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: 
The answer by user3491648 gives a complete characterization of the full set of values $x$ for which $32x+32$ is a square numbers. This is really just an observation on the set of values 

$$\{64,1600,53824,1827904,62094400,2109381184\}$$

reported by the OP. 

These are not actually values that $x$ takes on. (As user3491648 finds, it requires on *odd* value of $x$ to make $32x+32$ a square.) Rather, these seem to be the square values that correspond to values of $x$ that are themselves squares, namely of the following values:

$$\{1,7,41,239,1393,8119\}$$

For example, $32\cdot239^2+32=1827904$. It's unclear why these values are being singled out, but they do seem to obey a two-term recursion, namely

$$a\_{n+1}=6a\_n-a\_{n-1}$$

For example, $1393=6\cdot239-41$.
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)