BRUDNO’S FAMILIES OF EQUAL SUMS OF SIXTH POWERS AND BREMNER’S K3 SURFACE
Brudno (1970, 1976) and Brudno–Kaplansky (1974) found an elliptic family and a degree-four polynomial family of solutions of x^6 + y^6 + z^6 = u^6 + v^6 + w^6. We explain both in terms of the K3 surface V ⊂P5, the complete intersection of three quadrics studied by Bremner (1981) and Kuwata (2007). The Brudno–Kaplansky curve is Cremona’s curve 480f4 and is the fibre at λ=∞ of the elliptic pencil on V that Bremner calls the Brudno pencil; the curve z^2 = x^4 + 38x^2y^2 + y^4 of 1970 is 480f1, and the classical recursions of Lebesgue and Desboves that Brudno iterated are the multiplication maps…
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