# Save for 2,3,5,7, is there any known PAIR of twin prime numbers?
**URL:** https://boards.straightdope.com/t/save-for-2-3-5-7-is-there-any-known-pair-of-twin-prime-numbers/811393
**Category:** Factual Questions
**Created:** [March 29, 2018, 4:34am UTC](https://boards.straightdope.com/t/save-for-2-3-5-7-is-there-any-known-pair-of-twin-prime-numbers/811393 "2018-03-29T04:34:33Z")
**Posts on this page:** 16
**Page:** 1
### Author:  [@Steven\_Estes](https://boards.straightdope.com/u/Steven_Estes)
#### Post date: [March 29, 2018, 4:34am UTC](https://boards.straightdope.com/t/save-for-2-3-5-7-is-there-any-known-pair-of-twin-prime-numbers/811393/1 "2018-03-29T04:34:33Z")
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 ?
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### Author:  [@kk\_fusion](https://boards.straightdope.com/u/kk_fusion)
#### Post date: [March 29, 2018, 4:55am UTC](https://boards.straightdope.com/t/save-for-2-3-5-7-is-there-any-known-pair-of-twin-prime-numbers/811393/2 "2018-03-29T04:55:24Z")
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.
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### Author:  [@Cabbage](https://boards.straightdope.com/u/Cabbage)
#### Post date: [March 29, 2018, 4:56am UTC](https://boards.straightdope.com/t/save-for-2-3-5-7-is-there-any-known-pair-of-twin-prime-numbers/811393/3 "2018-03-29T04:56:01Z")
It’s not hard to prove it’s impossible: One of the number must be divisible by 3.
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### Author:  [@Great\_Antibob](https://boards.straightdope.com/u/Great_Antibob)
#### Post date: [March 29, 2018, 4:56am UTC](https://boards.straightdope.com/t/save-for-2-3-5-7-is-there-any-known-pair-of-twin-prime-numbers/811393/4 "2018-03-29T04:56:48Z")
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.
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### Author:  [@Novelty\_Bobble](https://boards.straightdope.com/u/Novelty_Bobble)
#### Post date: [March 29, 2018, 6:54am UTC](https://boards.straightdope.com/t/save-for-2-3-5-7-is-there-any-known-pair-of-twin-prime-numbers/811393/5 "2018-03-29T06:54:36Z")
What is meant by a “pair of twin primes”?
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### Author:  [@Richard\_Pearse](https://boards.straightdope.com/u/Richard_Pearse)
#### Post date: [March 29, 2018, 7:04am UTC](https://boards.straightdope.com/t/save-for-2-3-5-7-is-there-any-known-pair-of-twin-prime-numbers/811393/6 "2018-03-29T07:04:10Z")
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.
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### Author:  [@septimus](https://boards.straightdope.com/u/septimus)
#### Post date: [March 29, 2018, 7:53am UTC](https://boards.straightdope.com/t/save-for-2-3-5-7-is-there-any-known-pair-of-twin-prime-numbers/811393/7 "2018-03-29T07:53:17Z")
> [@kk\_fusion](#):
>
> 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.
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?
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### Author:  [@kk\_fusion](https://boards.straightdope.com/u/kk_fusion)
#### Post date: [March 29, 2018, 8:03am UTC](https://boards.straightdope.com/t/save-for-2-3-5-7-is-there-any-known-pair-of-twin-prime-numbers/811393/8 "2018-03-29T08:03:49Z")
> [@septimus](#):
>
> More generally, do all patterns not trivially impossible 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](https://en.wikipedia.org/wiki/Twin_prime#First_Hardy%E2%80%93Littlewood_conjecture).
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### Author:  [@septimus](https://boards.straightdope.com/u/septimus)
#### Post date: [March 29, 2018, 10:24am UTC](https://boards.straightdope.com/t/save-for-2-3-5-7-is-there-any-known-pair-of-twin-prime-numbers/811393/9 "2018-03-29T10:24:19Z")
> [@kk\_fusion](#):
>
> 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](https://en.wikipedia.org/wiki/Twin_prime#First_Hardy%E2%80%93Littlewood_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.
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### Author:  [@kk\_fusion](https://boards.straightdope.com/u/kk_fusion)
#### Post date: [March 29, 2018, 11:05am UTC](https://boards.straightdope.com/t/save-for-2-3-5-7-is-there-any-known-pair-of-twin-prime-numbers/811393/10 "2018-03-29T11:05:30Z")
> [@septimus](#):
>
> 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.
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### Author:  [@Novelty\_Bobble](https://boards.straightdope.com/u/Novelty_Bobble)
#### Post date: [March 29, 2018, 11:47am UTC](https://boards.straightdope.com/t/save-for-2-3-5-7-is-there-any-known-pair-of-twin-prime-numbers/811393/11 "2018-03-29T11:47:20Z")
> [@septimus](#):
>
> Thanks for the link! It noted that that (First) Hardy-Littlewood Conjecture is _incompatible_ with a Second Hardy-Littlewood Conjecture:
Oh my, it’s all gone a bit “Mornington Crescent”
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### Author:  [@digs](https://boards.straightdope.com/u/digs)
#### Post date: [March 29, 2018, 12:27pm UTC](https://boards.straightdope.com/t/save-for-2-3-5-7-is-there-any-known-pair-of-twin-prime-numbers/811393/12 "2018-03-29T12:27:05Z")
> [@Novelty\_Bobble](#):
>
> Oh my, it’s all gone a bit “Mornington Crescent”
😃
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### Author:  [@Chronos](https://boards.straightdope.com/u/Chronos)
#### Post date: [March 29, 2018, 1:42pm UTC](https://boards.straightdope.com/t/save-for-2-3-5-7-is-there-any-known-pair-of-twin-prime-numbers/811393/13 "2018-03-29T13:42:02Z")
But are we playing with or without the optional Axiom of Choice rule?
😉
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### Author:  [@septimus](https://boards.straightdope.com/u/septimus)
#### Post date: [March 29, 2018, 2:44pm UTC](https://boards.straightdope.com/t/save-for-2-3-5-7-is-there-any-known-pair-of-twin-prime-numbers/811393/14 "2018-03-29T14:44:09Z")
It’s been a long time since a _Mornington Theorem_ thread. Why doesn’t someone start one in Game Threads?
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### Author:  [@Ponderoid](https://boards.straightdope.com/u/Ponderoid)
#### Post date: [March 29, 2018, 3:50pm UTC](https://boards.straightdope.com/t/save-for-2-3-5-7-is-there-any-known-pair-of-twin-prime-numbers/811393/15 "2018-03-29T15:50:08Z")
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?
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### Author:  [@Ponderoid](https://boards.straightdope.com/u/Ponderoid)
#### Post date: [March 29, 2018, 3:56pm UTC](https://boards.straightdope.com/t/save-for-2-3-5-7-is-there-any-known-pair-of-twin-prime-numbers/811393/16 "2018-03-29T15:56:55Z")
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](https://boards.straightdope.com/sdmb/showthread.php?t=851862)
Too late for the edit window.