Square of 1 Less than Number Base

Theorem

Let $b \in \Z$ be an integer such that $b > 2$.

Let $n = b - 1$.

The square of $n$ is expressed in base $b$ as:

$n^2 = \left[{c1}\right]_b$

where $c = b - 2$.

Proof

\(\ds n^2\)

\(=\)

\(\ds \left({b - 1}\right)^2\)

\(\ds \)

\(=\)

\(\ds b^2 - 2 b + 1\)

\(\ds \)

\(=\)

\(\ds b \left({b - 2}\right) + 1\)

The result follows by definition of number base.

$\blacksquare$

Examples

Square of $5$ in Base $6$

The square of $5$ is expressed in base $6$ as:

$5^2 = \left[{41}\right]_6$