algebra precalculus - Any integer written as "polynomial" in irrational number

Does there exist an irrational number $x>2$ such that any positive integer $n$ can be written in the form $n=a_0+a_1x+a_2x^2+\dots$, where $a_i\in\{0,1,\dots,6\}$?

Some irrational numbers like $\varphi=\frac{\sqrt{5}+1}{2}$ combine well to give integers: $\varphi^2-\varphi=1$. But we need plus instead of minus and also $x>2$.

Answer

Let $x = \sqrt{7}$. Any positive integer $n$ has a base-$7$ representation, which is to say it can be written in the form
$$
n=b_0+b_1 \cdot 7^1+b_2 \cdot 7^2+\dots
$$
where each $b_k \in \{0, 1, \dots, 6\}$.

Now, take $a_{2k}=b_k$ and $a_{2k+1}=0$, and you have the desired representation of $n$.