<s>[INST] Find all intgers $a,b,c$ such that $a\cdot 4^n+b\cdot 6^n+c\cdot 9^n$ is a perfect square Find all intgers $a,b,c$ such that
$$a\cdot 4^n+b\cdot 6^n+c\cdot 9^n$$ is a perfect square for all sufficiently large $n$
This problem is creat by Vesselin Dimitrov,From the (Problems from the books chapter 17).
My approach:it is clear $a=1,b=2,c=1$ such it.lf let $a_{n}=a\cdot 4^n+b\cdot 6^n+c\cdot 9^n$,then we have
$$a_{n+3}=19a_{n+2}-114a_{n+1}+216a_{n}$$
and $a_{0}=a+b+c,a_{1}=4a+6b+9c,a_{2}=16a+36b+81c$, then this problem it suffices find $a,b,c$ such $a_{n}(n\to +\infty)$ is square
 [/INST] We’re given that $x_n = \sqrt{a4^n + b6^n + c9^n}$ is an integer for sufficiently large $n$. We have
\begin{multline*}
(6x_n + 5x_{n+1} + x_{n+2})(6x_n + 5x_{n+1} - x_{n+2}) \\ (-6x_n + 5x_{n+1} + x_{n+2})(-6x_n + 5x_{n+1} - x_{n+2}) = 900(b^2 - 4ac)36^n,
\end{multline*}
and since $x_n = \sqrt c⋅3^n\left[1 ± O{\left(\left(\tfrac23\right)^n\right)}\right]$, we can divide this by
\begin{align*}
6x_n + 5x_{n+1} + x_{n+2} &= 30\sqrt c⋅3^n\left[1 ± O{\left(\left(\tfrac23\right)^n\right)}\right], \\
6x_n + 5x_{n+1} - x_{n+2} &= 12\sqrt c⋅3^n\left[1 ± O{\left(\left(\tfrac23\right)^n\right)}\right], \\
-6x_n + 5x_{n+1} + x_{n+2} &= 18\sqrt c⋅3^n\left[1 ± O{\left(\left(\tfrac23\right)^n\right)}\right],
\end{align*}
leaving
$$y_n = -6x_n + 5x_{n+1} - x_{n+2} = k\left(\tfrac43\right)^n\left[1 ± O{\left(\left(\tfrac23\right)^n\right)}\right], \quad \text{where } k = \frac{5(b^2 - 4ac)}{36\sqrt{c^3}}.$$
So $y_n - k\left(\frac43\right)^n = ±O{\left(\left(\tfrac89\right)^n\right)}$ converges to zero.  Hence the integer $4y_n - 3y_{n+1}$ converges to zero and must eventually remain zero.  Since a nonzero integer is only divisible by finitely many powers of $3$, this can only happen if $y_n$ eventually remains zero, $k = 0$, and $b^2 = 4ac$.
We can therefore write $x_n = \sqrt{\frac{b^2}{4c}4^n + b6^n + c9^n} = \frac{\lvert b2^{n-1} + c3^n\rvert}{\sqrt c}$. If this is to be an integer, $c$ must be a perfect square $s^2$ and $a = \frac{b^2}{4c}$ must be a perfect square $r^2$.  So the solution must be of the form $x_n = \lvert r2^n + s3^n\rvert$, $a = r^2$, $b = 2rs$, $c = s^2$ for integers $r, s$, which always works.
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