Difference between Two Squares equal to Repunit/Mistake

Source Work

 * The Dictionary
 * $111$
 * $111$

Mistake

 * $111 = 20^2 - 17^2$, the third difference of $2$ squares equal to a repunit. The sequence of such squares starts $1, 0$; $6, 5$; $20, 17$; $56, 45$; $156, 115$; $344, 85$; $356, 125 \ldots$

The instances:

have been omitted.

This omission does not originate from -- but it does not appear (for reasons to be explained) in the list generated by  and.

It is necessary to point out that the work by and  started as an extension of the question:
 * Find all two- and three-digit squares such that when one is subtracted from each digit, the resulting number is a square.

This was apparently a grade-school exercise set by

and expanded the question to extend to $n$-digit squares. The question they formulated was:
 * For each $n$, find all positive integers $x$ and $y$ such that $x^2 - y^2 = 11 \cdots 1$ ($n$ ones) and $x^2$ contains no zero digits.

As such, this question is a flawed implementation of the problem in question, for the instance:
 * $56^2 - 55^2 = 111$

while it satisfies this condition, it is not the case that by subtracting $1$ from each digit of $56^2$ you get $55^2$, but you do get a repunit by subtracting $55^2$ from $56^2$.

So, on analysis of the paper by and, we find that while they have indeed found:
 * all the squares such that when one is subtracted from each digit, the resulting number is another square

they have not found:
 * all positive integers $x$ and $y$ such that $x^2 - y^2 = 11 \cdots 1$ ($n$ ones) and $x^2$ contains no zero digits.

In particular, they have glossed over those squares of the form $56^2 - 55^2 = 111$:

and the like, as well as specifically considering and excluding those of the form:

, however, includes $20^2 - 17^2 = 400 - 289 = 111$ in his list, suggesting that he is not deliberately excluding such where $x^2$ has a zero digit.

Consequently it appears that the incompleteness of his list is the result ofnot having completely worked through the content of the and  paper.