Just another 2017 problem

$\displaystyle x - y \mid x^n - y^n \ \forall \ n \in \mathbb{N}$

By generalizing we can see that this statement is true; we will use proof by induction.

$\displaystyle \textbf{1.}$ Show $\displaystyle P(1)$ is true.

$\displaystyle x^1 - y^1$ is divisible by $\displaystyle x - y$

$\displaystyle \textbf{2.}$ Assume statement is true for $\displaystyle P(k)$ .

$\displaystyle x - y \mid x^k - y^k$

$\displaystyle \textbf{3.}$ Show that the statement is true for $\displaystyle P(k + 1)$ .

$\displaystyle x^{k + 1} - y^{k + 1}$

$\displaystyle = (x - y + y)x^k - y \cdot y^k$

$\displaystyle = x \cdot x^k - y \cdot x^k + y \cdot x^k - y \cdot y^k$

$\displaystyle = x^k(x - y) + y(x^k - y^k)$

$\displaystyle x - y \mid x^k(x - y) + y(x^k - y^k)$

$\displaystyle P(k + 1)$ is true, so $\displaystyle P(n)$ is true for all $\displaystyle n \in \mathbb{N}$ .

Hence, $\displaystyle x - y \mid x^{2017} - y^{2017}$

Relevant wiki: Remainder Factor Theorem - Basic

Let f(x)=x^2017 - y^2017. If (x-y) is a factor then y is a root hence f(y)=0 according to the factor theorem. As f(y)=y^2017 - y^2017 = 0, the statement must be true. This also works for f(y)=x^2017 - y^2017 following the same steps.