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
Using Fermat's Little Theorem to solve a Diophantine equation.
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number-theory
elementary-number-theory
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3You might find a solution [here](https://docs.google.com/viewer?a=v&q=cache:XzxJYDz5GeIJ:www.math.uconn.edu/~kconrad/blurbs/gradnumthy/mordelleqn1.pdf+&hl=en&gl=uk&pid=bl&srcid=ADGEEShouNA-L01H_YNGLXaxOQfZ1TpY8LcXKrH7nRaKXqjU90rRnaNQhiH-8qnm5o8T6G_Jl2KblhYZ_Jx4OdtQ2Gpv28aqbq3sOLT2AQz9I74nnGIf1KOOePYHUvXeNDMsq1q6li7l&sig=AHIEtbQWaYmp-5s7ZGYZKll7CzP_BeB71Q) on page 2 - theorem 2.2 – 2012-10-29
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2@OldJohn: your link doesn't seem to work for me (Google docs login...) but I think that it's Conrad's paper available [here](http://ohkawa.cc.it-hiroshima.ac.jp/AoPS.pdf/Examples%20of%20Mordell's%20Equation%20-%20Keith%20Conrad.pdf) too. – 2012-10-29
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0Yes - that is the paper. Never really understood the Google docs system for links, I'm afraid. – 2012-10-29