Let us define the following functions:

$$f_n(x)=\sqrt{x^{n}-\sqrt{x^{n+1}- \sqrt{x^{n+2}-\cdots}}} $$ $$g_n(x)=\sqrt{x^{n}+\sqrt{x^{n+1}+ \sqrt{x^{n+2}+\cdots}}} $$

with $f(x)=f_1(x)$ and $g(x)=g_1(x)$. Very little is known about $f(x)$ and $g(x)$, except:

- The nested radical converges in both cases if $x > 1$
- $\lim_{x\rightarrow 1^+} f(x) = \frac{-1+\sqrt{5}}{2}$ and $g(1) = \frac{1+\sqrt{5}}{2}$
- $f(4)=1$ and $g(4)=3$

Let us now focus on the case where $x>1$ is an integer. The goal here is to obtain more advanced, interesting results about these nested radicals, maybe even a closed form or some asymptotic formulas.

**1. Integer part of the infinite nested radicals**

Let $\lfloor\cdot\rfloor$ denotes the integer part function. We have:

$$\lfloor f_{2n}(x) \rfloor = x^{n}-\phi(x)\\ \mbox{ } \lfloor g_{2n}(x) \rfloor = x^{n}+\psi(x)$$

with $\phi(x)=k$ if $x\in A_k$, $\psi(x)=k$ if $x\in B_k$. The sets $A_k, B_k$ are as follows:

- $A_1=[2,5[, A_2=[5,15[, A_3= [15,33[, A_4=[33,61[, A_5=[61,96[, \cdots$
- $B_0=[2,4[, B_1=[4,17[, B_2= [17,38[, B_3=[38,67[, B_4=[67,104[, \cdots$

**2. Fractional part of the infinite nested radicals**

Let $\{\cdot\}$ denotes the fractional part function. We seem to have:

$$\lim_{n\rightarrow\infty} \{ f_{2n}(x)\}=1-\Big\{\frac{\sqrt{x}}{2}\Big\}\\ \lim_{n\rightarrow\infty} \{ g_{2n}(x)\}=\Big\{\frac{\sqrt{x}}{2}\Big\} $$ Also, the roots of $\{g_1(x)\}$ have a very peculiar quadratic distribution. The first few ones, for $x>0$, are $\rho_1=4.0000$, $\rho_2=7.3370$, $\rho_3=11.6689$, $\rho_4=16.9982$, $\rho_5=23.3260$, $\rho_6=30.6526$, $\rho_7=38.9787$. Furthermore, it seems that

$$\lim_{k\rightarrow\infty} (\rho_{k+2}-2\rho_{k+1} +\rho_k) = 1.$$

Finally, values of $\{g_1(x)\}$ for large successive integers $x$ lying between two successive roots of $\{g_1(\cdot)\}$ tend to be equally spaced as $x\rightarrow\infty$. See table below.

**3. My question**

Actually a few related questions. Feel free to answer the one(s) you are most interested in.

Many of my results are experimental (thus I often use the word "it seems"). Can you prove some of them?

Obtain an explicit closed form for all sets $A_k,B_k$ used in the definition of $\phi(x)$ and $\psi(x)$ in section 1. Not sure if it is easy or not.

We focused on $n$ even. What happens for $n$ odd? Do we have interesting results? For instance, if $x=2$, the successive values of $\lfloor g_{2n+1}(x)\rfloor$ are $2, 3, 6, 12, 23, 45, 91, 181, 362, \cdots$ (for $n=0, 1, \cdots$). I did a reverse lookup on that sequence (see here) but it did not return any result despite the semi-obvious pattern.

What happens if $x$ is not an integer? Any interesting pattern or result?

Can you derive even more intriguing insights from the empirical results I presented here?

It looks like the larger $x$, the faster my limits are converging. Worth exploring.