Slim exceptional set for sums of mixed powers of primes

Abstract

Let N be a sufficiently large integer. In this paper, it is proved that, with at most O(N 4/27+ε ) exceptions, all even positive integers up to N can be represented in the form p 2 1 + p 2 2 + p 3 3 + p 3 4 + p 6 5 + p 6 6 , where p 1 , p 2 , p 3 , p 4 , p 5 , p 6 are prime numbers. This gives a large improvement of a recent result O(N 127/288+ε ) due to Liu (Proc. Indian Acad. Sci. (Math. Sci.) 130(1) (2020) Article ID. 8).