I was playing around with semi-prime numbers and I made two conjectures, which are:
Given any integer $a$, at least one of $a,(a+1),(a+2)$ or $(a+3)$ is not semi-prime.
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?