Does there exist a real number $0< x <1$, such that the decimal expansions of $x$ and $x^2$ are the same, starting from the millionth term, and neither expansion has an infinite tail of zeroes?
I was thinking $x=0.\overline{999}$, but does that work? Isn't that just equal to 1 which is not allowed.? If this works, how would I prove it?
We can concoct an example quite easily. Suppose we want the difference between $x$ and $x^2$ to be 0.1: $$x-x^2=0.1$$ where the order $x-x^2$ is mandated by $0<x<1$, so $x^2<x$. Solving this, we get two admissible values $x=\frac{1\pm\sqrt{0.6}}2$.
Thus (taking $x=\frac{1+\sqrt{0.6}}2$) we have $$x=0.88729833\dots$$ $$x^2=0.78729833\dots$$ so their decimal expansions agree after the first place, and indeed after the millionth place.
Any number $0<k<0.25$ with a terminating decimal expansion such that $\sqrt{1-4k}$ does not terminate can be used in place of the 0.1 in $x-x^2=0.1$.