Triples of sixth powers with equal sums

Abstract

The diophantine equation x 6 + y 6 + z 6 = u 6 + v 6 + w 6 {x^6} + {y^6} + {z^6} = {u^6} + {v^6} + {w^6} is shown to have a two-parameter solution which is homogeneous of degree four. The solution also satisfies x 2 + y 2 + z 2 = u 2 + v 2 + w 2 {x^2} + {y^2} + {z^2} = {u^2} + {v^2} + {w^2} ; and in addition, 3 x + y + z = 3 u + v + w 3x + y + z = 3u + v + w .