TITLE: Solve an equation with four nested radicals
QUESTION [5 upvotes]: I have to solve the equation $\sqrt{x-{\sqrt{x-{\sqrt{x-{\sqrt{x-5}}}}}}}=5$.
Repeated squaring of both sides of the equation makes it very complex.
Is there any substitution or something similar which can simplify the problem ?
Does the problem have a solution if "generalized" as $\sqrt{x-{\sqrt{x-{\sqrt{x-{\sqrt{x-n}}}}}}}=n$ ?

REPLY [2 votes]: For the moment, forget the question and consider this equation. 
$$
\sqrt{x-p}=p
$$
It is easy to see that :
$$
x-p=p^2
\\
\implies x=p(p+1)
$$
Now consider :
$$
x-\sqrt{x-p}=x-p=k
\\
x-\sqrt{x-\sqrt{x-p}}=x-\sqrt{k}=x-\sqrt{x-p}=x-p=p^2
$$
And so on. 
It is easy to see that the solution of $x$ always remains $p(p+1)$.