The last days I was playing around with two nested radicals which, as I learned here, can be simplified: $$u(x) =\sqrt{x + \sqrt{x +\sqrt{x +\sqrt{x +...}}}} = \frac{1}{2}(1+\sqrt{1+4x})$$ $$l(x) = \sqrt{x - \sqrt{x -\sqrt{x -\sqrt{x -...}}}} = \frac{1}{2}(-1+\sqrt{1+4x})$$
Whilst doing so I noticed something cool (maybe completely trivial or obvious but I think it's cool):
$u(x)$ and $l(x)$ are integers whenever $x = u(x)*l(x)$, where $u(x) = l(x)+1$.
I.e. when $x = k(k+1)$, e.g. $6, 12, 20, 30,\dots$
This can be generalized for $x = k(k+n)$:
$$\frac{1}{2}(\sqrt{1+4(x+(\frac{n^2-1}{4}))}\pm n)$$
My question: Is it possible to create a set of similiar functions $\{f_1(x),\dots,f_p(x)\}$, with the corresponding x-value being the product of their values, basicly extending the concept to higher roots? I.e: $$x = \prod^{p}_{k=1}{f_k(x)}$$