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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:
[email protected]
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
[email protected]
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!
***
On Feb 4, 2023, Natalia Makarova added:
Minimal 17-tuple found in Stop@home BOINC project
(April 2017)
159067808851610411: 0 42 60 96 102 186 210 240 246
252 282 306 390 396 432 450�492
***
�
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