90. Amazing Square. In which numeral system does the number 11111 represent a perfect square?
90. If the base of the number system is $B$, and $B>1$, then the number 11111 can be written as $B^{4}+B^{3}+B^{2}+B+1$. Further,

$$
\left(B^{2}+\frac{B}{2}\right)<B^{4}+B^{3}+B^{2}+B+1<\left(B^{2}+\frac{B}{2}+1\right)^{2}
$$

and if the middle term is a perfect square, then the equality must hold

$$
\left(B^{2}+\frac{B}{2}+\frac{1}{2}\right)^{2}=B^{4}+B^{3}+B^{2}+B+1
$$

from which

$$
\begin{array}{r}
\frac{B^{2}}{4}-\frac{B}{2}-\frac{3}{4}=0 \\
(B-3)(B+1)=0
\end{array}
$$

and for $B=3$ we get $11111=(102)^{2}$.

[V. Thébault A. M. M., 15, 149 (December 1940).]