Twin Prime Constellations

[robert44444uk]

I thought I would look at this again, using mart_r's table. The idea is that someone could create an entry to OEIS.

These are the first instance primes where the number of twins shown (of which the first instance prime was the smaller of a twin) are clustered in the narrowest range of integers. Excluded are non-admissible patterns. The list confirms the smallest 6 found by henryzz and lists the Jens Kruse Andersen 8 twin discovery.

#of twins First prime Twin pattern OEIS
3 179 {0,2,12,14,18,20} A253627
4 626597 {0,2,12,14,24,26,30,32} A253624
5 39713433671 {0,2,6,8,18,20,30,32,36,38} A256842
6 1256522812841 {0,2,6,8,18,20,30,32,36,38,48,50] Found in this group https://www.mersenneforum.org/showpo...94&postcount=5
7 1135141716537971 {0,2,6,8,18,20,30,32,36,38,48,50,60,62} Listed at http://www.balcro.com/fix5tp39.html as part of the search for 5-twins back in the day.
8 17625750738756291797 {0,2,12,14,24,26,30,32,42,44,54,56,72,74,84,86} Listed at https://primes.utm.edu/curios/page.p...756291797.html

I'm looking at all patterns, and have reserved 9 and 10 twins

[Bobby Jacobs]

The smallest admissible distance between 2 consecutive primes is 2. These are twin primes (p, p+2). The smallest admissible distance between 2 consecutive pairs of twin primes is 6. These are twin twin primes, or prime quadruplets (p, p+2, p+6, p+8). The smallest admissible distance between 2 consecutive sets of twin twin primes is 30. These are twin twin twin primes, or twin prime quadruplets (p, p+2, p+6, p+8, p+30, p+32, p+36, p+38). The smallest set of twin twin twin primes is 1006301, 1006303, 1006307, 1006309, 1006331, 1006333, 1006337, 1006339. What is the smallest admissible distance between 2 consecutive sets of twin twin twin primes? What is the smallest set of twin twin twin twin primes?

[robert44444uk]

I'm up to 6e19 in the search for 9 twins (using all 4 patterns) in 105 digits without a result.

[mart_r]

Originally Posted by Bobby Jacobs

What is the smallest admissible distance between 2 consecutive sets of twin twin twin primes? What is the smallest set of twin twin twin twin primes?

420. 11281963036964038421 + [0, 2, 6, 8, 30, 32, 36, 38, 420, 422, 426, 428, 450, 452, 456, 458]. (J. Waldvogel / P. Leikauf, 2008)

Playing the game further, the smallest admissible distance between two of those is 2310. You probably won't live long enough for witnessing the discovery of the first occurrence.

[a1call]

3# = 6
5# = 30
7# = 210
11# = 2310

Is the 7# distance impossible, or just that none is found so far?

ETA: the pattern can be extended towards the lower side for 3, 5, 7 with 2#
(p, p+2, p+2, p+4)

ETA II: “half” twin primes are very finite.
Twin primes are very much probably infinite.
Is it known if Twin-Twin primes are finite? (I assume the knowledge to their infinity would definitely be unknown since even infinity of twins is unknown).

[Dr Sardonicus]

Originally Posted by a1call

3<snip>
Is the 7# distance impossible, or just that none is found so far?

ETA: the pattern can be extended towards the lower side for 3, 5, 7 with 2#
(p, p+2, p+2, p+4)
<snip>

? v=[0, 2, 6, 8, 30, 32, 36, 38, 420, 422, 426, 428, 450, 452, 456, 458];w=vector(#v,i,x+ v[i]);w2=vector(#v, i,x + v[i]+210);f=prod(i=1,#v,w[i]*w2[i]);forprime(p=2,32,fp=Mod(1,p)*f;if(Mod(fp,x^p-x)==0,print(p)))
11
19

For any integer x, at least one of the 32 numbers x, x + 2, ..., x + 456, x + 458; x + 210, x + 212, ..., x + 668, x + 670 is divisible by 11, and at least one is divisible by 19.

[ATH]

Originally Posted by mart_r

420. 11281963036964038421 + [0, 2, 6, 8, 30, 32, 36, 38, 420, 422, 426, 428, 450, 452, 456, 458]. (J. Waldvogel / P. Leikauf, 2008)

There are extra primes and even a pair of twin primes in that pattern at:
11281963036964038421 + [80,122,318,330,332,402]

Is it theoretically possible with only those 16 primes between p and p+458 ?

Edit: Yes, it is possible I found out myself, but probably much harder to find an example.

[mart_r]

Originally Posted by ATH

There are extra primes and even a pair of twin primes in that pattern at:
11281963036964038421 + [80,122,318,330,332,402]

Is it theoretically possible with only those 16 primes between p and p+458 ?

