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

Factual Questions
#1

Save for 2,3,5,7, is there any known PAIR of twin prime numbers? (I.e., If 9 were prime, 3.5.7.9 would be an example.)
Or has it been proven that there couldn’t be ?

#2

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.

#3

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

#4

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.

#5

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

#6

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.

#7

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?

#8

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.

#9

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.

#10

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.

#11

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

#12

:smiley:

#13

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

#14

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

#15

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?

#16

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

https://boards.straightdope.com/sdmb/showthread.php?t=851862

Too late for the edit window.