Problems & Puzzles: Problems

Problem 62. Symmetric k-tuples of consecutive primes

Natalia Makarova sent the following nice puzzle.

Definition 1

A prime k-tuple is a finite collection of values (p + a1, p + a2, p + a3, …, p + ak),

where p, p + a1, p + a2, p + a3, …, p + ak  are prime numbers, (a1, a2, a3, …, ak) are pattern. Typically the first value in the pattern is 0 and the rest are distinct positive even numbers. [1]

We consider the k-tuple, where p + a1, p + a2, p + a3, ..., p + ak are consecutive primes.

Definition 2

k-tuple (p + a1, p + a2, p + a3, ..., p + a [k / 2], p + a [k / 2+1], ..., p + a [k-2], p + a [k-1], p + ak) for k even, is called symmetric, if the following condition is satisfied:

a1 + ak = a2 + a[k-1] =  a3 + a[k-2] = … = a[k/2] + a[k/2+1]

Example
symmetric 8-tuple

 (17 + 0, 17 + 2, 17 + 6, 17 + 12, 17 + 14, 17 + 20, 17 + 24, 17 + 26)

Shortened we write this:

17: 0, 2, 6, 12, 14, 20, 24, 26

Definition 3

k-tuple (p + a1, p + a2, p + a3, ..., p + a [(k-1) / 2], p + a [(k-1) / 2 + 1], p + a [(k-1) / 2 + 2], ..., p + a [k-2], p + a [k-1], p + ak) for k odd called symmetric, if the following condition is satisfied:

a1 + ak = a2 + a[k-1] = a3 +a [k-2] =…= a[(k-1)/2] + a[(k-1)/2+2] = 2 a[(k-1)/2+1]

Example
symmetric 5-tuple

18713: 0, 6, 18, 30, 36

(See in [2])

Definition 4

The diameter d of k-tuple is the difference of its largest and smallest elements. [1] 

Example

8-tuple
17: 0, 2, 6, 12, 14, 20, 24, 26
It has a diameter d = 26.

Known solutions with a minimal diameter and a minimal value of p for k = 2, 4, 6, 8 see in [1]

Further,

k=5 ( d, p minimal)

18713: 0, 6, 18, 30, 36

k=7 (d, p minimal)

12003179: 0, 12, 18, 30, 42, 48, 60

k=9 (d, p minimal)

1480028129: 0, 12, 24, 30, 42, 54, 60, 72, 84 

k=10 (d, p minimal)

13: 0, 4, 6, 10, 16, 18, 24, 28, 30, 34

k=11 (possible minimal ?)

660287401247651: 0, 6, 30, 42, 60, 66, 72, 90, 102, 126, 132 

k=12 (not minimal)

137: 0, 2, 12, 14, 20, 26, 30, 36, 42, 44, 54, 56 

k=13 (not minimal)

5348080416833699: 0, 12, 30, 42, 48, 72, 90, 108, 132, 138, 150, 168, 180

k=14 (not minimal)

19636011281690651: 0, 2, 8, 12, 18, 26, 30, 38, 42, 50, 56, 60, 66, 68

k=15 (not minimal)

5348080416833681: 0, 18, 30, 48, 60, 66, 90, 108, 126, 150, 156, 168, 186, 198, 216

k=16 (not minimal)

19636011281690647: 0, 4, 6, 12, 16, 22, 30, 34, 42, 46, 54, 60, 64, 70, 72, 76

k=18 (not minimal)

49549273441123: 0, 4, 24, 40, 46, 54, 58, 66, 70, 84, 88, 96, 100, 108, 114, 130, 150, 154

k=20 (not minimal)

11785542108641839: 0, 4, 10, 18, 24, 30, 52, 70, 72, 84, 118, 130, 132, 150, 172, 178, 184, 192, 198, 202

k=22 (not minimal)

18620445306703861: 0, 10, 36, 46, 66, 76, 82, 96, 102, 130, 136, 162, 168, 196, 202, 216, 222, 232, 252, 262, 288, 298 

k=24 (not minimal)

22930603692243271: 0, 70, 76, 118, 136, 156, 160, 178, 202, 222, 238, 250, 378, 390, 406, 426, 450, 468, 472, 492, 510, 552, 558, 628

(See [3]) 

For k = 17, 19, 21, 23 solutions no found. 

