How to Prove Square Number?

Question

Suppose that we have these numbers and their radix is $b$. How to prove that these numbers are square number?

1. $(12321)_b $for every $b > 3$
2. $(14641)_b$ for every $b > 6$
3. $(1234321)_b$ for every $b > 4$

Answer 1

\begin{eqnarray*} [(111)_b]^2 =(12321)_b \\ [(11)_b]^4= (14641)_b \\ [(1111)_b]^2=(1234321)_b. \end{eqnarray*}

Answer 2

Hint: For example with part 1, $(12321)_b=b^4+2b^3+3b^2+2b+1$. You could then try to find a polynomial (of degree $2$) that squares to this value.

Answer 3

Just to help you started out with, here's how you prove 1.

$(12321)_b = b^4+2b^3+3b^2+2b+1 =(b^2+b+1)^2 $

And hence the given number in base $b \geq 4$ is a square.The rest are similar to the one I just did, you shouldn't have any problems.