On $2xy+x+y+1 = 2ab+a+b$

Question

Let us suppose that $2xy+x+y = n$, where $x,y,n$ are positive integers and $x≥2$ and $y≥2$.

Let us choose x and y, so that $n+1 = 2ab+a+b$ , where $a, b$ are also positive integer numbers.

I was wondering what we can say about the relation between $x,y$ and $a,b$. As $n$ and $n+1$ are very close, and as $xy > x+y$, there must be a maximal difference between $x$ and $a$, also between $y$ and $b$.

Did anyone read a paper about it, or have any suggestion on how much this difference can be?

Answer 1

If you multiply by $2$ and add $1$ you get:$$(2x+1)(2y+1)+2=(2a+1)(2b+1)$$ so you are looking at the factorisation of successive odd numbers with the factors on the left both at least $5$ and the factors on the right both at least $3$.

If we say $x=y=2$ we get $27$ and to get the maximum difference (in fact this is the only option here) we choose $a=1, b=4$ corresponding to $3\times 9$.

This should be enough clue to allow you to explore further.

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There are other cases to consider and you might want $x$ and $y$ to be different. But $x=y=3m$ and $a=1, b=6m^2+2m$ are enough to show that for $x=y$ the differences can be arbitrarily large.