Save for 2,3,5,7, is there any known PAIR of twin prime numbers?

Save for 2,3,5,7, is there any known PAIR of twin prime numbers? (I.e., If 9 were prime, would be an example.)
Or has it been proven that there couldn’t be ?

With an inner difference of 6 or less, no. One of the numbers has to be a multiple of 3.

With an inner difference of 8 (quadruple primes), there are many examples. 5,7,11,13 or 11,13,17,19. It is indeed conjectured that there are infinitely many such quadruples.

It’s not hard to prove it’s impossible: One of the number must be divisible by 3.

There can’t be any (and 2,3 aren’t really twin primes, by the way).

The proof is pretty simple, too. Of any three consecutive odd whole numbers (all primes greater than 2 will be odd), one of them will be divisible by 3.

What is meant by a “pair of twin primes”?

A twin prime is two prime numbers separated by just one number, eg 3 and 5. The op is defining a pair of twin primes as a run of four primes.

More generally, do all patterns not trivially impossible occur infinitely often? For example, here’s an octuplet prime pattern that occurs more than once:
17 19 23 29 31 37 41 43
224008217 224008219 224008223 224008229 224008231 224008237 224008241 224008243
1433889617 1433889619 1433889623 1433889629 1433889631 1433889637 1433889641 1433889643
Does it occur infinitely often?

Yes (conjecturally), and there are even estimates how dense these patterns are expected to be distributed as you move along the number line: The Hardy-Littlewood conjecture.

Thanks for the link! It noted that that (First) Hardy-Littlewood Conjecture is incompatible with a Second Hardy-Littlewood Conjecture:
π(x + y) ≤ π(x) + π(y)
I didn’t follow the whole discussion but it sounds like the Second Conjecture is expected to be false, but with the smallest counterexamples bigger than googol.

Thanks in return for that lead; I hadn’t stumbled on that before.

Basically, the second Hardy-Littlewood-Conjecture says that the smallest interval containing k primes starts at the beginning of the number line (at 1 or 2, don’t know if it matters).

However, the first HLC states that there is an interval of 3159 numbers somewhere out there that contains 447 primes, while the interval from 1 to 3159 does not.

Oh my, it’s all gone a bit “Mornington Crescent”


But are we playing with or without the optional Axiom of Choice rule?

It’s been a long time since a Mornington Theorem thread. Why doesn’t someone start one in Game Threads?

Has anyone ever proved that some possible prime gap k where k>1 does not appear an infinite number of times? For any particular k, or just that some k with that property exists?

Oops, I meant to put that question over in this other thread about primes; please respond over there.

Too late for the edit window.