Hua's Theorem With Five Almost Equal Prime Variables

Abstract

It is proved that each sufficiently large integer N ≡ 5 (mod24) can be written aswhere pj are primes. This result, which is obtained by an iterative method and a hybrid estimate for Dirichlet polynomial, improves the previous results in this direction.

On the representation of large integers as sums of four almost equal squares of primes

“…Because the procedure is very similar to that used in [8], we only give the outline for the proof of Lemma 2. where δ χ = 1 or 0 according as the Dirichlet character χ is principal or not. Therefore, we can express the difference between S(α) and the expected major arc approximation to it as follows:…”

“…In the last integral, we take out |W (χ 1 , λ)| and |W (χ 2 , λ)|, and then use Cauchy's inequality, to get Thus our Lemma 2.1 follows form the following lemmas and an iterative approach (see [8] for a similar argument). The proofs of Lemmas 4.1-4.3 are standard.…”

“…The proofs of Lemmas 4.1-4.3 are standard. We shall omit the details and refer to [8] for a similar argument.…”

Exponential sums over primes in short intervals

“…[10]; Δ = 1 35 in ref. [11]. In this paper, we apply Theorem 1.1 in conjunction with the estimate in ref.…”

On a generalization of Hua’s theorem with five squares of primes

“…Later Liu, Wooley, and Yu [10] showed that the exponent 2/5 can be improved to 3/8, and very recently, Harman and Kumchev [3] further reduced the exponent 3/8 to 5/14. There are also many other interesting approximations (see for example [1], [4], [9], [8], [13], [12] and their references).…”