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Suppose we define $A= (8+\sqrt{x})^{1/3} + (8-\sqrt{x})^{1/3}$. How can we find, algebraically, all values of x for which $A$ is an integer?

I was not able this problem save for with Mathematica. How can we solve this using the tool of our brains?

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    $A^3 = 16+3A \sqrt[3]{64-x}$, $A^3 = 16+3A y$.2012-11-10
  • 0
    for each $A \ne 0$ we have $y=\frac{A^3-16}{3A}$, and $x=64-y^3$.2012-11-10

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