The Exceptional Sets For Waring-Goldbach Problem

Number theory is an important field of mathematics.Additive prime number theory is the intersection field of the distribution of prime numbers and Diophantine equation,and it was motivated by two seminal works:Vinogadov s remarkable proof of the three primes theorem and Hua’s work on nonlinear cases of the Waring-Goldbach problem.Waring-Goldbach problem is the core issue of additive prime number theory.The Waring-Goldbach problem is concerned with the solvability of representing positive integers which satisfy some appropriate congruence conditions as the sums of powers of prime numbers.The congruence condition needed here is to avoid the degenerate solutions.Let k be a positive integer,the Waxing-Goldbach problem of degree k considers the solvability of the equation n=p1k+p2k+…+psk,(2)where p1,P2,…,ps are prime numbers,and n satisfies some congruence conditions.It is the famous binary Goldbach Conjecture when k=1 and s=2,that is,every even number which is no less than 6 is the sum of two odd primes.This conjecture is still an open problem.When k=1 and s=3,this is the ternary Goldbach Conjecture which is implied by the binary Goldbach Conjecture,and it was already completely solved,that is,every odd number which is no less than 9 is the sum of three odd primes.For the nonlinear cases of Waring-Goldbach problem,mathematicians are inter-ested in the upper bound of H(k)which is the smallest s for which all sufficiently large positive integers n satisfying some congruence conditions can be written as the form(2).It is conjectured that H(k)=k+1 for all k≥ 1,while this conjecture has not been proved for any integer k.Lookeng Hua gave a number of remarkable results for H(k),he proved that H(k)≤2k+1 for k≥ 1.This result is still the best bound for k≤3.For k≥ 4,Hua’s result has been improved dramatically.Vinogradov,Dav-enport,Thanigasalam,Kawada,Kumchev,Wooley and many number theorists did remarkable work on H(k).One can further reduce the number of variables needed to solve(2)by trying to represent almost all n instead of all suffiiently large n,whence mathematicians are interested in exceptional sets.For large positive integer N,let Ek.s(N)be the number of integers satisfying some necessary congruence conditions not exceeding N for which(2)cannot be solved in primes p1,p2,…,ps.Mathematicians mainly consider the upper bound of Ek,s(N).In this paper,we discuss the relations between exceptional sets,and estimate exceptional sets of Waring-Goldbach problem and Waring-Goldbach problem for unlike powers.In Chapter one,we describe the history of Waring’s problem and Waring-Goldbach problem.In Chapter two,we consider the relations between exceptional sets following the idea of Kawada and Wooley.Given a set A and a set B which is not too sparse.Putting one or two elements of B into A,one can get a new set C.In this chapter,we get some relations between the exceptional set of A and the exceptional set of C,and give a new way of estimating the upper bound of exceptional set.In Chapter three,we estimate the exceptional set of Waring-Goldbach problem for fourth powers.Using the Hardy-Littlewood method,Zhao’s method and the relations obtained in chapter two to estimate E4,s(N).In fact,we obtain the best result E4,7(N) that the Zhao’s lemma can reach for Waring-Goldbach problem for fourth powers with 7 variables.For 9 to 12 variables,we get the corresponding result for Waring’s problem for fourth powers given by Kawada and Wooley.In Chapter four,we investigate the exceptional sets of Waring-Goldbach problem for fifth to tenth powers.When the number of variables is small,we use the Hardy-Littlewood method and the new estimate of exponent sum given by Kumchev and Wooley to estimate Ek,s(N).When the number of variables is big,we use the relations obtained in Chapter two to estimate Ek,s(N).Our results improve the works of Kumvhev and Zhixin LiuIn Chapter five,we consider the Waring-Goldbach problem for unlike powers.We use Hardy-Littlewood method,and give new mean value estimates for unlike powers to reduce the number of variables in the integral.Our work improve the results of Schwarz,Brudern,Zhixin Liu,Horffman and Gang Yu and so on.