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

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## Abstract

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.

## Keywords

Chen’s theorem Estermann’s theorem Sieve method## Mathematics Subject Classification

11P32 11N36 11N80## Notes

### Acknowledgements

The author is very grateful to Jim Brown, Luke Giberson, Kevin James, Daozhou Zhu and the reviewer for their helpful discussions and suggestions.

## References

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