On Waring’s problem for larger powers

Abstract

Abstract Let G ⁢ ( k ) {G(k)} denote the least number s having the property that every sufficiently large natural number is the sum of at most s positive integral k-th powers. Then for all k ∈ ℕ {k\in\mathbb{N}} , one has G ⁢ ( k ) ⩽ ⌈ k ⁢ ( log ⁡ k + 4.20032 ) ⌉ . G(k)\leqslant\lceil k(\log k+4.20032)\rceil. Our new methods improve on all bounds available hitherto when k ⩾ 14 {k\geqslant 14} .