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.

From Integer as Difference between Two Squares:


 * $R_n$ has at least two distinct divisors of the same parity that multiply to $R_n$.

Then from Difference of Two Squares:
 * $x = \dfrac {a + b} 2$
 * $y = \dfrac {a - b} 2$

where:
 * $R_n = a b$

for all $a, b$ where:
 * $a b = R_n$
 * $a$ and $b$ are of the same parity.

Here we have that $R_n$ is odd.

So both $a$ and $b$ are always odd and therefore always of the same parity.

It remains to perform the calculations and evaluate the examples.