<s>[INST] Show that $x^3-x^2+8=y^2$ has no integer solution 
Show that $x^3-x^2+8=y^2$ has no integer solution.

I spent several hours on this problem but I couldn't figure out how to solve it. A hint from the professor was to find a proper field where to find the solutions, or use Gaussian integers, but I still can't find the proof. I only showed that $y$ must be even, and I tried to factorize $(x^3+8)$ and equalize the two members with $x^2+y^2$, but nothing came out of it.
This problem was given at the Competencia Interuniversitaria Matemática Argentina (CIMA). 
 [/INST] Here is an official solution: https://drive.google.com/file/d/1qOPYHK3p_S8tlwlewQ91X6JbDrZ-oKrc/view. Following is an English version (not literal translation, also little reordered):
If $x$ is even, say $x=2a$, we have $y^2=x^3-x^2+8=8a^3-4a^2+8$, implying $2 \mid y^2$, and so $2 \mid y$. Let $y=2b$, giving $8a^3-4a^2+8=4b^2$, and after simplification $2a^3-a^2+2=b^2$. Now if $a$ is even, then we have $b^2 \equiv 2 \pmod 4$, and if $a$ is odd, then $b^2 \equiv 3 \pmod 4$, either way impossible.
If $x \equiv 3 \pmod{4}$, then left side $x^3-x^2-8\equiv 2 \pmod{4}$. This implies $y^2 \equiv 2 \pmod {4}$, impossible.
For final case $x \equiv 1 \pmod {4}$, consider
$$(x+2)(x^2-2x+4)=x^2+y^2.$$
Since we have $x+2 \equiv 3 \pmod 4$, there must be prime $p \equiv 3 \pmod {4}$ such that $p \mid x+2$. From equation above $p \mid x^2+y^2$, or $x^2 \equiv -y^2 \pmod {p}$. So $p \mid x$ iff $p \mid y$. We also have
$$
x^{p-1} \equiv (x^2)^{\frac{p-1}{2}} \equiv (-y^2)^{\frac{p-1}{2}}\equiv (-1)^{\frac{p-1}{2}}y^{p-1} \pmod{p}.
$$
Now clearly $p \nmid x$ (otherwise we would have $p=2$, a contradiction). So we have $p\nmid x, p \nmid y$. From Fermat's Little Theorem we have $x^{p-1}\equiv y^{p-1} \equiv 1 \pmod p$, and so from above equation $(-1)^{\frac{p-1}{2}}\equiv 1 \pmod{p}$. This implies $x\equiv 3 \pmod{p}$, a contradiction.
So there is no integer solution to the equation.
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