Is $357911$ the only perfect power that is obtained by concatenating $5$ or more consecutive odd numbers(in decimal)?
I noticed that while memorizing all perfect cubes from $1$ to $10^{6}$, $357911=71^{3}$ is a perfect cube that is obtained by concatenating $5$ consecutive odd numbers(namely $3, 5, 7, 9,$ and $11$).
But is $357911$ the only perfect power is obtained by concatenating $5$ or more consecutive odd numbers?
I know these things:
- If a odd number is a perfect square, then the number must be $1\pmod{8}$, so if we want to make a perfect square that is obtained by concatenating $n$ odd numbers, then the number must be (either) form of $40k+1$ or $40k+9$.
- If we want our number that we concatenated to be a perfect 5th power, then the last two digits must either one of these:$01, 07, 25, 43, 49, 51, 57, 75, 93, 99$.
I have no idea for other perfect odd prime powers.