Difference between Two Squares equal to Repunit

Theorem
The sequence of differences of two squares that each make a repunit begins:

Proof
Let $x^2 - y^2 = R_n$ for some $n$, where $R_n$ denotes the $n$-digit repunit.

Then from Difference of Two Squares:
 * $\left({x + y}\right) \left({x - y}\right) = R_n$.

Let $R_n = a b$ for some $a, b \in \Z$.

Then:
 * $x + y = a$
 * $x - y = b$

which leads to:
 * $x = \dfrac {a + b} 2$
 * $y = \dfrac {a - b} 2$

Thus all instances of $x^2 - y^2 = R_n$ can be achieved by:
 * extracting the divisors of each repunit for all $n$
 * calculating the mean $x$ of each pair of those divisors whose product is $R_n$
 * calculating the midpoint $y$ between that same pair
 * returning $x^2 - y^2$.

It remains to perform the calculations and evaluate the examples.