1. Base Numbers

The number of possible bases $\displaystyle b$ for which $\displaystyle 12321(base b)$ is a perfect square is:
a] 1
b] 2
c] 3
d] 4
e] more than 4

2. Hello, foreverbrokenpromises!

I found the answer by experimenting . . .

The number of possible bases $\displaystyle b$ for which $\displaystyle 12321_b$ is a perfect square is:

. . $\displaystyle (a)\;1 \qquad (b)\;2 \qquad (c)\;3 \qquad (d)\;4 \qquad (e)\text{ more than 4}$

The number $\displaystyle 12321_b$ is equal to: .$\displaystyle b^4 + 2b^3 + 3b^2 + 2b + 1$

. . which is already a perfect square! . $\displaystyle (b^2 + b + 1)^2$

Since the number uses {1, 2, 3}, the base must be at least 4.

Therefore, $\displaystyle 12321_b$ is a perfect square for any base $\displaystyle b \ge 4$ . . . answer (e)

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Examples:

. . $\displaystyle \begin{array}{cccccc}12321_4 &=& 441 &=& 21^2 \\ 12321_5 &=& 961 &=& 31^2 \\ 12321_6 &=& 1849 &=& 43^2 \\ 12321_9 &=& 8281 &=& 91^2 \\ 12321_{12} &=& 24,\!649 &=& 157^2 \end{array}$