Symmetrical compositions of consecutive twin primes

[Natalia Makarova]

We consider the consecutive twin primes:

(p1, p1+2), (p2, p2+2), … , (pn, pn+2)

where n > 2 and p1 < p2 < … < pn

This composition should be symmetrical.
Required for each n > 2 find the composition with a minimal value of p1.

Example:

n = 3
(5, 7), (11, 13), (17, 19)

Symmetry has the following property:

5 + 19 = 7 + 17 = 11 + 13 = 24

You can record a solution briefly:

5: 0, 2, 6, 8, 12, 14

I found the solutions for n = 4, 5, 6.

n = 4 (minimal)
663569: 0, 2, 12, 14, 18, 20, 30, 32

n = 5 (minimal)
3031329797: 0, 2, 12, 14, 42, 44, 72, 74, 84, 86

n = 6 (minimal)
17479880417: 0, 2, 30, 32, 42, 44, 60, 62, 72, 74, 102, 104

Jaroslaw Wroblewski found a solution for n = 8, but it maybe a not minimal solution:

119890755200639999: 0, 2, 42, 44, 78, 80, 90, 92, 120, 122, 132, 134, 168, 170, 210, 212

See puzzle
http://www.primepuzzles.net/puzzles/puzz_813.htm

Dear colleagues!
You can send your solutions to the website primepuzzles.net

• Questo topic è stato modificato 8 anni, 9 mesi fa da Natalia Makarova.

[Natalia Makarova]

Of symmetrical composition for n = 8 can sometimes be made pandiagonal square of order 4.

Example

119890755200639999: 0,2,42,44,78,80,90,92,120,122,132,134,168,170,210,212

119 890 755 200 639 999 +
0 170 44 210
134 120 90 80
168 2 212 42
122 132 78 92

I’ll show all similar compositions by J. Wroblewski:

119890755200639999: 0,2,42,44,78,80,90,92,120,122,132,134,168,170,210,212
1025519173619653079: 0,2,42,44,78,80,90,92,120,122,132,134,168,170,210,212
1709642327471063801: 0,2,30,32,60,62,90,92,96,98,126,128,156,158,186,188
1759943151645258947: 0,2,12,14,42,44,54,56,120,122,132,134,162,164,174,176
1960984050584219159: 0,2,30,32,42,44,48,50,72,74,78,80,90,92,120,122
3808061696393625101: 0,2,30,32,60,62,90,92,138,140,168,170,198,200,228,230
4018288550284158077: 0,2,12,14,42,44,54,56,90,92,102,104,132,134,144,146
5512467165717387017: 0,2,30,32,42,44,72,74,132,134,162,164,174,176,204,206
6118066623221589779: 0,2,30,32,42,44,72,74,78,80,108,110,120,122,150,152
6868687010299798889:
0,2,60,62,102,104,162,164,168,170,228,230,270,272,330,332
7214261446565240399: 0,2,48,50,120,122,132,134,168,170,180,182,252,254,300,302

Very interesting solutions!

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

[Natalia Makarova]

We have new solutions.

n = 7 (minimal, author D. Petukhov)

1855418882807417: 0, 2, 12, 14, 30, 32, 72, 74, 114, 116, 132, 134, 144, 146

n = 8 (possibly not minimal? Authors A. Belyshev & N. Makarova)

2640138520272677: 0, 2, 12, 14, 30, 32, 54, 56, 90, 92, 114, 116, 132, 134, 144, 146

See
http://www.primepuzzles.net/puzzles/puzz_813.htm
http://dxdy.ru/post1070606.html#p1070606

Dear colleagues!
I ask you to check the minimal solution for n = 8.
Please tell us about your check.

[Natalia Makarova]

I found some theoretical patterns for n = 9:

0  2  18  20  30  32  42  44  60  62  78  80  90  92  102  104  120  122
0  2  12  14  30  32  42  44  72  74  102  104  114  116  132  134  144  146 
0  2  12  14  30  32  54  56  72  74  90  92  114  116  132  134  144  146 
0  2  12  14  42  44  54  56  72  74  90  92  102  104  132  134  144  146 
0  2  30  32  42  44  54  56  72  74  90  92  102  104  114  116  144  146
0  2  12  14  42  44  48  50  90  92  132  134  138  140  168  170  180  182

This approximation to the solution for n = 9, wherein only two elements are not prime numbers:

2640138520272677: 0, 2, 12, 14, 30, 32, 54, 56, 72*, 74*, 90, 92, 114, 116, 132, 134, 144, 146

[Autore]

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