K-Tuples of Primes. Contest

[Natalia Makarova]

Dear Colleagues!

I invite everyone to take part in this competition.
At the forum in Russia you can participate in the debate
http://dxdy.ru/topic100750.html

You can write your message in English.

I offer to help the contestants theoretical patterns with a minimal diameter
(task 2):

k=12
0 4 6 10 12 22 24 34 36 40 42 46
k=14
0 2 6 12 14 20 26 30 36 42 44 50 54 56
k=15
0 6 24 30 54 66 84 90 96 114 126 150 156 174 180
k=16
0 6 8 14 18 24 26 36 38 48 50 56 60 66 68 74 
0 6 8 14 20 24 26 36 38 48 50 54 60 66 68 74
k=17
0 6 24 36 66 84 90 114 120 126 150 156 174 204 216 234 240 
0 12 18 30 42 72 78 102 120 138 162 168 198 210 222 228 240 
0 12 30 42 60 72 78 102 120 138 162 168 180 198 210 228 240
k=18
0 4 10 12 18 22 28 30 40 42 52 54 60 64 70 72 78 82
k=19
0 6 12 30 42 72 90 96 120 126 132 156 162 180 210 222 240 246 252
k=20
0 4 6 10 16 18 24 28 30 34 60 64 66 70 76 78 84 88 90 94 
0 4 6 10 16 18 24 28 34 36 58 60 66 70 76 78 84 88 90 94 
0 4 6 10 16 18 24 28 36 46 48 58 66 70 76 78 84 88 90 94 
0 4 6 10 16 18 24 30 34 46 48 60 64 70 76 78 84 88 90 94 
0 4 6 10 16 18 24 34 36 46 48 58 60 70 76 78 84 88 90 94 
0 6 10 16 18 24 28 34 36 46 48 58 60 66 70 76 78 84 88 94
k=21
0 12 30 42 54 60 72 84 114 120 162 204 210 240 252 264 270 282 294 312 324 
0 12 30 42 54 60 84 114 120 144 162 180 204 210 240 264 270 282 294 312 324
k=22
0 6 10 12 16 22 24 30 34 42 52 54 64 72 76 82 84 90 94 96 100 106 
0 6 10 12 16 22 24 30 40 42 52 54 64 66 76 82 84 90 94 96 100 106 
0 6 12 16 22 24 30 34 40 42 52 54 64 66 72 76 82 84 90 94 100 106
k=23
0 6 30 36 42 60 72 102 120 132 162 186 210 240 252 270 300 312 330 336 342 366 372 
0 6 30 36 42 60 102 120 126 132 162 186 210 240 246 252 270 312 330 336 342 366 372 
0 6 30 36 42 72 102 120 132 156 162 186 210 216 240 252 270 300 330 336 342 366 372 
0 6 30 36 42 90 102 120 132 156 162 186 210 216 240 252 270 282 330 336 342 366 372 
0 6 36 42 60 90 102 120 126 132 156 186 216 240 246 252 270 282 312 330 336 366 372
k=24
0 6 12 16 18 22 28 30 36 40 48 58 60 70 78 82 88 90 96 100 102 106 112 118 
0 6 12 18 22 28 30 36 40 46 48 58 60 70 72 78 82 88 90 96 100 106 112 118
k=25
0 6 24 36 60 66 84 120 126 150 186 204 210 216 234 270 294 300 336 354 360 384 396 414 420 
0 6 24 36 66 84 120 126 144 150 186 204 210 216 234 270 276 294 300 336 354 384 396 414 420 
0 6 24 60 66 84 90 120 126 144 186 204 210 216 234 276 294 300 330 336 354 360 396 414 420 
0 6 30 84 90 96 114 126 156 174 180 204 210 216 240 246 264 294 306 324 330 336 390 414 420 
0 12 30 42 48 78 120 132 162 168 180 198 210 222 240 252 258 288 300 342 372 378 390 408 420 
0 12 30 48 78 90 120 132 162 168 180 198 210 222 240 252 258 288 300 330 342 372 390 408 420 
0 24 30 54 60 66 84 96 126 144 156 186 210 234 264 276 294 324 336 354 360 366 390 396 420 
0 24 30 54 60 66 84 126 144 150 156 186 210 234 264 270 276 294 336 354 360 366 390 396 420 
0 24 30 54 60 66 114 126 144 156 180 186 210 234 240 264 276 294 306 354 360 366 390 396 420 
0 24 30 60 66 84 114 126 144 150 156 180 210 240 264 270 276 294 306 336 354 360 390 396 420
k=26
0 6 8 14 20 24 26 30 36 38 44 48 66 68 86 90 96 98 104 108 110 114 120 126 128 134 
0 6 8 14 20 24 26 30 36 38 48 50 66 68 84 86 96 98 104 108 110 114 120 126 128 134 
0 6 8 14 20 24 26 30 36 44 48 50 66 68 84 86 90 98 104 108 110 114 120 126 128 134 
0 6 8 14 20 24 26 30 36 48 50 54 66 68 80 84 86 98 104 108 110 114 120 126 128 134 
0 6 8 14 24 26 30 36 38 44 48 50 66 68 84 86 90 96 98 104 108 110 120 126 128 134 
0 8 14 20 26 30 36 38 44 48 54 56 66 68 78 80 86 90 96 98 104 108 114 120 126 134
k=27
0 6 12 30 42 66 72 90 126 132 156 192 210 216 222 240 276 300 306 342 360 366 390 402 420 426 432 
0 6 12 30 42 72 90 126 132 150 156 192 210 216 222 240 276 282 300 306 342 360 390 402 420 426 432 
0 6 12 36 90 96 102 120 132 162 180 186 210 216 222 246 252 270 300 312 330 336 342 396 420 426 432
k=28
0 4 10 12 18 24 28 30 34 40 42 48 52 70 72 90 94 100 102 108 112 114 118 124 130 132 138 142 
0 4 10 12 18 24 28 30 34 40 42 52 54 70 72 88 90 100 102 108 112 114 118 124 130 132 138 142 
0 4 10 12 18 24 28 30 34 40 48 52 54 70 72 88 90 94 102 108 112 114 118 124 130 132 138 142 
0 4 10 12 18 24 28 30 34 40 52 54 58 70 72 84 88 90 102 108 112 114 118 124 130 132 138 142 
0 4 10 12 18 28 30 34 40 42 48 52 54 70 72 88 90 94 100 102 108 112 114 124 130 132 138 142
k=29
0 30 36 42 60 72 96 102 120 156 162 186 222 240 246 252 270 306 330 336 372 390 396 420 432 450 456 462 492 
0 30 36 42 60 72 102 120 156 162 180 186 222 240 246 252 270 306 312 330 336 372 390 420 432 450 456 462 492
k=30
0 2 6 12 14 20 26 30 32 36 42 44 50 54 72 74 92 96 102 104 110 114 116 120 126 132 134 140 144 146 
0 2 6 12 14 20 26 30 32 36 42 44 54 56 72 74 90 92 102 104 110 114 116 120 126 132 134 140 144 146 
0 2 6 12 14 20 26 30 32 36 42 54 56 60 72 74 86 90 92 104 110 114 116 120 126 132 134 140 144 146 
0 2 6 12 14 20 30 32 36 42 44 50 54 56 72 74 90 92 96 102 104 110 114 116 126 132 134 140 144 146

