I was playing around with semi-prime numbers and I made two conjectures, which are:

1. Given any integer $a$, at least one of $a,(a+1),(a+2)$ or $(a+3)$ is not semi-prime.

2. There are infinitely many integers $a$, such that $a,(a+1)$ and $(a+2)$ are semi-primes.

I've written a computer program to verify the conjectures for values up to 700,000 (so, there's a high chance that they are both true).

Can anyone give a proof or counter example for any of these problems, or a link to a paper on the subject?

Thanks!

-
Conjecture (1) is easy: every four consecutive integers contains a multiple of four. Conjecture (2) has an associated OEIS entry but I'm not aware of any results or research on it. – anon Feb 10 '12 at 21:01
Yes. Thanks! I didn't even think about that ;) Since 5 and 3 are not semi-primes the prof becomes trivial. :) – Obinna Okechukwu Feb 10 '12 at 21:06

There are standard, but hopeless, conjectures in Number Theory that would imply, for example, that there are infinitely many $a$ such that $a=3p$, $a+1=2q$, and $a+2=5r$, where $p,q,r$ are all prime. A search for prime $k$-tuples conjecture should get you started on the literature.