structural:summarization
Other elementary properties

Every third odd number is divisible by 3, and therefore no three successive odd numbers can be prime unless one of them is 3. Therefore, 5 is the only prime that is part of two twin prime pairs. The lower member of a pair is by definition a Chen prime.

It has been proven that the pair (m, m + 2) is a twin prime if and only if

If m − 4 or m + 6 is also prime then the three primes are called a prime triplet.

For a twin prime pair of the form (6 n − 1, 6 n + 1) for some natural number n > 1, n must end in the digit 0, 2, 3, 5, 7, or 8 (OEIS: A002822).