I found these solutions on its program. It is not difficult.
Unfortunately, I have not found a solutions for k > 30.

• Questo topic è stato modificato 8 anni, 7 mesi fa da Natalia Makarova.

[Natalia Makarova]

Interesting solutions with a maximal diameter:

k=12 (my solution)
996794298566998363: 0, 40, 106, 154, 196, 256, 438, 498, 540, 588, 654, 694
d=694
Perhaps more?

k=16 (solution by D. Petukhov)
13319464281880157: 0, 42, 140, 272, 294, 360, 372, 440, 486, 554, 566, 632, 654, 786, 884, 926
d=926
Perhaps more?

I suggest a mini-contest
find solutions with a maximal diameter for k > 11.

Please submit your solutions here
or in forum
http://dxdy.ru/topic100750.html

[Natalia Makarova]

It records from colleagues

k=12
7318133876391253: 0,84,108,234,238,268,516,546,550,676,700,784
k=13
6486808502428973: 0,24,78,120,168,234,294,354,420,468,510,564,588
k=14
13319464281880199: 0,98,230,252,318,330,398,444,512,524,590,612,744,842
k=15
4956528381450799: 0,18,60,90,132,180,222,240,258,300,348,390,420,462,480
k=16
13319464281880157: 0,42,140,272,294,360,372,440,486,554,566,632,654,786,884,926
k=18
23524137017378423: 0,30,38,84,104,180,294,336,338,546,548,590,704,780,800,846,854,884
k=20
15392696329764619: 0,10,24,48,178,220,222,342,378,420,472,514,550,670,672,714,844,868,882,892
k=22
12241378636561883: 0,44,54,98,110,168,200,224,264,308,330,344,366,410,450,474,506,564,576,620,630,674
k=24
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

