"Show that (3x2+5xy−5y2)3+(4x2−4xy+6y2)3+(5x2−5xy−3y2)3=(6x2−4xy+4y2)3 and find other quadratic expressions satisfying similar relations."
There are in fact infinitely many such quadratic expressions. For general a3+b3+c3+d3=0 use the identity A3+B3+C3+D3=(a3+b3+c3+d3)(x2+wy2)3
and A,B,C,D are quadratic forms, A=ax2−v1xy+bwy2B=bx2+v1xy+awy2C=cx2+v2xy+dwy2D=dx2−v2xy+cwy2
where (v1,v2,w)=(c2−d2,a2−b2,(a+b)(c+d)).
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