On Representation Of Integers By Sums Of A Cube And Four Cubes Of Primes

In the additive theory of prime numbers, one studies the representation of positive integers by powers of primes. The Waring-Goldbach problem seeks to represent positive integers satisfying necessary congruence conditions by powers of primes. The ternary and binary Goldbach problems[1] are just liner examples of the Waring-Goldbach prob-lem.The circle method of Hardy and Littlewood in combination with the estimates of Vinogradov for exponential sums over primes gives an affirmative answer to the general Waring-Goldbach problem, and the results before 1965 was summarized in Hua's book "Additive Theory on Primes Numbers". After that, especially in resent yeas, new ideas in the circle method,sieves, and exponential sums are incorporated into the Waring-Goldbach problem, and hence give remarkable advance.On the other hand, the additive theory of prime numbers with certain conditions appeals to many researchers to study. It is conjectured that all sufficiently large integers satisfying some necessary congruence conditions are sums of four cubes of primes. Such a strong result is out of reach at present. The best result in this direction is due to Hua[2] and dates back to 1938:●All sufficiently large integers are sums of nine cubes of primes;●Almost all integers n in the set n={n≥1:n(?)0,±2 (mod 9)} can be represented as sums of five cubes of primes, i.e. (0.1) To be more precise, let E(N) denote the number of integers n∈n not exceeding N which cannot be written as the formula(0.1) mentioned above. Then Hua's second result actually states that where A> 0 is arbitrary.With the help of circle method, we can get advanced result for this question. And Ren[3] gave the following result:To get a result of this strength, we have to deal with rather large major arcs. in 1998, Liu and Zhan[4] found a new approach to treat the enlarged major arcs in the Waring-Glodbach problem, in which the possible existence of the Sicgel zero does not have a special influence, and hence the Deuring-Heilbronn phenomenon can be avoided.In this paper, with the help of circle method, we study the generalization of the problem. Hua in [2] proved that every sufficiently large integer satisfying necessary conditions is the sums of five cubes of primes. We replace a prime with a integer, studying the representation of integers by sums of a cube and four cubes of primes, that is. if n∈n, then where m1 is a positive integer.We shall prove the following theorem.Theorem If E(N) is as above, then...