[Natalia Makarova]

For task #3
Example

0, 2, 12, 14, 30, 32, 42, 44, 102, 104, 114, 116, 132, 134, 144, 146
0, 2, 12, 14, 42, 44, 54, 56, 90, 92, 102, 104, 132, 134, 144, 146
0, 6, 14, 20, 30, 36, 44, 50, 96, 102, 110, 116, 126, 132, 140, 146
0, 6, 14, 20, 36, 42, 50, 56, 90, 96, 104, 110, 126, 132, 140, 146
0, 6, 20, 26, 30, 36, 50, 56, 90, 96, 110, 116, 120, 126, 140, 146
0, 6, 24, 30, 56, 60, 62, 66, 80, 84, 86, 90, 116, 122, 140, 146
0, 6, 26, 30, 32, 36, 56, 62, 84, 90, 110, 114, 116, 120, 140, 146
0, 6, 36, 42, 50, 54, 56, 60, 86, 90, 92, 96, 104, 110, 140, 146
0, 12, 14, 26, 30, 42, 44, 56, 90, 102, 104, 116, 120, 132, 134, 146
0, 12, 30, 42, 44, 56, 60, 72, 74, 86, 90, 102, 104, 116, 134, 146
0, 14, 30, 42, 44, 56, 60, 72, 74, 86, 90, 102, 104, 116, 132, 146
0, 20, 24, 42, 44, 60, 62, 66, 80, 84, 86, 102, 104, 122, 126, 146
0, 20, 30, 42, 50, 54, 62, 72, 74, 84, 92, 96, 104, 116, 126, 146

This theoretical patterns from which you are sure to get pandiagonal magic squares of order 4.
Now try to find a k-tuples according to these patterns.

[Natalia Makarova]

More examples of theoretical patterns to produce pandiagonal magic squares of order 4.

d=136

0, 6, 24, 30, 46, 52, 60, 66, 70, 76, 84, 90, 106, 112, 130, 136 
0, 10, 12, 22, 30, 40, 42, 52, 84, 94, 96, 106, 114, 124, 126, 136 
0, 10, 12, 22, 42, 52, 54, 64, 72, 82, 84, 94, 114, 124, 126, 136 
0, 10, 18, 28, 48, 58, 60, 66, 70, 76, 78, 88, 108, 118, 126, 136 
0, 10, 24, 34, 36, 46, 60, 66, 70, 76, 90, 100, 102, 112, 126, 136 
0, 10, 30, 40, 42, 52, 54, 64, 72, 82, 84, 94, 96, 106, 126, 136 
0, 12, 30, 40, 42, 52, 54, 66, 70, 82, 84, 94, 96, 106, 124, 136 
0, 24, 30, 40, 42, 54, 64, 66, 70, 72, 82, 94, 96, 106, 112, 136

d=172

0, 4, 18, 22, 60, 64, 78, 82, 90, 94, 108, 112, 150, 154, 168, 172 
0, 10, 12, 22, 42, 52, 54, 64, 108, 118, 120, 130, 150, 160, 162, 172 
0, 10, 18, 24, 28, 34, 42, 52, 120, 130, 138, 144, 148, 154, 162, 172 
0, 10, 18, 28, 60, 70, 78, 84, 88, 94, 102, 112, 144, 154, 162, 172 
0, 10, 24, 34, 54, 64, 78, 84, 88, 94, 108, 118, 138, 148, 162, 172 
0, 10, 24, 34, 60, 70, 78, 84, 88, 94, 102, 112, 138, 148, 162, 172 
0, 10, 36, 46, 60, 66, 70, 76, 96, 102, 106, 112, 126, 136, 162, 172 
0, 12, 18, 30, 42, 54, 60, 72, 100, 112, 118, 130, 142, 154, 160, 172 
0, 12, 18, 30, 70, 72, 82, 84, 88, 90, 100, 102, 142, 154, 160, 172 
0, 12, 22, 34, 48, 60, 70, 82, 90, 102, 112, 124, 138, 150, 160, 172 
0, 12, 28, 30, 40, 42, 58, 70, 102, 114, 130, 132, 142, 144, 160, 172 
0, 12, 30, 42, 58, 70, 72, 84, 88, 100, 102, 114, 130, 142, 160, 172 
0, 12, 40, 42, 52, 54, 78, 82, 90, 94, 118, 120, 130, 132, 160, 172 
0, 18, 22, 40, 42, 60, 64, 82, 90, 108, 112, 130, 132, 150, 154, 172 
0, 18, 24, 42, 60, 70, 78, 84, 88, 94, 102, 112, 130, 148, 154, 172 
0, 18, 30, 40, 48, 58, 70, 84, 88, 102, 114, 124, 132, 142, 154, 172 
0, 18, 30, 48, 54, 70, 72, 84, 88, 100, 102, 118, 124, 142, 154, 172 
0, 22, 30, 42, 52, 64, 72, 78, 94, 100, 108, 120, 130, 142, 150, 172 
0, 22, 42, 48, 60, 64, 70, 82, 90, 102, 108, 112, 124, 130, 150, 172 
0, 28, 30, 54, 58, 60, 82, 84, 88, 90, 112, 114, 118, 142, 144, 172

