Article in Proceedings of the American Mathematical Society · October 005 doi: 10. 2307/4097907 citations reads 10 72 author

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On the Waring-Goldbach problem for seventh powers
Article in Proceedings of the American Mathematical Society · October 2005
DOI: 10.2307/4097907

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Angel Kumchev
Towson University
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ON THE WARING–GOLDBACH PROBLEM FOR SEVENTH POWERS
ANGEL V. KUMCHEV
Abstract. We use sieve theory and recent estimates for Weyl sums over almost primes to prove that every sufficiently large even integer is the sum of 46 seventh powers of prime numbers.
1. Introduction
Let k be a natural number and let H(k) denote the least integer s such that the diophantine equation
(1.1)
is soluble in primes p₁,...,p_s for all sufficiently large integers n satisfying certain local conditions. The local conditions are designed to exclude degenerate cases, in which (1.1) reduces to a similar equation in fewer unknowns. For example, since every representation of an even integer n as the sum of three primes reduces to a representation of n − 2 as the sum of two primes, we study (1.1) with k = 1 and s = 3 only when n is odd.
In 1937 I. M. Vinogradov [16] found a new method for estimating sums over primes, which he used to prove that H(1) ≤ 3. Hua [4] used Vinogradov’s method to establish the bound
(1.2) H(k) ≤ 2^k + 1 (k ≥ 1),
which is still the best result known for k ≤ 3. Later, work of Vinogradov, Hua, and Davenport from the 1940s and 1950s (see Hua [5]) and a technique in Waring’s problem developed in the mid-1980s by Thanigasalam [11, 12] and Vaughan [14] led to a series of improvements on (1.2) for k ≥ 4. In particular, it was known by the late 1980s that
H(4) ≤ 15, H(5) ≤ 23, H(6) ≤ 33, H(7) ≤ 47, H(8) ≤ 63, H(9) ≤ 83.
Recently, Kawada and Wooley [7] showed that
H(4) ≤ 14 and H(5) ≤ 21.
The main innovation in [7] is the use of minor arc estimates stemming directly from sharp estimates for exponential sums over primes rather than from estimates for artificially introduced exponential sums over consecutive integers. The purpose of the present paper is to obtain a similar result for seventh powers of primes. We establish the following theorem.