) +
>     (n^(4/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*
>           Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n)*n^2))]))^ 
> (2/3)*
>       ((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2) -
>         3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1  
> + n)]*(
>                 -1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3))^2)/
>      (1 + n)^(14/3) +
>     (3*n^(2/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*
>           Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n)*n^2))]))^ 
> (1/3)*
>       (-((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2)) +
>        3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 +  
> n)]*
>               (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3)))/(1 + n)^ 
> (7/3) +
>     (3*(-((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2)) +
>        3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 +  
> n)]*
>               (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3)))/
>      ((-1 + n)^(2/3)*(n*(1 + n)*(9*(1 + n)^2 +
>          Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n) 
> *n^2))]))^
>        (1/3)) +
>     9*n*(-1 + (n^(2/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*
>             Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n) 
> *n^2))]))^(1/3)*
>         (-((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2)) +
>          3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1  
> + n)]*
>                 (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3)))/
>        (3*(1 + n)^(7/3)))*(-((1 + n)^2/((-1 + n)*n)) +
>       (n^(2/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*
>               (27*(1 + n) + (1 + n)^4/((-1 + n)*n^2))]))^(1/3)*
>         (-((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2)) +
>          3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1  
> + n)]*
>                 (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3)))/
>        (3*(1 + n)^(7/3)))))/
>   (3*((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2) -
>      3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 + n)]*
>             (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3))^2) +
>  ((1 + n)*(((1 + n)^4*(9*(1 + n)^2 + Sqrt[3]*
>         Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n)*n^2))]))/
>      ((-1 + n)*n^2))^(2/3)*
>    (9 - (3*n^(2/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*
>           Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n)*n^2))]))^ 
> (1/3)*
>       ((2*3^(2/3)*(1 + n)^3)/n - (2*3^(1/3)*(1 + n)^(7/3)*
>          (Sqrt[3] + 3*Sqrt[3]*n - 15*Sqrt[3]*n^2 + 19*Sqrt[3]*n^3 -
>           6*n*Sqrt[(-1 + n)*(1 + 7*n)] + 12*n^2*Sqrt[(-1 + n)*(1 +  
> 7*n)]))/
>         ((-1 + n)^(1/6)*n^(4/3)*(-1 + 2*n)*Sqrt[1 + 7*n]*
>          (9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 + n)]*(-1 + n +  
> 2*n^2))/n)^
>           (1/3))))/(1 + n)^(7/3) -
>     (n^(5/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*
>           Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n)*n^2))]))^ 
> (4/3)*
>       (-((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2)) +
>        3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 +  
> n)]*
>               (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3))*
>       ((Sqrt[3]*(1 + n)^5*Sqrt[(-1 + n)/(1 + 7*n)])/(n*(1 - 3*n +  
> 2*n^2)) +
>        2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
>              (1 + n)^4/((-1 + n)*n^2))])))/(1 + n)^(13/3))*
>    ((-9*(1 + n)^2)/((-1 + n)*n) +
>     (n^(4/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*
>           Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n)*n^2))]))^ 
> (2/3)*
>       ((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2) -
>         3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1  
> + n)]*(
>                 -1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3))^2)/
>      (1 + n)^(14/3) +
>     (3*n^(2/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*
>           Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n)*n^2))]))^ 
> (1/3)*
>       (-((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2)) +
>        3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 +  
> n)]*
>               (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3)))/(1 + n)^ 
> (7/3) +
>     (3*(-((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2)) +
>        3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 +  
> n)]*
>               (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3)))/
>      ((-1 + n)^(2/3)*(n*(1 + n)*(9*(1 + n)^2 +
>          Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n) 
> *n^2))]))^
