1. Exceptional sets for sums of almost equal prime cubes
- Author
-
Mengdi Wang
- Subjects
Combinatorics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Waring–Goldbach problem ,010103 numerical & computational mathematics ,0101 mathematics ,01 natural sciences ,Prime (order theory) ,Mathematics - Abstract
In this paper, we continue to investigate the exceptional sets for sums of five and six almost equal cubes of primes. We would also like to establish that almost all natural numbers n, subjected to certain congruence conditions, can be written as n = p 1 3 + ⋯ + p s 3 {n=p_{1}^{3}+\cdots+p_{s}^{3}} ( s = 5 , 6 {s=5,6} ) with | p j - ( n / s ) 1 / 3 | ≤ n θ s / 3 + ε {|p_{j}-(n/s)^{1/3}|\leq n^{\theta_{s}/3+\varepsilon}} ( 1 ≤ j ≤ s {1\leq j\leq s} ), where θ s {\theta_{s}} is as small as possible. The main result of this paper is to improve θ 6 = 5 / 6 + ε {\theta_{6}=5/6+\varepsilon} , which is proven in [M. Wang, Exceptional sets for sums of five and six almost equal prime cubes, Acta Math. Hungar. 156 2018, 2, 424–434], to θ 6 = 9 / 11 + ε {\theta_{6}=9/11+\varepsilon} , as well as prove θ 5 = 8 / 9 + ε {\theta_{5}=8/9+\varepsilon} in another way.
- Published
- 2019