Sum of Two Fifth Powers

Theorem

 * $x^5 + y^5 = \paren {x + y} \paren {x^4 - x^3 y + x^2 y^2 - x y^3 + y^4}$

Proof
From Sum of Two Odd Powers:
 * $a^{2 n + 1} + b^{2 n + 1} = \paren {a + b} \paren {a^{2 n} - a^{2 n - 1} b + a^{2 n - 2} b^2 - \dotsb + a b^{2 n - 1} + b^{2 n} }$

We have that $5 = 2 \times 2 + 1$.

The result follows directly by setting $n = 2$.