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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

1 Answer 1

Yes, I know how to obtain infinitely many solutions to that equations. I'll tell you how to do it, as soon as you get your accept rate up over 50%.

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This at least deserves a "Great Answer" badge. We deserve some respect. –  Patrick Da Silva Dec 19 '11 at 5:00
-1. I strongly disapprove of this pressure to accept answers the OP is not happy with. Already posting the same comment under every single one of his questions was over the top. Now posting "non-answers" with the same content is - hmm - well over the top. This not only doesn't deserve any badges, this should be flagged as not an answer, which I am now doing. If you fear that your effort will not be duly appreciated or rewarded, then simply don't answer. –  Alex B. Dec 19 '11 at 5:40
@Alex B.: please don't do that (flag as "not an answer"), unless you think that answer contributes content but only should be a comment. From your comment I suspect you don't actually think this answer should be a comment at all. –  Willie Wong Dec 19 '11 at 10:23
@Gerry: is posting this as an answer really necessary? –  Willie Wong Dec 19 '11 at 10:24
The question was, "Does anyone know how..." "Yes," all by itself, is a precise and truthful answer to that question. I even went beyond that by naming someone who knows how. How is this not an answer to the question? As to whether posting my answer as an answer was really necessary, that is setting the bar very high; I have yet to come across any answer on this website whose posting was absolutely necessary. But perhaps I miss the point. Feel free to delete my answer, or move it elsewhere, or display it prominently in advertisements for the site, as you see fit. –  Gerry Myerson Dec 19 '11 at 21:17

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