Edit: Yes, it is possible I found out myself, but probably much harder to find an example.

You're right, I wasn't paying appropriate attention to the word "consecutive" in Bobby's post.

The currently largest known 16-tuplet has 35 digits, log is 78.56. The gap between the two double quadruplets is 382, which equals a merit of 4.86 in that region. (Also don't forget the possibility of up to two primes between the quadruplets.) Finding such a pattern without any prime in-between will be very hard, at least for now. At the least, it might be interesting to figure out the trade-off between number size and theoretical number of 16-tuplets to be found until the gap appears, for a possible future computation on a quantum chip.

[ATH]

Even the next one is possible with no primes in between with only those 32 primes between p and p+2768.

p + [0, 2, 6, 8, 30, 32, 36, 38, 420, 422, 426, 428, 450, 452, 456, 458, 2310, 2312, 2316, 2318, 2340, 2342, 2346, 2348, 2730, 2732, 2736, 2738, 2760, 2762, 2766, 2768]

[a1call]

Originally Posted by Dr Sardonicus

? v=[0, 2, 6, 8, 30, 32, 36, 38, 420, 422, 426, 428, 450, 452, 456, 458];w=vector(#v,i,x+ v[i]);w2=vector(#v, i,x + v[i]+210);f=prod(i=1,#v,w[i]*w2[i]);forprime(p=2,32,fp=Mod(1,p)*f;if(Mod(fp,x^p-x)==0,print(p)))
11
19

For any integer x, at least one of the 32 numbers x, x + 2, ..., x + 456, x + 458; x + 210, x + 212, ..., x + 668, x + 670 is divisible by 11, and at least one is divisible by 19.

If the Primorial pattern holds then:

Twin => p, p+2
1/2 => p, p+2(2#), p,4
Twin-Twin => p, p+2, p+6(3#), p+8
Twin-Twin-Twin => p, p+2, p+6, p+8, p+30(5#), p+32, p+36, p+38
Twin-Twin-Twin-Twin => p, p+2, p+6, p+8, p+30, p+32, p+36, p+38, p+210(7#), p+212, p+216, p+218, p+240, p+242, p+246, p+248
...
So, 11 & 19 or any other prime less than 210 might very well be missed since the constellation does not increment by 2 only.
That is if I am not missing something.

ETA:
If I don't have the usual errors in my code, then for p = 10531061
p, p+2, p+6, p+8, p+30, p+32, p+36, p+38, p+210 are all primes:

p=10531061
%1 = 10531061
(18:20) gp > isprime(p)
%2 = 1
(18:21) gp > isprime(p+2)
%3 = 1
(18:21) gp > isprime(p+6)
%4 = 1
(18:21) gp > isprime(p+8)
%5 = 1
(18:21) gp > isprime(p+30)
%6 = 1
(18:21) gp > isprime(p+32)
%7 = 1
(18:21) gp > isprime(p+36)
%8 = 1
(18:22) gp > isprime(p+38)
%9 = 1
(18:22) gp > isprime(p+210)
%10 = 1

[a1call]

Originally Posted by a1call

If the Primorial pattern holds then:

Twin => p, p+2
1/2 => p, p+2(2#), p,4
Twin-Twin => p, p+2, p+6(3#), p+8
Twin-Twin-Twin => p, p+2, p+6, p+8, p+30(5#), p+32, p+36, p+38
Twin-Twin-Twin-Twin => p, p+2, p+6, p+8, p+30, p+32, p+36, p+38, p+210(7#), p+212, p+216, p+218, p+240, p+242, p+246, p+248
...
So, 11 & 19 or any other prime less than 210 might very well be missed since the constellation does not increment by 2 only.
That is if I am not missing something.

ETA:
If I don't have the usual errors in my code, then for p = 10531061
p, p+2, p+6, p+8, p+30, p+32, p+36, p+38, p+210 are all primes:

p=10531061
%1 = 10531061
(18:20) gp > isprime(p)
%2 = 1
(18:21) gp > isprime(p+2)
%3 = 1
(18:21) gp > isprime(p+6)
%4 = 1
(18:21) gp > isprime(p+8)
%5 = 1
(18:21) gp > isprime(p+30)
%6 = 1
(18:21) gp > isprime(p+32)
%7 = 1
(18:21) gp > isprime(p+36)
%8 = 1
(18:22) gp > isprime(p+38)
%9 = 1
(18:22) gp > isprime(p+210)
%10 = 1

Couldn't edit my last post, but as a concept of feasibility of Twin-Twin-Twin-Twin with the 7#=210 distance, these are prime also for p+212 which would make a pair of twins with p+210:

7503957281
9812361071
13083135641