Bhargava's Theorem

Author

Eric W. Weisstein
April 13, 2004

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Algebra/BhargavasTheorem.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/BhargavasTheorem.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Check

cyc[f_,{a_,b_,c_}]:=f[{a,b,c}]+f[{b,c,a}]+f[{c,a,b}]
norm1[{a_,b_,c_}]:=a+b+c
norm2[{a_,b_,c_}]:=a^4+b^4+c^4

f[{a_,b_,c_}]:={a(b-c),b(c-a),c(a-b)}
g1[{a_,b_,c_}]:=a^2 b
g2[{a_,b_,c_}]:=a b^2
g[{a_,b_,c_}]:={cyc[g1,{a,b,c}],cyc[g2,{a,b,c}],3a b c}

ans1[{a_,b_,c_},n_,m_]:=norm2[Nest[f,Nest[g,{a,b,c},n],m]]
ans2[{a_,b_,c_},n_,m_]:=2(a b+b c+c a)^(2^(m+1) 3^n)
ans3[{a_,b_,c_},n_,m_]:=norm2[Nest[g,Nest[f,{a,b,c},m],n]]

Table[Expand[ans3[{a,b,-a-b},k,n]-ans1[{a,b,-a-b},k,n]],
{k,0,3},{n,0,3}]//Timing

Table[Expand[ans2[{a,b,-a-b},k,n]-ans1[{a,b,-a-b},k,n]],
{k,0,3},{n,0,3}]//Timing