>        (1/3)) +
>     9*n*(-1 + (n^(2/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*
>             Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n) 
> *n^2))]))^(1/3)*
>         (-((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2)) +
>          3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1  
> + n)]*
>                 (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3)))/
>        (3*(1 + n)^(7/3)))*(-((1 + n)^2/((-1 + n)*n)) +
>       (n^(2/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*
>               (27*(1 + n) + (1 + n)^4/((-1 + n)*n^2))]))^(1/3)*
>         (-((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2)) +
>          3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1  
> + n)]*
>                 (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3)))/
>        (3*(1 + n)^(7/3)))))/
>   (18*((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2) -
>      3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 + n)]*
>             (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3))^2) +
>  ((1 + n)*(((1 + n)^4*(9*(1 + n)^2 + Sqrt[3]*
>         Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n)*n^2))]))/
>      ((-1 + n)*n^2))^(2/3)*(-((1 + n)^2/((-1 + n)*n)) +
>     (n^(2/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*
>           Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n)*n^2))]))^ 
> (1/3)*
>       ((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2) -
>        3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 +  
> n)]*
>               (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3)))/
>      (3*(1 + n)^(7/3)))*
>    (27 + (9*n^(2/3)*((-1 + n)/(9*(1 + n)^2 +
>          Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n) 
> *n^2))]))^
>        (1/3)*((2*3^(2/3)*(1 + n)^3)/n - (2*3^(1/3)*(1 + n)^(7/3)*
>          (Sqrt[3] + 3*Sqrt[3]*n - 15*Sqrt[3]*n^2 + 19*Sqrt[3]*n^3 -
>           6*n*Sqrt[(-1 + n)*(1 + 7*n)] + 12*n^2*Sqrt[(-1 + n)*(1 +  
> 7*n)]))/
>         ((-1 + n)^(1/6)*n^(4/3)*(-1 + 2*n)*Sqrt[1 + 7*n]*
>          (9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 + n)]*(-1 + n +  
> 2*n^2))/n)^
>           (1/3))))/(1 + n)^(7/3) +
>     (9*((2*3^(2/3)*(1 + n)^3)/n - (2*3^(1/3)*(1 + n)^(7/3)*
>          (Sqrt[3] + 3*Sqrt[3]*n - 15*Sqrt[3]*n^2 + 19*Sqrt[3]*n^3 -
>           6*n*Sqrt[(-1 + n)*(1 + 7*n)] + 12*n^2*Sqrt[(-1 + n)*(1 +  
> 7*n)]))/
>         ((-1 + n)^(1/6)*n^(4/3)*(-1 + 2*n)*Sqrt[1 + 7*n]*
>          (9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 + n)]*(-1 + n +  
> 2*n^2))/n)^
>           (1/3))))/((-1 + n)^(2/3)*
>       (n*(1 + n)*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
>              (1 + n)^4/((-1 + n)*n^2))]))^(1/3)) -
>     (9*n^(2/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*
>           Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n)*n^2))]))^ 
> (1/3)*
>       (-((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2)) +
>        3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 +  
> n)]*
>               (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3)))/(1 + n)^ 
> (7/3) +
>     (6*n^(4/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*
>           Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n)*n^2))]))^ 
> (2/3)*
>       ((2*3^(2/3)*(1 + n)^3)/n - (2*3^(1/3)*(1 + n)^(7/3)*
>          (Sqrt[3] + 3*Sqrt[3]*n - 15*Sqrt[3]*n^2 + 19*Sqrt[3]*n^3 -
>           6*n*Sqrt[(-1 + n)*(1 + 7*n)] + 12*n^2*Sqrt[(-1 + n)*(1 +  
> 7*n)]))/
>         ((-1 + n)^(1/6)*n^(4/3)*(-1 + 2*n)*Sqrt[1 + 7*n]*
>          (9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 + n)]*(-1 + n +  
> 2*n^2))/n)^
>           (1/3)))*(-((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2)) +
>        3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 +  
> n)]*
>               (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3)))/(1 + n)^ 
> (14/3) +
>     (2*((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2) -
>         3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1  
> + n)]*(
>                 -1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3))^2*
>       ((Sqrt[3]*(1 + n)^5*Sqrt[(-1 + n)/(1 + 7*n)])/(n*(1 - 3*n +  
> 2*n^2)) +
>        2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
>              (1 + n)^4/((-1 + n)*n^2))])))/
>      (n*(((1 + n)^4*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 +  
> n) +
>               (1 + n)^4/((-1 + n)*n^2))]))/((-1 + n)*n^2))^(5/3)) +
>     (3*n^(5/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*
>           Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n)*n^2))]))^ 
> (4/3)*
>       (-((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2)) +
