Determine all integers x,y that satisfy the equation $x^3 = y^2 +2 $
My attempt to the solution
Taking mod 2 we get that x an y can't be even. How to proceed? By trial and error there is a solution $y =\pm5$ and $x =3$.
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Hope to wake up to somebody having written enough that I don't need to. The standard proof uses the fact that $\mathbb{Z}[\sqrt{-2}]$ is a Euclidean domain. Then one factors $x^2+2$ as $(x+\sqrt{-2})(x+\sqrt{-2})$, and does some calculations. There must be a proof that mentions only ordinary integers, but I do not recall seeing one. – André Nicolas Jul 06 '13 at 07:16
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1$x^3-3^3=y^2-5^2$ – lab bhattacharjee Jul 06 '13 at 07:31
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its different .. – maths lover Jul 06 '13 at 10:00