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

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 is likely to be a maximal difference between $xy$ and $ab$.

Am I right to suppose that to fulfil the equations, $xy = ab$ or $xy + 1 = ab$ ? So this maximum difference is 1 ?

Answer 1

apparently not

 a: 4 b: 1 x: 2 y: 2 |a*b - x*y| : 0
 a: 5 b: 3 x: 7 y: 2 |a*b - x*y| : 1
 a: 5 b: 5 x: 8 y: 3 |a*b - x*y| : 1
 a: 6 b: 2 x: 4 y: 3 |a*b - x*y| : 0
 a: 6 b: 4 x: 11 y: 2 |a*b - x*y| : 2
 a: 7 b: 4 x: 9 y: 3 |a*b - x*y| : 1
 a: 8 b: 1 x: 3 y: 3 |a*b - x*y| : 1
 a: 8 b: 3 x: 6 y: 4 |a*b - x*y| : 0
 a: 8 b: 5 x: 18 y: 2 |a*b - x*y| : 4
 a: 8 b: 7 x: 11 y: 5 |a*b - x*y| : 1
 a: 8 b: 8 x: 20 y: 3 |a*b - x*y| : 4
 a: 9 b: 1 x: 5 y: 2 |a*b - x*y| : 1
 a: 9 b: 4 x: 6 y: 6 |a*b - x*y| : 0
 a: 9 b: 5 x: 11 y: 4 |a*b - x*y| : 1
 a: 9 b: 6 x: 24 y: 2 |a*b - x*y| : 6
 a: 9 b: 6 x: 17 y: 3 |a*b - x*y| : 3
 a: 10 b: 3 x: 14 y: 2 |a*b - x*y| : 2
 a: 10 b: 4 x: 8 y: 5 |a*b - x*y| : 0
 a: 10 b: 8 x: 35 y: 2 |a*b - x*y| : 10

Answer 2

Let for instance $a=1$ so we get that $3 b =n$ , and assume that $x =y$ so we get that $2 x^2 +2 x =n = 3b $ which leads to $x = \frac{-2+\sqrt{4+24b}}{4}$,

so for instance : for $b=28$ gives that $x = y = 6$ and $a=1$, so $| x y -a b| = 8$ , and you can get the maximum difference as large as you want.

Answer 3

Not bounded at all.

With $a \equiv 3,4 \pmod 5,$ we can take $b = a-3,$ then $y=2$ and $$ x = \frac{2(a-3)(a+1)}{5} $$

With $a \equiv 0,1 \pmod 5,$ we can take $b = a-2,$ then $y=2$ and $$ x = \frac{2a^2 -2a-5}{5} $$

 a: 275 b: 273 x: 30139 y: 2 |a*b - x*y| : 14797
 a: 276 b: 274 x: 30359 y: 2 |a*b - x*y| : 14906
 a: 278 b: 275 x: 30690 y: 2 |a*b - x*y| : 15070
 a: 279 b: 276 x: 30912 y: 2 |a*b - x*y| : 15180
 a: 280 b: 278 x: 31247 y: 2 |a*b - x*y| : 15346
 a: 281 b: 279 x: 31471 y: 2 |a*b - x*y| : 15457
 a: 283 b: 280 x: 31808 y: 2 |a*b - x*y| : 15624
 a: 284 b: 281 x: 32034 y: 2 |a*b - x*y| : 15736
 a: 285 b: 283 x: 32375 y: 2 |a*b - x*y| : 15905
 a: 286 b: 284 x: 32603 y: 2 |a*b - x*y| : 16018
 a: 288 b: 285 x: 32946 y: 2 |a*b - x*y| : 16188
 a: 289 b: 286 x: 33176 y: 2 |a*b - x*y| : 16302
 a: 290 b: 288 x: 33523 y: 2 |a*b - x*y| : 16474
 a: 291 b: 289 x: 33755 y: 2 |a*b - x*y| : 16589
 a: 293 b: 290 x: 34104 y: 2 |a*b - x*y| : 16762
 a: 294 b: 291 x: 34338 y: 2 |a*b - x*y| : 16878
 a: 295 b: 293 x: 34691 y: 2 |a*b - x*y| : 17053
 a: 296 b: 294 x: 34927 y: 2 |a*b - x*y| : 17170
 a: 298 b: 295 x: 35282 y: 2 |a*b - x*y| : 17346
 a: 299 b: 296 x: 35520 y: 2 |a*b - x*y| : 17464
 a: 300 b: 298 x: 35879 y: 2 |a*b - x*y| : 17642