A natural number n is called semiprime if Omega(n)=2. Twin semiprimes are those differ by 2. For examples, (4,6), (33,35), (49,51), (55,57), (85,87) and so on.
The question is related to a lot of other similar problems (very difficult): twin prime conjecture,erdos–Mirsky conjecture,Parity problem,... and so on, Maybe the well known result about your question is that there are infinitely many semi-primes for which the difference is less than 6 (Erdös Centennial
By László Lovász, Imre Ruzsa, Vera T. Sós) which is a consequence of another more general theorem related to this problem, so without copying other's work I suggest you reading these articles :
Small gaps between primes or almost primes :http://arxiv.org/abs/math/0506067
Small gaps between almost primes, the parity problem, and some conjectures of Erdos on consecutive integers :http://arxiv.org/abs/0803.2636
It is an open question, though one which is 'almost' solved. Some of the results are:
There are infinitely many prime numbers p such that p+2 is either prime or semiprime (Chen, 1966). This was later strengthened to: For any fixed positive even integer k, there are infinitely many prime numbers p such that p+k is either prime or semiprime (Chen, 1973).
There are infinitely many pairs of semiprime numbers such that the difference between them is less than or equal to 26 (Goldston, Graham, Pintz and Yildirim, 2007).
@Elaqqad Mohamed: I am aware of both the paper. Thanks for your suggestion.
@Petar Marković : Do you know whether any results are know for the following general problem:
For any positive integer k, there are infinitely many pairs of semiprimes which differ by exactly k.
I believe this should be true and as difficult as twin prime conjecture.
- Badges
- Science topic
- Similar topics
- Correction