the number of integer solutions to $y^p = x^2 +4$

Question

Let $p>2$ be prime, investigate the number of integer solutions to $$y^p = x^2 +4$$. The first part of the question was find solutions to the equation $y^3 = x^2 +4$, I could do this and I see the same method works for this one ie factorize in $Z[i]$, show that the factors are coprime and so they're associate to a cube or in this case a pth power. But I don't know how to find out how many solutions the resulting simultaneous equations have. Any ideas?

Answer 1

As you noted, you can prove that $x$ is odd and hence $\gcd(x+2i,x-2i) = 1$, and now you get $4i = a^p-b^p = (a-b) \sum_{k=0}^{p-1} \binom{p-1}{k} a^k b^{p-1-k}$ for some gaussian integers $a,b$. Immediately we get $|a-b| \in \{1,2,4\}$. I think it should now be possible to slowly eliminate the cases one by one for sufficiently large $|a|$.