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.
2026-04-30 10:21:23.1777544483
integer solutions of an equations
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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 $k \in \mathbb{Z}$.