Repunit cannot be Square

Theorem
A repunit (apart from the trivial $1$) cannot be a square.

Proof
Let $m$ be a repunit with $r$ digits such that $r > 1$.

By definition, $m$ is odd.

Thus from Square Modulo 4, if $m$ were square it would be of the form:
 * $m \equiv 1 \pmod 4$.

$m$ is of the form $\displaystyle \sum_{k \mathop = 0}^{r - 1} 10^k$ where $r$ is the number of digits.

Thus for $r \ge 2$:

Hence:
 * $m \equiv 3 \pmod 4$

and so cannot be square.