Curious decimal behaviour

Loosely speaking the problem is asking us if we can find at least one $\rho$ with this property: all the digits following the dot of $\rho$ are the same digits following the dot of $\rho^2$ . Now let's write this $\rho - \lfloor \rho \rfloor=\rho^2 - \lfloor \rho^2 \rfloor \quad \Leftrightarrow \quad \rho^2 - \rho= \lfloor \rho^2 \rfloor - \lfloor \rho \rfloor.$ $\varphi=\frac{1+\sqrt{5}}{2}$ The golden ratio $\varphi$ will do the trick for $\rho$ because it's the positive solution for $\rho^2-\rho-1=0$ so $1<\varphi<2 \Rightarrow \lfloor \varphi \rfloor=1$ $\lfloor \varphi^2 \rfloor=\lfloor \varphi+1 \rfloor=2$ and hence $\varphi^2-\varphi=1=\lfloor \varphi^2 \rfloor - \lfloor \varphi \rfloor=2-1=1.$