Question: Let $X$ be a (possibly T1, that is, singleton subsets are closed) topological space. Suppose that there is a *proper* subset $D \subset X$ such that:

* $D$ is dense in $X$;
* $D$ is homeomorphic to $X$.

What can be said about $X$?

Answer: 
$$32(x + 1) = y^2 \implies 32 \mid y^2 \implies 4 \mid y$$ so let $y = 4z$. $$2(x+1) = z^2 \implies 2 \mid z$$ so let $z = 2w$ $$x + 1 = 2w^2 $$ so given any $w$ we can solve for $x$.

So the set of solutions is
$$\{(2w^2 - 1, 8w) \mid w \in \Bbb Z\}$$
Solution for x such that 32x+32 is a square as follows:
 For integral solution, x= (n2 -2)/2 such that x is in Natural number.(n is in natural number)