Solutions to x^3 + y^3 + z^3 = 6xyz

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Theorem

The Diophantine equation:

$x^3 + y^3 + z^3 = 6 x y z$

has exactly two kinds of solutions in the integers:

\(\ds \set {x, y, z}\) \(=\) \(\ds \set {n, 2 n, 3 n}\) for some $n \in \Z$
\(\ds \set {x, y, z}\) \(=\) \(\ds \set {0, n, -n}\) for some $n \in \Z$


Proof



Sources

  • 1995: Erik DofsSolutions of $x^3 + y^3 + z^3 = n x y z$ (Acta Arith. Vol. 73: pp. 201 – 213)