# integer solutions of an equations

Now I came with an equation to find the solutions in integers. Not aonly that, I would like to know other types of solutions (if exists). Find the solutions and method of solving the equation $p^3 - 2pqr = q^3 + r^3$. Where the $p, q, r$ may be integers.

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either two are odd and one is even, or all three are even – user31280 Nov 3 '12 at 5:31
Why do you want to solve this Diophantine equation? – Jonah Sinick Nov 3 '12 at 5:36
@F'OlaYinka! there is no restriction on even and odd. I am looking integral solutions and the method of solving such equations. Kindly help me in this regard. – vmrfdu123456 Nov 3 '12 at 5:38
@vmr, F'Ola is saying every solution satisfies those restrictions. Can't you see why there can't possibly be a solution with all three odd? – Gerry Myerson Nov 3 '12 at 5:59
@GerryMyerson!I know it. I said, how you can determine those solutions in numerical. The solutions may be odd, does not matter. How to solve such equation to list all the solutions (may be odd). – vmrfdu123456 Nov 3 '12 at 6:29

This defines a cubic curve with a rational point and hence an elliptic curve. It is birational to Cremona's 19A1 which has rank $0$ and precisely $3$ rational points. Tracing these back to the original equation leaves one with precisely the trivial solutions, $$(p,q,r)=(k,0,k), (k,k,0) \mbox{ and } (0,k,-k)$$ for $k \in \mathbb{Z}$.