We have the following solution for d = 172

23653934725904299: 0, 12, 22, 34, 48, 60, 70, 82, 90, 102, 112, 124, 138, 150, 160, 172

23653934725904299 +
  0  160   60  124
 82  102   22  138
112   48  172   12
150   34   90   70

S = 94615738903617540

Please find other solutions.

[Natalia Makarova]

I got the formula for a known solution:

12829 + 30030n

A known solution is obtained for n = 787 676 814 049.
Perhaps there are other solutions according to this formula.

[Natalia Makarova]

More theoretical formula for the pattern:

0, 12, 22, 34, 48, 60, 70, 82, 90, 102, 112, 124, 138, 150, 160, 172

17029+30030n
14719+30030n
12409+30030n
10099+30030n
3379+30030n
1069+30030n
28789+30030n
26479+30030n
19759+30030n
17449+30030n
15139+30030n

Can there be a solutions?

• Questa risposta è stata modificata 8 anni, 7 mesi fa da Natalia Makarova.

[Natalia Makarova]

Interesting solutions

“There exist exactly 3 numbers n below 192*47# such that all the 16 numbers n+d where d = 0 10 12 18 22 28 30 40 42 52 54 60 64 70 72 82 are prime:

78830573871633653539 (20 digits)
94505039351105832919 (20 digits)
110732011215202177249 (21 digits)”

See http://dxdy.ru/post751870.html#p751870

d = 82 it a minimal diameter of 16-tuples, of which you can make pandiagonal square of order 4.

I found the theoretical formula for this pattern:

19489+30030n
24109+30030n
5839+30030n
10459+30030n

For example, we have the solution by the first formula:

78830573871633653539=19489+30030*2625060734986135

[Natalia Makarova]

Big congratulations Jaroslaw Wroblewski!
And thank you for unique results!

This is the best solutions:

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.

Already there are two sequences in OEIS

https://oeis.org/A266511
https://oeis.org/A266512

[Natalia Makarova]

Dear colleagues!

The project “Symmetrical tuples of consecutive primes” is in testing for BOINC
http://inferia.ru

I invite everyone to participate!

[Natalia Makarova]

Dear colleagues!

You can participate in the discussion of the project at a forum in Russia
http://mathhelpplanet.com/viewtopic.php?f=57&t=52906

You can write messages in English.

[Natalia Makarova]

Dear colleagues!

The project «Symmetrical tuples of consecutive primes» changed the name and address.
See here
http://forum.boinc.ru/default.aspx?g=posts&m=86435#post86435

The new address of the project
http://stop.inferia.ru

The new name of the project
Stop@home

• Questa risposta è stata modificata 7 anni, 1 mese fa da Natalia Makarova.

[Natalia Makarova]

Here a database, obtained in my project (before the BOINC-project)
http://forum.boinc.ru/default.aspx?g=posts&m=86353#post86353

Let me remind you:
my project “Symmetrical tuples of consecutive primes” started in forum dxdy.ru (Russia, 9 February 2015)
http://dxdy.ru/topic93581.html

[Natalia Makarova]

Dear colleagues!

You can take part in the discussion of the BOINC-project Stop@home here
http://stop.inferia.ru/forum_forum.php?id=3

You can also take part in the calculations.

[Autore]

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