表大偶數為一個不超過三個素數的乘積及一個不超過四個素數的乘積之和

<正> V.Brun最初在1920年證明了:每一充分大的偶數可表為兩個各不超過9個素數的乘積之和.簡記之為(9,9).後來,不少數學家改進與簡化了Brun方法,因此,Brun的結果也得到相應的改進,

ON THE REPRESENTATION OF LARGE EVEN INTEGER AS A SUM OF A PRODUCT OF AT MOST 3 PRIMES AND A PRODUCT OF AT MOST 4 PRIMES

Professor L. K. Hua has pointed out to me that we may improve result on Goldbach problem by the combination of the methods of Selbergand Brun-The purpose of this paper is to carry out this idea and to prove the following two results:Theorem 1. Every sufficiently large even integer is a sum of a product of at most 3 primes and a product of at most 4 primes.Theorem 2. There are infinitely many integers n, such that n is a product of at most 3 primes and n+2 is a product of at most 4 primes.It can be expected that by the present method with some complicated numerical calculations we can prove that every sufficiently large even integer is a sum of two products each of which has at most 3 primes.In succeeding papers, the author will give the proofs of the following results.1°. If F(x) denotes a irreducible integral valued polynomial of degree k without any fixed prime divisor, then there are infinitely many integers x, such that F(x) is a product of at most 2.1 k] primes.2°. Let π(N; F(x)) be the number of primes represented by F(x) as x=1, 2,…,N. Then we have π(N; F(x))≤2e~γ μF N/log N + o(N/log N ), where μF is a constant depending on F(x) only and γ denotes the Euler's constant.Assuming the truth of grand Riemann hypothesis, that is, assuming the real parts of all zeros of all Diricblet's L-functions L(s, X) are ≤1/2, we have the following.3°. Every sufficiently large even integer is a sum of a prime and a product of at most 4 primes.4°. Let Z_2 (N) be the number of twin primes (p, p+2)not exceeding N. Then where ε is any given positive number and the constant implied by the symbol "O" depends on εonly.