# Are there any known special properties of a number located between twin primes?

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?

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?

• FYI : Average of twin prime pairs (OEIS : A014574) Commented Apr 23, 2015 at 12:23
• @mathlove: There are some interesting properties there, and perhaps even more interesting is the fact that they have been determined relatively recently. Thanks. Commented Apr 23, 2015 at 12:25
• Maybe useful Commented Apr 23, 2015 at 12:25
• @Elaqqad: Thank you, but the answer to this question shows that it implies nothing more than these numbers being divisible by $6$ (which as I've mentioned, is obvious). Thanks anyway. Commented Apr 23, 2015 at 12:27
• @barak manos, just in case, I asked a related question about twin primes, it might give you another point of view, it is about twin primes in the vicinity of twin primes. math.stackexchange.com/questions/1005852/… Commented Apr 23, 2015 at 13:12

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).
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.
• they are Also the only numbers not describable in one of the forms : $6kl+k+l, 6kl-k-l , 6kl+k-l$ for aome $k,l$ where both are positive integers ... Commented Apr 17, 2021 at 23:34