Proof that $\sqrt[n]{a+1}$ and $\sqrt[n]{a-1}$ cannot be both rationals

Question

Let $a \neq 0$ be a natural number. How can be proved that $\sqrt[n] {a+1}$ and $\sqrt[n]{a-1}$ cannot be both rational numbers?

Answer 1

I presume $n$ is an integer $> 1$.

If $x = (a+1)^{1/n}$ and $y = (a-1)^{1/n}$ are rational, we have $$ x^n - 1 = a = y^n + 1$$ $x$ and $y$ are algebraic integers which are rational, thus they are (ordinary) integers, and $$ 2 = x^n - y^n$$ Now $x^n - y^n$ is divisible by $x-y$, so $x-y$ can only be $1$ or $2$.
And then it's easy to prove that the only possibility is $x=1$, $y=-1$, $a=0$, $n$ odd.