Ars Mathematica: Solución del problema 36

\begin{matrix} (x + y)^{2} & = & x^{2} + 2 x y + y^{2} \Rightarrow x^{2} + y^{2} = s^{2} - 2 p \\ (x^{2} + y^{2})^{2} & = & x^{4} + 2 x^{2} y^{2} + y^{4} \Rightarrow x^{4} + y^{4} = (s^{2} - 2 p)^{2} - 2 p^{2} \\ (x + y)^{4} & = & x^{4} + 4 x^{3} y + 6 x^{2} y^{2} + 4 x y^{3} + y^{4} \\ (x - y)^{4} & = & x^{4} - 4 x^{3} y + 6 x^{2} y^{2} - 4 x y^{3} + y^{4} \end{matrix}

(x + y)^{4} + (x - y)^{4} = 2 (x^{4} + 6 x^{2} y^{2} + y^{4})

s^{4} + (x - y)^{4} = 2 ((s^{2} - 2 p)^{2} - 2 p^{2}) + 12 p^{2}

(x - y)^{4} = s^{4} - 8 p s^{2} + 16 p^{2}

Ars Mathematica