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?