Exceptional Sets In The Waring-Goldbach Problem

Waring-Goldbach problem asks whether every sufficiently large natural number n can be represented as (?) where p1,…,ps are prime numbers.We consider the exceptional sets in the WaringGoldbach problem in this paper.At first,we will give the exceptional sets in Waring-Goldbach problem forfifth powers.When s satisfies 12 ≤s ≤ 20,we can get the exceptional sets in the Waring-Goldbach problem for fifth powers.Let E5,s(N)be the number of n<N,which satisfies n≡s(mod 2)but cannot be represented as the s fifth powers of primes.The improvement comes from the application of the sieve method and one can refer to Kumchev[21]for such method.In order to obtain the estimates of E5,19(N)and E5.20(N),we apply the method of Kawada and Wooley[19]to establish a relation between E5,s(N)and E5,s-4(N).Let θ1=27/3200,we have (?)(?)(?)Our result can be compared with the previous results (?)(?)(?) where θ2=73/9600.Moreover,we can also consider the exceptional sets in short interval.We have two results.When Y>>N17/63,the number of n in the interval[N-18N2/3Y,N+18N2/3Y],which satisfies n≡0(mod 2),n(?)±1(mod 9),but cannot be represented as the s primes of cube where the primes satisfy |pi-N1/3|≤Y(1≤i≤s),is o(N2/3Y).Our result is smaller than the previous result Y>>N3/11.When M>N1/36+∈,we can also get that the number of n in the interval[NM,N],which satisfy n ≡ 0(mod 2)but cannot be represented as the 8 primes of cube,is o(M).