>        3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 +  
> n)]*
>               (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3))*
>       ((Sqrt[3]*(1 + n)^5*Sqrt[(-1 + n)/(1 + 7*n)])/(n*(1 - 3*n +  
> 2*n^2)) +
>        2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
>              (1 + n)^4/((-1 + n)*n^2))])))/(1 + n)^(13/3) +
>     (3*(-1 + n)^(1/3)*n^(2/3)*(-((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2)) +
>        3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 +  
> n)]*
>               (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3))*
>       ((Sqrt[3]*(1 + n)^5*Sqrt[(-1 + n)/(1 + 7*n)])/(n*(1 - 3*n +  
> 2*n^2)) +
>        2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
>              (1 + n)^4/((-1 + n)*n^2))])))/((1 + n)^(7/3)*
>       (9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
>             (1 + n)^4/((-1 + n)*n^2))])^(4/3)) +
>     3*n*(-1 + (n^(2/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*
>             Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n) 
> *n^2))]))^(1/3)*
>         (-((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2)) +
>          3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1  
> + n)]*
>                 (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3)))/
>        (3*(1 + n)^(7/3)))*
>      (9 + (3*n^(2/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*
>             Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n) 
> *n^2))]))^(1/3)*
>         ((2*3^(2/3)*(1 + n)^3)/n - (2*3^(1/3)*(1 + n)^(7/3)*
>            (Sqrt[3] + 3*Sqrt[3]*n - 15*Sqrt[3]*n^2 + 19*Sqrt[3]*n^3 -
>             6*n*Sqrt[(-1 + n)*(1 + 7*n)] + 12*n^2*Sqrt[(-1 + n)*(1  
> + 7*n)]))/
>           ((-1 + n)^(1/6)*n^(4/3)*(-1 + 2*n)*Sqrt[1 + 7*n]*
>            (9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 + n)]*(-1 + n +  
> 2*n^2))/n)^
>             (1/3))))/(1 + n)^(7/3) +
>       (n^(5/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*
>               (27*(1 + n) + (1 + n)^4/((-1 + n)*n^2))]))^(4/3)*
>         (-((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2)) +
>          3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1  
> + n)]*
>                 (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3))*
>         ((Sqrt[3]*(1 + n)^5*Sqrt[(-1 + n)/(1 + 7*n)])/(n*(1 - 3*n +  
> 2*n^2)) +
>          2*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) + (1 +  
> n)^4/
>                 ((-1 + n)*n^2))])))/(1 + n)^(13/3)) -
>     (3*(1 + n)*(-((1 + n)^2/((-1 + n)*n)) +
>        (n^(2/3)*((-1 + n)/(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27* 
> (1 + n) +
>                 (1 + n)^4/((-1 + n)*n^2))]))^(1/3)*
>          (-((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2)) +
>           3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/ 
> (-1 + n)]*
>                  (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3)))/
>         (3*(1 + n)^(7/3)))*(-((-((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2)) +
>           3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/ 
> (-1 + n)]*
>                  (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3))*
>          ((Sqrt[3]*(1 + n)^5*Sqrt[(-1 + n)/(1 + 7*n)])/
>            (n*(1 - 3*n + 2*n^2)) + 2*(9*(1 + n)^2 +
>             Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) + (1 + n)^4/((-1 + n)*
>                   n^2))]))) - (2*3^(1/3)*(1 + n)^4*(-1 + n^2)*
>          (9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
>               (1 + n)^4/((-1 + n)*n^2))])*(3*3^(1/3) -
>           ((Sqrt[3]*(1 + n)^4*Sqrt[(-1 + n)/(1 + 7*n)])/
>              (n*(1 - 3*n + 2*n^2)) + 2*(9 + 9*n + (Sqrt[3]*Sqrt[(1  
> + 7*n)/
>                   (-1 + n)]*(-1 + n + 2*n^2))/n))/
>            (((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 + n)]*
>                  (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(1/3)))/
>         ((-1 + n)^2*n^2)))/
>      (((1 + n)^4*(9*(1 + n)^2 + Sqrt[3]*Sqrt[(1 + n)^3*(27*(1 + n) +
>              (1 + n)^4/((-1 + n)*n^2))]))/((-1 + n)*n^2))^(4/3)))/
>   (6*((3^(2/3)*(1 + n)^5)/((-1 + n)*n^2) -
>      3^(1/3)*(((1 + n)^5*(9 + 9*n + (Sqrt[3]*Sqrt[(1 + 7*n)/(-1 + n)]*
>             (-1 + n + 2*n^2))/n))/((-1 + n)*n^2))^(2/3))^2)
>
>
> --- \/   FROM Andrzej Kozlowski AT 05.05.25 12:42 (Today)   \/ ---
>
>
>> *This message was transferred with a trial version of CommuniGate 
>> (tm) Pro*
>>
>> On 25 May 2005, at 10:00, Daniel Reeves wrote:
>>
>>
>>> *This message was transferred with a trial version of CommuniGate
>>> (tm) Pro*
>>> wow, this is huge progress.  medium prize, at the least, is  
>>> cinched :)
>>>
>>> eager to hear how you establish that the derivative has <= 2  
>>> roots...
