TITLE: Sets of Prime and Composite Numbers
QUESTION [12 upvotes]: We know that all primes are of the form $ 6k ± 1 $ with the exception of 2 and 3.
We also know that not all numbers of the form $ 6k ± 1 $ are prime.
This leads to four distinct sets of pairs adjacent to a multiple of six:

Twin Primes, Example: $ 5, 7 $ (prime followed by a prime)
Twin Composites, Example: $ 119, 121 $ (composite followed by a composite)
Prime-Composite, Example: $ 23, 25 $ (prime followed by a composite)
Composite-Prime, Example: $ 35, 37 $ (composite followed by a prime)

The Twin Prime Conjecture states that there are infinitely many Twin Primes, but has yet to be proven.
Could it be proven that any of these four sets are infinite?

REPLY [3 votes]: It is very simple to construct an infinite sequence for the Prime-Composite case:
$$23+60n,25+60n$$
$25+60n$ will generate an infinite amount of composite numbers, all of which are divisible by $5$ (in fact, it will generate only composite numbers).
$23+60n$ will generate an infinite amount of prime numbers, since $23$ and $60$ are coprime integers (according to Dirichlet's theorem on arithmetic progressions).

It is very simple to construct an infinite sequence for the Composite-Prime case:
$$35+60n,37+60n$$
$35+60n$ will generate an infinite amount of composite numbers, all of which are divisible by $5$ (in fact, it will generate only composite numbers).
$37+60n$ will generate an infinite amount of prime numbers, since $37$ and $60$ are coprime integers (according to Dirichlet's theorem on arithmetic progressions).