<s>[INST] Integer solutions for $x^3=y^2+5$ I have been able to solve the equation $x^3=y^2+a$ for integers where $1 \leq a \leq 4$ by splitting in $\Bbb Z [i\sqrt a]$. However as a natural continuation I would like to know whether the equation $x^3=y^2+5$ can be solved using elementary methods. Seems this case is much tougher though .I would like some hints on how to proceed.
 [/INST] Working mod 4 we see that $y$ must be even and $x=1\bmod4$. Now we have $$y^2+4=x^3-1=(x-1)(x^2+x+1)$$ But $x^2+x+1=3\bmod4$, so $x^2+x+1$ is odd. It is also positive, so it must be at least 3. Hence it must have a prime factor $p=3\bmod4$. So $y^2+4=0\bmod p$, in other words -4 is a quadratic residue of $p$. But that implies that $p=1\bmod 4$. Contradiction.
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