<s>[INST] Find sufficient and necessary conditions in which $f(x)$ is a natural number Let us consider the function $$f(x)=(-√3+2)^{2^{x-2}}+(√3+2)^{2^{x-2}}$$ where $x≥3$. 
We know for example that if $x$ is an integer, then $f(x)$ is also an integer.
My question is: Find sufficient and necessary conditions in which $f(x)$ is a natural number.
 [/INST] 1st claim :

Let $k \in \mathbb{N}$ with $k \geq 2$. Then :
$$ \Big( 2\cosh(x) = k \Big) \, \Leftrightarrow \, x = \ln \Big( \frac{k+\sqrt{k^2-4}}{2} \Big).  $$

Let $k \in \mathbb{N}$ with $k \geq 2$. We have :
$$ \begin{align*}
2\cosh(x) = k &\Leftrightarrow {} e^{2x} - ke^{x} + 1 = 0 \\
\end{align*}
$$
Let $u=e^{x} > 0$. We solve $u^2 - k u + 1 =0$ and find that $u=e^{x}=k+\sqrt{k^2-4}$. 
2nd claim : 

For all $x \in \mathbb{R}$ with $x \geq 3$, 
$$ f(x) = 2 \cosh \big( 2^{x-2} \ln(\sqrt{3}+2) \big). $$

Just note that : $\ln(-\sqrt{3}+2) = - \ln(\sqrt{3}+2)$.
3rd claim :

Let $k \in \mathbb{N}$ with $k \geq 2$. Then :
$$f(x) =k \; \Leftrightarrow \; x = \frac{1}{\ln(2)} \Bigg( (\ln \circ \ln) \Big( \frac{k+\sqrt{k^2-4}}{2} \Big) - (\ln \circ \ln)(\sqrt{3}+2) \Bigg) + 2. $$

We have :
$$\begin{align*}
f(x) = k &\Leftrightarrow {} 2^{x-2}\ln(\sqrt{3}+2) = \underbrace{\ln\Big( \frac{k+\sqrt{k^2-4}}{2} \Big)}_{>0} \\[2mm]
 &\Leftrightarrow (x-2)\ln(2) + (\ln \circ \ln)(\sqrt{3}+2) = (\ln \circ \ln)\Big( \frac{k+\sqrt{k^2-4}}{2} \Big) \\[2mm]
\end{align*}
$$
Which gives the expected value for $x$.
Then, you can verify that if $x$ is given as in the third claim, then $f(x)$ is an integer.
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