On the representation of a large integer as the sum of a prime and a square-free number with at most three prime divisors

In this paper we prove that every sufficiently large odd integer can be written as a sum of a prime and 2 times a product of at most two distinct odd primes. Together with Chen’s theorem and Ross’s observation, this shows every sufficiently large integer can be written as a sum of a prime and a square-free number with at most three prime divisors.