Square Root Begins with the Same Digits

Question

Are there any natural numbers $n$ such that $\sqrt n$, when expressed as a decimal, begins with the same string of numerals? For example, $\sqrt {100} = 10.00000.$ Obviously even powers of $10$ work, but what about non-trivial solutions? What about other bases?

Answer 1

No.

Write $n=a 10^m$ where $1\le a <10$.

If $m=2k$, $\sqrt n = \sqrt a 10^k$ so we want $a \approx \sqrt a$ or $a \approx 1$.

If $m=2k+1$, $\sqrt n = \sqrt{10a}10^k$ so we want $a \approx \sqrt{10a}$ or $a \approx 10$.

However you can for cube root. $\sqrt[3]{31.6} \approx 3.16$.

Answer 2

Depending on how long you want the leading string to be, consider $10^k+1$. It starts with a $1$ followed by $k-1$ zeros. $\left(10^k+1\right)^2=10^{2k}+2\cdot 10^k+1$ so it starts with a $1$ followed by $k-1$ zeros. This is not quite your trivial solution and works in any base.