[Math] Are there any known special properties of a number located between twin primes

Question

With the exception of $4$, every number located between twin primes is divisible by $6$.

This one is obvious, but are there any other properties that can be ascribed to such numbers?

A property may be ascribed either to each number individually or to the entire sequence.

For example, consider the amount of prime factors of such numbers:

• Is there any known restriction on this amount per each number?
• Is there any known restriction on this amount as a function of the sequence-index?

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The context in which I am asking this question:

What attempts have been made towards proving that there are infinitely many pairs of twin primes, by proving that there are infinitely many numbers located between twin primes?

Answer 1

About the sequence you're after, I would say that it is possibly more interesting to consider the quotients when they're divided by 6. The result is the OEIS sequence A002822.

I'm not sure whether this counts as "special", but there seems to be a way to associate bijectively A002822 to the twin prime pairs. I actually found a few pointers in other questions here, so you could simply check them. In Does proving the following statement equate to proving the twin prime conjecture? you can find a pointer to an arXiv paper from 2011, and in About a paper by Gold & Tucker (characterizing twin primes) you will find a link to a small, 20 years older article which also proves the same bijective relationship and derives a similar formula (even if it's hard to distinguish because of their $G(n)$ function).

My preferred formulation of the theorem is as follows:

Any twin prime pair greater than $(3;5)$ is of the form $(6z-1;6z+1)$ where $z\inℕ$ and satisfies following system of inequalities: $$6xy+5x+5y+4 \neq z$$ $$6xy+7x+5y+6 \neq z$$ $$6xy+7x+7y+8 \neq z$$ for all $(x,y) \in (\mathbb{N}\cup\{0\})^2$; and conversely, for any such integer $z$, the pair $(6z-1;6z+1)$ is a twin prime pair.

In response to your contextual question, you can see above that a few people did try to find a proof using this theorem. And there are regularly similar attempts posted on the arXiv. For the sake of citing an example, I'll mention this one: Twin Prime Sieve.