>>>
>>
>> First, I only think I could probably prove this, but that does not
>> mean I actually have a rigorous proof. I have a sketch of a possible
>> proof that I will describe, but it may not work. What is worse, even
>> if it is essentially correct, it is probably the easy part of the
>> problem. Checking the two "simple identities" for every integer may
>> turn out to be awfully hard, perhaps as hard as Fermat's Last  
>> Theorem?
>> So I don't think that my contribution so far deserves any prize :-(
>>
>> Anyway, this is my idea for a argument proving that D[f[x,n]]==0 has
>> only two roots, for every n. For each complex number n, think of the
>> expression D[f[x,n]] as an element of  the field Al of "algebraics"
>> over the complex numbers, that is the algebraic closure of the  
>> field C
>> [x] of rational functions in one variable with coefficients in the
>> complex numbers C. This is not just an algebraic object but also a
>> topological space. Each element f[x] of Al can be thought as a
>> function on C given by a radical expression like the one in your
>> problem. We know that for each f[x] there is a polynomial p[x] such
>> that all the roots of f[x]==0 are also roots of p[x]==0. As Daniel
>> Lichtblau pointed out in a recent posting the function
>> First[GroebnerBasis[f[x],x]]  gives such a polynomial. I think one
>> can probably make sure that the polynomial is monic, that is, its
>> highest degree term is 1. So consider the mapping f[x]->First
>> [GroebnerBasis[f[x],x]] from the topological space Al to the space of
>> all polynomials in one variable. Now, here comes the point I am most
>> uncertain about: I think this mapping can be made continuous. In
>> other words, if we change our radical expression f[x] only slightly
>> than the corresponding polynomial will only change slightly. In
>> particular, it's degree will not jump. If that is so  we are nearly
>> done. We need only to check only one more thing, which is that any
>> expression D[f[x,n],x] can be connected to D[f[x,m],x] by a
>> continuous path. Looking at the monstrous expression it seems to me
>> that it is so, anyway, as long as we keep n>1 .  But since we can
>> compute the polynomial for a number of integers n and we know that it
>> has degree 2, it would seem to follow that it must always have  
>> degree 2.
>>
>> To really be sure the argument is correct would require very good
>> understanding of what the GrobenerBasis algorithm really does in such
>> situations and I have only a vague understanding. In particular I
>> have never seen anywhere the issue of continuity being discussed, but
>> then I have not looked for it.
>> But one must remember even if we could prove this we could still be
>> almost as far from proving the full statement as we were at the  
>> start.
>>
>> Andrzej Kozlowski
>>
>>
>>
>>
>>
>>
>>>
>>>
>>> --- \/   FROM Andrzej Kozlowski AT 05.05.25 09:16 (Tomorrow)   \/  
>>> ---
>>>
>>>
>>>
>>>> I am now pretty sure that I could now prove the general result
>>>> provided that I could establish two "simple" facts, which are that:
>>>>
>>>>
>>>> FullSimplify[D[f[x, n], x] /. x -> (n - 1)/n,
>>>>    Element[n,Integers] && n > 2]
>>>>
>>>> FullSimplify[D[f[x, n], x] /.
>>>>     x -> (-n^2 - 2*n - 1)/((n - 1)*n),
>>>>    Element[n,Integers] && n > 2]
>>>>
>>>> are both zero. In other words, I think I can prove (there are some
>>>> details that I would have to check but I am pretty sure they are
>>>> fine)   that the derivative can have no more than two roots, so if
>>>> the above are the roots everything is done. But unfortunately after
>>>> 24 hours Matheamtica has not returned any answer to the first of  
>>>> the
>>>> above. I have not even tried the second.
>>>> Perhaps a more specialized algebra program for this sort of thing
>>>> might do better?
>>>>
>>>> Andrzej Kozlowski
>>>>
>>>>
>>>
>>> --
>>> http://ai.eecs.umich.edu/people/dreeves  - -  google://"Daniel  
>>> Reeves"
>>>
>>> "I have enough money to last me the rest of my life, unless I
>>> buy something." -- Jackie Mason
>>>
>>>
>>>
>>
>>
>
> -- 
> http://ai.eecs.umich.edu/people/dreeves  - -  google://"Daniel Reeves"
>
> "Engineers think of their equations as approximating reality.
>  Scientists think of reality as approximating their equations.
>  Mathematicians don't care."
>
>


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