<s>[INST] Not a perfect square of the form for any integer x. Now a days, I become good fan of this site, as this site making me to learn more math..hahaha.
Okay! 
Can we prove that $x^3 + 7$ cannot be perfect square for any positive/negative or odd/even integer of $x$.
I checked with number up to x = 1,...1000. I realized that, not a perfect. But, how do we prove the statement without further checking in place of 1001, ...
 [/INST] Rewrite our Diophantine equation as $y^2+1=x^3+8$.  Note that $x$ cannot be even, for $y^2+1$ cannot be divisible by $4$. 
So $x$ must be odd. We have
$$y^2+1=x^3+8=(x+2)(x^2-2x+4).$$
Suppose $x\equiv 1\pmod{4}$. Then $x+2\equiv 3\pmod{4}$, which is impossible, since a positive number of the form $4k+3$ cannot divide $y^2+1$.
Suppose $x\equiv 3\pmod{4}$. Then $x^2-2x+4\equiv 1-2(3)\equiv 3\pmod{4}$, again impossible.  
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