The exceptional sets of a Waring-Goldbach problem with unequal powers

Abstract

The representability of positive odd number n=p1+P32+p3k(k∈N and k≥4) was studied. The circle method in additive number theory and the iterative method in the circle method were used to deal with the main arcs and the exponential sum method was used to deal with the minor arcs. The final conclusions were reached: Let N be a sufficiently large integer, then for any ε>0, it is proved that with at most O(N 31/60+2/k-2/3k×2k-1+ε) exceptions, all odd positive integers up to N can be represented in the form p1+p32+pk3, where p1, p2, p3 are prime numbers.

Keywords

• circle method
• the singular series
• exceptional set
• main arcs
• minor arcs