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

From ProofWiki
Jump to navigation Jump to search

Source Work

1997: David Wells: Curious and Interesting Numbers (2nd ed.):

The Dictionary
$6$


Mistake

The equation $x^3 + y^3 + z^3 = 6 x y z$ has the unique solution $x = 1$, $y = 2$, $z = 3$.


Correction

The correct answer is as given in Solutions to $x^3 + y^3 + z^3 = 6 x y z$

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 $n \in \Z$


Sources