I thought I had a reasonable understanding of Diophatine Equations (or atleast I thought I was fairly competent in solving them) until I saw this example that had been worked through in my lecture notes. I would normally email my lecturer if I had a problem though this one isn't particularly good at replying! Any help would be greatly appreciated as I have an exam on the topic looming.
Anyway, the problem is as follows: find the integer solutions to the equation X^3 = Y^2 + 5.
The solution first noted that Y^2 + 5 = (Y + (-5)^(1/2))(Y - (-5)^(1/2)) and so the highest common factor of (Y + (-5)^(1/2)), (Y - (-5)^(1/2)) will devide the difference 2(-5)^(1/2). Now from my understanding of the topic this would give rise to us considering the behaviour of the prime 2 in Q[(-5)^(1/2)] however, the solution considered also the prime 5. Why is this so?
Thanks for any help...