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How to prove that the equation $x^2+5=y^3$ has no integer solutions? I have proved the case when $x$ is odd. I used the fact $x^2\equiv 1 \pmod 4$ but how would you do for even $x$: the mod 4 analysis becomes useless. The problem is from Fermat Little Theorem section. But I do not know how apply it. Thanks

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You might find a solution here on page 2 - theorem 2.2 – Old John Oct 29 '12 at 22:09
@OldJohn: your link doesn't seem to work for me (Google docs login...) but I think that it's Conrad's paper available here too. – Raymond Manzoni Oct 29 '12 at 22:14
Yes - that is the paper. Never really understood the Google docs system for links, I'm afraid. – Old John Oct 29 '12 at 22:17

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