TITLE: Nested radicals of 1000 square roots
QUESTION [25 upvotes]: Find integer solutions to the equation
$$\underbrace{\sqrt{x+\sqrt{x+\sqrt{x+\cdots+\sqrt{x+\sqrt{x}}}}}}_{1000}=y$$
The first thing that popped to my mind was the infinitely nested radical and the recurrence relation $$a_{n+1}=\sqrt{x+a_n}$$
That lead me nowhere so I tried to show there aren't any such $(x,y)$ except for the trivial $(0,0)$ I think I proved that $$\sqrt{a_0}\leq a_{n}\leq \sqrt{a_0}+1$$ but the proof itself was pretty vague and even if it is correct doesn't help me.I have no idea what else to do,any hints would be appreciated.

REPLY [3 votes]: Another problem where starting with the simpler version of the problem does wonders:
$\sqrt{x+\sqrt{x}}=y$
We need both $x+\sqrt{x}$ and $\sqrt{x}$ to be squares; say $x=k^2$, $k\in \mathbb{N}$.
Then $x+\sqrt{x}=k^2+k<(k+1)^2$, so it cannot be a perfect square.
Taking further square roots is not going to remedy this.