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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: [mg57388] Re: Solve or Reduce on a monstrosity of an expresssion (and a prize!)
  • From: Daniel Reeves <dreeves at umich.edu>
  • Date: Thu, 26 May 2005 04:31:35 -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>
  • Sender: owner-wri-mathgroup at wolfram.com

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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