# Diofantine equations of the sum of cubes equal to square

Does anyone know how to obtain infinite solutions of the following diofantine equation $X^2=DY^3+K^3$ all numbers non zero natural.

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Your question is as unclear as ever. Do you want to find solutions for any fixed $D$ and $K$, or are those also variables (in which case the question is trivial)? In what domain are you looking for solutions? –  Alex B. Dec 19 '11 at 5:47
@Alex: D is variable K is fixed all numbers natural. –  Vassilis Parassidis Dec 19 '11 at 6:56
@Vassili: If you are simply asking for infinitely many solutions with $D$ variable, that is too easy. Pick $K=1$, $X$ anything bigger than $1$, $Y=1$, $D=X^2-1$. One can also arrange for $Y$ arbitrarily large. I would have expected $D$ fixed. –  André Nicolas Dec 19 '11 at 7:50
@AndréNicolas $K$ begin fixed means you can't just "pick $K=1$". It is god-given to you. Vassili, you are looking for integer solutions in a family of quadratic twists of a given elliptic curve. While for a given elliptic curve, there are only finitely many integer solutions, and they can be found algorithmically, I am pretty sure that to find infinitely many solutions in a family of quadratic twists is outside current number theoretic technology. –  Alex B. Dec 19 '11 at 13:28
@Alex B.: I thought that the OP wanted a family of examples. O.K., god has provided a $K$. Pick any $X^2>K^3$, pick $Y=1$, $D=X^2-K^3$. –  André Nicolas Dec 19 '11 at 14:47