<s>[INST] Prove that $\sqrt{m-1}$ is an integer 
Let $k$ be a fixed odd positive integer. Let $m,n$ be positive integers such that $(2+\sqrt{3})^k=m+n\sqrt{3}$. Prove that $\sqrt{m-1}$ is an integer.

Let $(2+\sqrt{3})^n = a_n+b_n\sqrt{3}$. From $$a_{n+1}+b_{n+1}\sqrt{3} = (a_n+b_n\sqrt{3})(2+\sqrt{3}) = (2a_n+3b_n)+(a_n+2b_n)\sqrt{3},$$ we get $a_{n+1} = 2a_n+3b_n$ and $b_{n+1} = a_n+2b_n$ with $a_1 = 2$ and $b_1 = 1$.
How can we show that $\sqrt{a_n-1}$ is an integer for odd $n$?
 [/INST] A first step:
Define integer sequences $x_n$ and $y_n$ by
$$ (2+\sqrt{3})^{2n+1} = x_n+y_n\sqrt{3}. $$
Then $x_0=2$ and $y_0=1$, and furthermore you get the recursion
\begin{eqnarray*}
x_{n+1} &=& 7x_n+12y_n\\
y_{n+1} &=& 4x_n+7y_n
\end{eqnarray*}.
The first equation gives $12y_n=x_{n+1}-7x_n$, and plugging this into the second equation gives
$$ y_{n+1} = 4x_n+7y_n = 4x_n+\frac{7}{12}(x_{n+1}-7x_n) = 
\frac{1}{12}(7x_{n+1}-x_n).$$
This yields $12y_n=7x_n-x_{n-1}$, and plugging it into the first recursion yields
$$ x_{n+1} = 14x_n-x_{n-1}. $$
Together with $x_0=2$ and $x_1=26$, this nicely describes the sequence of odd-indexed numbers $a_{2n+1}=x_n$.  Hence we want to show that every $x_n-1$ is a square.
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