 Questions:

 1. Find solutions with a minimal diameter and a minimal value of p for 10 < k < 17, k = 18, 20, 22, 24.

 2. Find solutions for the remaining k, minimal or not minimal.

Links

[1] https://en.wikipedia.org/wiki/Prime_k-tuple

[2] http://oeis.org/A055380

[3] http://oeis.org/A081235


Contributions came from two Russian puzzlers but I have no names from them:

dima2000@mail.ru

k=11, 1542186111157: 0 6 30 42 60 66 72 90 102 126 132 (minimal)
k=12, 41280160361347: 0 4 6 10 12 22 24 34 36 40 42 46 (minimal)
k=13, 660287401247633: 0 18 24 48 60 78 84 90 108 120 144 150 168 (minimal, found by I at 08 aug 2014)
k=14, 10421030292115097: 0 2 6 12 14 20 26 30 36 42 44 50 54 56 (minimal)
k=16, 996689250471604163: 0 6 8 14 18 24 26 36 38 48 50 56 60 66 68 74 (minimal)
See http://dxdy.ru/post1047313.html#p1047313

and

gogolmogol16@mail.ru

Known solutions with a minimal diameter and a minimal value of p for k = 2, 4, 6, 8 see in [1]

solutions with a minimal diameter and a minimal value
k = 2 - p=2,d=1 A081235 (1)=(2,3), no (3,5)
k = 6 - p=5,d=14 A081235 (3)=(5,7,11,13,17,19), no (7, 11, 13, 17, 19, 23)


***

Natalia Makarova wrote on Set. 12, 2015

“solutions with a minimal diameter and a minimal value

k = 2 - p=2,d=1 A081235 (1)=(2,3), no (3,5)

k = 6 - p=5,d=14 A081235 (3)=(5,7,11,13,17,19), no (7, 11, 13, 17, 19, 23)”

This is the wrong solutions.

 In the sequence  http://oeis.org/A008407  we see:

k=2, d=2

k=6, d=16

It is right.

Solutions in Wikipedia 

[1] https://en.wikipedia.org/wiki/Prime_k-tuple

k=2    d=2     (0, 2)    (3, 5)

k=6    d=16   (0, 4, 6, 10, 12, 16)    (7, 11, 13, 17, 19, 23)

It is right.

***

On Oct-18-2015, Natalia sent the following note:

I and my colleague S. Tognon organized a contest on this problem. We invite everyone to the contest here: http://primesmagicgames.altervista.org/wp/competitions/

***

On Dec 30, 2015, Natalia Makarova wrote again:

I have a addition to problem # 62

http://www.primepuzzles.net/problems/prob_062.htm

 

k=15 (minimal; J. Wroblewski)

3112462738414697093: 0, 6, 24, 30, 54, 66, 84, 90, 96, 114, 126, 150, 156, 174, 180

 

k=17 (minimal; J. Wroblewski)

258406392900394343851: 0, 12, 30, 42, 60, 72, 78, 102, 120, 138, 162, 168, 180, 198, 210, 228, 240

 

k=18 (not minimal p? J. Wroblewski)

824871967574850703732309:0, 4, 10, 12, 18, 22, 28, 30, 40, 42, 52, 54, 60, 64, 70, 72, 78, 82

 

k=20 (not minimal p? J. Wroblewski & N. Makarova)

824871967574850703732303: 0, 6, 10, 16, 18, 24, 28, 34, 36, 46, 48, 58, 60, 66, 70, 76, 78, 84, 88, 94

All these solutions have a minimal diameter.

 Jaroslaw Wroblewski, thank you for participating in the contest and these wonderful results!

***

 


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