# 121 is Square Number in All Bases greater than 2

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## Theorem

Let $b \in \Z$ be an integer such that $b \ge 3$.

Let $n$ be a positive integer which can be expressed in base $b$ as $121_b$.

Then $n$ is a square number.

## Proof

Consider $11_b$.

By the Basis Representation Theorem:

- $11_b = b + 1$

Thus:

\(\displaystyle {11_b}^2\) | \(=\) | \(\displaystyle \paren {b + 1}^2\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle b^2 + 2 b + 1\) | Square of Sum | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 121_b\) |

Thus:

- $121_b = {11_b}^2$

and so is a square number.

$\blacksquare$

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $121$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $121$