论筛法及其有关的若干应用——殆素数的分布问题

<正> 本文的宗旨在于证明作者在1]内所提及的全部结果,现在将本文的强果详述于下:定理1.命 F(x)表一无固定素因子的 k 次既约整值多项式.命(?)此处 w 是适合下面不等式的最小正整数(?)则在叙列{F(x)}中存在无限多个不超过 n 个素数的乘积.例如存在无限多个 x,使 x~3+2的素因子个数(包括相同的与相异的)不多于4.与此相类似,有定理2.设 k 为一正整数,命 n 适合(1)及(2),则当 x 充分大时,区间 x

ON SIEVE METHODS AND SOME OF THEIR APPLICATIONS

In this paper,we give the details of the proofs of the following three theorems(Cf.Science Record,Academia Sinica,New Ser.Vol.I,No.3,1957,pp.1—5).Theorem 1.Let F(x)be an irreducible integral valued polynomial of degree k withoutany fixed prime divisor.Let(?)where w is the least integer satisfying(?)Then there are infinitely many integers x such that F(x)is a product of at most n primes.Theorem 2.Let k be a positive integer.Let n be an integer satisfying(1)and(2).Then for sufficiently large x,there is always an integer between x and x+x~(1/k)which has at most n prime factors.Theorem 3.For sufficiently large x