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