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.

***

Natalia wrote again on Marcjh, 30, 2022:

The following solutions for n = 8 were found in the T. Brada Experimental Grid BOINC project
 
16197229696176289: 0 4 18 22 48 52 78 82 90 94 120 124 150 154 168 172 (minimal)
62579867832657739: 0 4 78 82 84 88 120 124 168 172 204 208 210 214 288 292
135187756300435507: 0 4 30 34 96 100 102 106 180 184 186 190 252 256 282 286
197328576729627247: 0 4 42 46 60 64 90 94 102 106 132 136 150 154 192 196
215124623515323877: 0 4 6 10 42 46 90 94 102 106 150 154 186 190 192 196
216921082582106233: 0 4 66 70 126 130 144 148 156 160 174 178 234 238 300 304
224885572097732083: 0 4 6 10 66 70 114 118 126 130 174 178 234 238 240 244
257626291111063729: 0 4 24 28 78 82 84 88 144 148 150 154 204 208 228 232
486851014920118249: 0 4 30 34 48 52 84 88 114 118 150 154 168 172 198 202
632878876266807217: 0 4 60 64 90 94 126 130 150 154 186 190 216 220 276 280
704802573734987239: 0 4 24 28 54 58 84 88 120 124 150 154 180 184 204 208
757375608578469967: 0 4 6 10 72 76 132 136 150 154 210 214 276 280 282 286
770012695877078233: 0 4 24 28 84 88 90 94 150 154 156 160 216 220 240 244
817667281755374287: 0 4 42 46 66 70 72 76 150 154 156 160 180 184 222 226
848886677583996589: 0 4 24 28 90 94 150 154 174 178 234 238 300 304 324 328
854979407997444577: 0 4 12 16 42 46 96 100 126 130 180 184 210 214 222 226
909016555400879797: 0 4 42 46 72 76 126 130 156 160 210 214 240 244 282 286
1341390859265265253: 0 4 60 64 96 100 126 130 180 184 210 214 246 250 306 310
1684157406173732383: 0 4 24 28 60 64 84 88 150 154 174 178 210 214 234 238
1900838610092677369: 0 4 30 34 48 52 114 118 144 148 210 214 228 232 258 262
2347914877554064963: 0 4 84 88 96 100 114 118 126 130 144 148 156 160 240 244
2892090328244566069: 0 4 18 22 60 64 84 88 234 238 258 262 300 304 318 322
2931589102242721507: 0 4 66 70 90 94 96 100 126 130 132 136 156 160 222 226
3087867117156553099: 0 4 30 34 54 58 78 82 120 124 144 148 168 172 198 202
3118809000878091889: 0 4 54 58 84 88 180 184 198 202 294 298 324 328 378 382
 
The puzzle is not completed! It is required to find solutions for n > 8.

***

 

 

 

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