The Exceptional Set Of Representations Of Two Prime Squares And One κth Power Of Prime

The Waring-Goldbach problem is a classical problem in additive prime number theory. In 1938, Hua [6] proved that almost all integers satisfying necessary congruence conditions can be written as n=p12+p22+p3k (0.3) where k≥2. Let Ak denote the set of all positive integers n satisfying local congruence conditions and Ek(N) the number of positive integers n∈Ak, not exceeding N. that cannot be written as (0.3). Then Hua’s result actually states that Ek(N)<< NL-A (0.4) for some positive constant A. Later Schwarz [16] proved that Hua’s estimate holds for arbitrary A> 0. In 1993, Leung and Liu [13] consider the solvability of the following problem: n=α1p12+α2p22+α3p3k, where α1,α2,α3 are fixed positive integers which are relatively prime. They proved that there exists ε(k)>0 such that the cardinality of the corresponding exceptional set is O(N1-ε(k). In 2014, Lu and Tang [9] studied this problem with pi taken values in short intervals, and proved that Hua’s result is true in this condition.In this paper, based on the results of Hua, Leung and Liu, applying circle method, we consider the exceptional set of (0.3) with k≥3. The main result we obtain is the following: Ek(N)<<N1-δ(k)+ε where We will use circle method to prove the above result. Here we will apply the idea in Zhao [18] and the new estimate for exponential sums to treat the minor arcs.There are four sections in this paper. In the first section, we briefly introduce the research background and the main result. In the second section, we introduce the outline of the proof of Theorem 1.1. The last two sections will be devoted to the estimates of integrals on the major arcs and the minor arcs, respectively.