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Does anyone know for which values of $D$ the equation $X^3=DY^3+A^3$ has solutions? All numbers non-zero naturals.

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The answer is that nobody knows! An answer to your question is equivalent to asking what numbers $D$ can be expressed as the sum of two rational cubes $u^3+v^3=D$, where $u=X/Y$ and $v=-A/Y$, but the classification of such numbers $D$ is not known. (If one believes the Birch and Swinnerton-Dyer conjecture, then there is an analytic method to test that will check whether $D$ is a sum of rational cubes, though.)

This article by J. H. Silverman is a great introduction to the subject. I can't recommend it highly enough.

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    If I may add, cube-free $D$ such that $u^3+v^3=D$ has rational solutions is given by OEIS [A020898](https://oeis.org/A020898) as $D=2, 6, 7, 9, 12, 13, 15,\dots$2017-08-13
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For the equation:

$X^3=DZ^3+Y^3$

If the number of $D$ can be represented in this form $D=\frac{3p^2+1}{4}$

Then primitive solution can be written:

$X=p^2+p$

$Y=p^2-p$

$Z=2p$

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    Actually, since $p$ must be an odd integer, then the smallest integer solution is $X = \frac{p+1}{2}$ $Y = \frac{p-1}{2}$ $Z=1$2015-03-17