{"id":"hexagon/2610.00191","pdf":"https://hexagonmath.org/pdf/2610.00191v1","url":"https://hexagonmath.org/abs/2610.00191v1","tags":["math.AG","msc:11D72","Diophantine equations","Elliptic Curves","Sextic"],"score":0,"title":"BRUDNO’S FAMILIES OF EQUAL SUMS OF SIXTH POWERS AND BREMNER’S K3 SURFACE","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Alon Amit"],"reviews":0,"subject":"math.NT","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Fable"],"discussed":0,"published":"2026-10-11","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}