# On two nested radicals and divisibility

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: $$x = \prod^{p}_{k=1}{f_k(x)}$$

• I'm confused; $x$ is always the product of $u$ and $l$. You can verify this by just multiplying the expressions on the right. So it seems like what you're asking just boils down to when $\sqrt{1+4x}$ is an odd integer – TomGrubb Apr 2 at 21:20
• I think TomGrubb is right in that the property of $u_n$, $l_n$ should not be that they are integers iff $x=u_n(x)l_n(x)$. This is true for any $x$, also when $u$ and $l$ are not integers. I think what you want is $u_n(x)\in \mathbb Z \iff l_n(x)\in \mathbb Z \iff x=k(k+n)$ for some integer $k$. (This might be what you already meant). – Milten Apr 2 at 22:38
• Yes you are both correct, thanks. I removed the uneccesarry condition. – SmallestUncomputableNumber Apr 3 at 9:49