On the Representation of Large Even Integers as Sums of Two Almost Primes. I

Abstract

The classical Goldbach problem, which still survives unsolved, is to prove that every even integer $\geqq 6$ is a sum of two prime numbers. In 1948 A. R\'enyi [4] succeeded, by making use of his refinement of the large sieve of Yu. V. Linnik, in proving that every even integer $\geqq 6$ is a sum of a prime and of an almost prime. Here an almost prime is a positive integer $(>1)$ the total number of prime factors of which is bounded by a certain constant. Recently this result was sharpened in part by Ch.-D. Pan [3], who showed that every sufficiently large even integer can be represented as a sum of a prime and of an almost prime possessing at most five prime factors.On the other hand, A. A. Buhstab [1] has proved that every large even integer can be written as a sum of two almost primes, each of which is composed of at most four prime factors. The purpose of the present paper is to improve this result of Buh\v{s}tab. Indeed, we shall prove the following Theorem. Every sufficiently large even integer is representable as

On the Representation of Large Even Integers as Sums of Two Almost Primes. Ii

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In a previous paper [3] the writer has given with Miss A. Togashi an elementary proof for the fact that every sufficiently large even integer is representable as a sum of two almost primes, each of which has at most three prime factors, a result first obtained by A. I. Vinogradov. On the other hand, we are able to prove by a rather transcendental method that every large even integer is representable as a sum of a prime and an almost prime composed of at most four prime factors (see [4]).

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“…Let $q,$ $(q, N)=1$ , be a fixed prime number in the interval $z<q\leqq z_{1}$ , where $z=N^{\frac{1}{9}}$ , $z_{1}=N^{\frac{5}{9}}$ , and let $S(q)$ denote the number of those integers $a_{n}=n(N-n)(1\leqq n\leqq N$, $(n, N)=1)$ which are not divisible by any prime $p\leqq z$ and are divisible by the prime $q$ . Then we find as in [3] that…”

“…Let $N$ be a sufficiently large but fixed even integer. -We $cons^{t}ider$ as in [3] the $\varphi(N)$ in,tegers $a_{n}=n(N-n)(1\leqq n\leqq N, (n, N)=1)$.…”