Sets of Prime and Composite Numbers 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?
 A: 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).
A: The composite-composite case is easy.  By the Chinese remainder theorem there are infinitely many solutions of, for example,
$$x\equiv0\pmod6\ ,\quad x\equiv1\pmod5\ ,\quad x\equiv-1\pmod7\ .$$
And for any such $x$ greater than $6$ we have $x-1,x+1$ are adjacent to a multiple of $6$, and $x-1$ is a multiple of $5$ and hence composite, and $x+1$ is a multiple of $7$ and hence composite.

The composite-prime case follows from Dirichlet's theorem (which is not easy to prove).  Consider the numbers $x=30k+6$.  Then the numbers $x-1,x+1$ are adjacent to a multiple of $6$; and the numbers $x+1$ are prime infinitely often; and the numbers $x-1$ are always composite.

Similarly, $x=30k-6$ covers the prime-composite case.

And as you mentioned, the prime-prime case is still unsolved.

Alternative proof for the composite case: consider the numbers
$$x=6\times119\times121k+120\ .$$
Then $x-1$ is always a multiple of $119$, and $x+1$ is always a multiple of $121$.

Or to keep the numbers a bit smaller, $x=6\times7\times11k+120$.
A: Case $2$ is answered so I will address the others.  For case $3$, by Dirichlet's theorem there are an infinite number of primes of the form $10 \cdot n + 3$, and $10 \cdot n + 5$ is always divisible by $5$.  There is a similar example for case $4$.
