Problems & Puzzles: Puzzles

Puzzle 813. Symmetrical compositions of consecutive twin primes

Natalia Makarova sent the following nice puzzle:

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

Q. find the minimal solutions for n > 6.

 


Natalia Makarova wrote on Jan 7, 2016

n=7 (minimal, author D. Petukhov)

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

See

http://dxdy.ru/post1070606.html#p1070606

 

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

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Vasiliy Danilov wrote on Jan 19, 2016:

Dmitry found the minimum values for pairs of twins in September 2015

n=8 (minimal, author D. Petukhov)

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

See ( only a number of prime numbers rather than pairs of twins )
http://dxdy.ru/post1050824.html#p1050824

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Natalia Makarova is looking for the minimal for n=8 in order to confirm or discard the Petukhov claim

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