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
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Sources
- 1995: Erik Dofs: Solutions of $x^3 + y^3 + z^3 = n x y z$ (Acta Arith. Vol. 73: pp. 201 – 213)
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $6$