Waring’s problem for cubes

Abstract

In his book Meditationes Algebraicae, published in 1770, Edward Waring stated without proof that every nonnegative integer is the sum of four squares, nine cubes, 19 fourth powers, and so on. Waring’s problem is to prove that, for every k ≥ 2, the set of nonnegative kth powers is a basis of finite order.

Omnis integer numerus vel est cubus; vel e duobus, tribus, 4,5,6,7,8, vel novem cubus compositus: est etiam quadratoquadratus; vel e duobus, tribus &c. usque ad novemdecim compositus &sic deinceps.¹

E. Waring [138]