Problems & Puzzles: Puzzles

 

Puzzle 841. Symmetrical compositions of consecutive pairs of cousin primes

Natalia Makarova sent the following puzzle:

We consider symmetrical composition of consecutive pairs of cousin primes:

(p1, p1+4), (p2, p2+4), , (pn, pn+4)

where n >1 and p1 < p2 < < pn

Required for each n >1 find the composition with a minimal value of p1.

Example:

n = 2
(7, 11), (13, 17)

Symmetry has the following property:
7 + 17 = 11 + 13 = 24

You can record a solution briefly:
7: 0, 4, 6, 10

I found the minimal solutions for n = 3 6.

n = 3
7: 0, 4, 6, 10, 12, 16

n = 4
853: 0, 4, 6, 10, 24, 28, 30, 34

n = 5
1286220583: 0, 4, 36, 40, 60, 64, 84, 88, 120, 124

n = 6
178706126107: 0, 4, 6, 10, 36, 40, 90, 94, 120, 124, 126, 130


Q. find the minimal solutions for n> 6.


Natalia wrote on Feb 5, 2020:

I found a puzzle solution
http://www.primepuzzles.net/puzzles/puzz_841.htm
for n = 7

1939807184636677: 0 4 6 10 42 46 66 70 90 94 126 130 132 136

But I'm not sure if this is the minimum solution.
I would be grateful if someone confirms the minimality.

Later she added:

I have new puzzle solutions
http://www.primepuzzles.net/puzzles/puzz_841.htm

The following solutions were found in the T. Brada Experimental Grid BOINC project

888895528231807: 0 4 6 10 42 46 66 70 90 94 126 130 132 136
1346390969722159: 0 4 30 34 144 148 204 208 264 268 378 382 408 412

https://boinc.tbrada.eu/

We are waiting for confirmation of the minimality of these solutions.

I found the following solutions

1939807184636677: 0 4 6 10 42 46 66 70 90 94 126 130 132 136
2054905758322603: 0 4 60 64 84 88 120 124 156 160 180 184 240 244
2068740148286083: 0 4 24 28 30 34 90 94 150 154 156 160 180 184

We do not yet know if there are more solutions in the interval between these solutions.

The puzzle is not completed! It is required to find solutions for n>7.

I take this opportunity to invite everyone to join the T. Brada Experimental Grid project.
This is a very interesting project.

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