Consider an arithmetic sequence $4n+3$. This sequence contains infinitely many primes and infinitely many composites. It is clear that there cannot be $3$ consecutive primes in the sequence as every third number in the sequence is divisible by $3$.
Two consecutive primes, on the other hand, appear quite often in the sequence. My first question is:
Are there infinitely many pairs of consecutive members of the sequence that are both prime?
Also, it appears to me that composite numbers in the sequence tend to stack up quite a lot. Is there any limit to this? In particular, is it true that for any $n$, we can find $n$ consecutive members of the sequence such that they are all composite?