Differences of squares

$(a + 1)^2 - (b + 1)^2$

$= a^2 + 2a + 1 - (b^2 + 2b + 1)$

$= a^2 - b^2 + 2(a - b) = 2193$

$a^2 - b^2 = 2159 \Longrightarrow 2159 + 2(a - b) = 2193$

$2(a - b) = 34$

$a - b = 17$

We can factor $a^2 - b^2$ as $(a+b)(a-b)$ and similarly, $(a+1)^2-(b+1)^2 = ((a+1)+(b+1))((a+1)-(b+1))$ , or, combining constants, $(a+b+2)(a-b)$ . We then have $(a+b)(a-b) = 2159$ and $(a+b+2)(a-b) = 2193$ . Subtracting the first equation from the second, $2(a-b) = 34$ , so $a-b = 17$ .

Expanding the second equation, we get

$(a+1)^2-(b+1)^2=2193$

$a^2+2a+1-(b^2+2b+1)=2193$

$a^2+2a+1-b^2-2b-1=2193$

$a^2-b^2+2a-2b+1-1=2193$

$a^2-b^2+2a-2b=2193$

We know that the first equation is $a^2-b^2=2159$ , now we substitute

$2159+2a-2b=2193$

$2a-2b=34$

$2(a-b)=34$

$a-b=17$