2
$\begingroup$

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.

  • 0
    A maple search didn't find any solutions where $p,q,r$ are between $-100$ and $100$ (except for the trivial solutions where $pqr=0$ and the nonzero have same magnitude).2013-01-23

1 Answers 1

3

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 $k \in \mathbb{Z}$.