Mar 252014
Which integers can be expressed as \(a^3+b^3+c^3-3abc\)? \(a\), \(b\), \(c\in\Bbb Z\).
\[(a\pm1)^3+a^3+a^3-3(a\pm1)a^2=3a\pm1\]
\[(a-1)^3+a^3+(a+1)^3-3a(a+1)(a-1)=9a\]
\[2(a^3+b^3+c^3-3abc)=3(a+b+c)(a^2+b^2+c^2)-(a+b+c)^3\]
If \(3\mid(a^3+b^3+c^3-3abc)\), then \(3\mid(a+b+c)^3\), \(3\mid(a+b+c)\). so \(9\mid(a^3+b^3+c^3-3abc)\).
All \(n\) such that \(3\nmid n\) or \(9\mid n\).