Symmetrical compositions of twin primes

[Natalia Makarova]

We consider the twin primes

(p1, p1+2), … (p2, p2+2), …, (p3, p3+2), … , (pn, pn+2)
where n > 2 and p1 < p2 < p3 < … < pn

Between the twin primes may be other primes for which we do not pay attention.
Then formed composition (p1, p2, p3, …, pn), which should be symmetrical.
Required for each n > 2 find the composition with a minimal value of p1.

Examples:

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

Symmetrical composition:

[5, 11, 17]
5+17=2*11

n=4
(29, 31), (41, 43), (59, 61), (71, 73)

Symmetrical composition:

[29, 41, 59, 71]
29+71=41+59

I found the following solutions:

n=5
[155861, 155891, 156059, 156227, 156257]
n=6
[59, 71, 101, 107, 137, 149]
n=7
[227927459, 227927597, 227927639, 227927699, 227927759, 227927801, 227927939]
n=8
[41387, 41411, 41519, 41609, 41759, 41849, 41957, 41981]
n=9
[54793185527, 54793185659, 54793185989, 54793186169, 54793186559, 54793186949, 54793187129, 54793187459, 54793187591]
n=10
[34623805211, 34623805421, 34623805787, 34623806249, 34623806771, 34623807017, 34623807539, 34623808001, 34623808367, 34623808577]

You can record solutions briefly in the following form:
n=10
34623805211: 0, 210, 576, 1038, 1560, 1806, 2328, 2790, 3156, 3366

Solutions for n = 9 are required to solve the puzzle
http://www.primepuzzles.net/puzzles/puzz_769.htm
for n=3.

I have another solutions for n = 9:

354584248349: 0, 132, 372, 678, 900, 1122, 1428, 1668, 1800
388743941039: 0, 42, 240, 282, 450, 618, 660, 858, 900
403147629431: 0, 126, 420, 750, 768, 786, 1116, 1410, 1536
463060598321: 0, 390, 906, 1116, 1218, 1320, 1530, 2046, 2436
584591273177: 0, 372, 744, 1122, 1152, 1182, 1560, 1932, 2304

But I have not got a magic square of order 3.

Dear Colleagues!
Please take part in solving the problem.
Required:
1. Find the minimal solutions for n > 10
2. Find more solutions for n = 9.

[Natalia Makarova]

Should you find many solutions of the problem for n = 8, you can make pandiagonal squares of order 4.
For example, symmetrical composition

71580585467: 0, 180, 420, 600, 1194, 1374, 1614, 1794

transform into the next symmetrical composition:

71580585467: 0, 2, 180, 182, 420, 422, 600, 602, 1194, 1196, 1374, 1376, 1614, 1616, 1794, 1796

This symmetrical composition gives the following pandiagonal square of order 4:

71580585467 +

   0 1794  422 1376
 602 1196  180 1614
1374  420 1796    2
1616  182 1194  600

[Natalia Makarova]

I found another solution for n = 9

1110317288231: 0, 450, 648, 756, 1038, 1320, 1428, 1626, 2076

But the magic square of order 3 of these numbers will not be.

I continue to search.

[Natalia Makarova]

The problem is published here
http://www.primepuzzles.net/puzzles/puzz_807.htm

You can send your solutions to this site.

I found another solution for n = 9
2007253835681: 0, 6, 420, 1896, 1938, 1980, 3456, 3870, 3876

But the magic square of order 3 is not received.

[Natalia Makarova]

I found another solution for n = 9
2188700058659: 0, 792, 1038, 1428, 1590, 1752, 2142, 2388, 3180

But the magic square of order 3 is not received.

[Natalia Makarova]

Jaroslaw Wroblewski (Jarek) said at a forum in Russia:

«I have found the following solutions to the problem of 3×3 magic square of consecutive twins:

204860134660098317297: 0, 42, 60, 84, 102, 120, 144, 162, 204
422229725797687239077: 0, 42, 84, 120, 162, 204, 240, 282, 324
5646440666838544810187: 0, 42, 84, 210, 252, 294, 420, 462, 504
6082062789438398013049: 0, 12, 24, 240, 252, 264, 480, 492, 504

Diameter 204 is the smallest possible, while my approach cannot find minimal solution (with respect to size of primes).»

See
http://dxdy.ru/post1071800.html#p1071800

Very interesting solutions! However, the problem remains minimal solution.

• Questa risposta è stata modificata 8 anni, 10 mesi fa da Natalia Makarova.

[Natalia Makarova]

Refinement
6082062789438398013047: 0, 12, 24, 240, 252, 264, 480, 492, 504

[Autore]

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