Problems & Puzzles: Puzzles

Puzzle 44.- “Enoch Haga Puzzle about Consecutive Primes”

A. Find sets of k consecutive odd primes such that:

P1 + Pk + 1 = prime
P2 + Pk-1 + 1 = prime
P3 + Pk-2 + 1 = prime
etc. 

Observe that when k is odd the central prime is simply ignored.

Just to offer an interesting point of start, I have calculated the first elementary examples:

k: primes

2 : 5  7
3 : 3  5  7
4 : 5  7  11  13
5 : 3  5  7  11  13
6 : 7  11  13  17  19  23
7 : 11  13  17  19  23  29  31
8 : 17  19  23  29  31  37  41  43
9 : 19  23  29  31  37  41  43  47  53
10 : 13  17  19  23  29  31  37  41  43  47
11 : 557  563  569  571  577  587  593  599  601  607  613
12 : 137  139  149  151  157  163  167  173  179  181  191  193
13 : 10009  10037  10039  10061  10067  10069  10079  10091  10093  10099  1010 3  10111  10133
14 : 373  379  383  389  397  401  409  419  421  431  433  439  443  449
15 : 10007  10009  10037  10039  10061  10067  10069  10079  10091  10093  10099  10103  10111  10133  10139
16 : 823  827  829  839  853  857  859  863  877  881  883  887  907  911  919  929
17 : ???????????
18 : 821  823  827  829  839  853  857  859  863  877  881  883  887  907  911  919  929  937
19 : ???????????
20 : 811  821  823  827  829  839  853  857  859  863  877  881  883  887  907  911  919  929  937  941

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B. Now,  let’s add the condition that:

P1 + Pk + 1 = P2 + Pk-1 + 1 = P3 + Pk-2 + 1 = … the same prime

In this case I have obtained solutions for k=2 to 13:

k: primes (‘the same prime’)

2 : 5  7 (the same prime = 13)
3 : 3  5  7 (11)
4 : 5  7  11  13 (19)
5 : 3  5  7  11  13 (17)
6 : 7  11  13  17  19  23 (31)
7 : 61  67  71  73  79  83  89 (151)
8 : 17  19  23  29  31  37  41  43 (61)
9 : 563  569  571  577  587  593  599  601  607 (1171)
10 : 13  17  19  23  29  31  37  41  43  47 (61)
11 : 557  563  569  571  577  587  593  599  601  607  613 (1171)
12 : 137  139  149  151  157  163  167  173  179  181  191  193 (331)
13 : 93911  93913  93923  93937  93941  93949  93967  93971  93979  93983  9399 7  94007  94009 (187921)
14 and more: ?????

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Question to A : Find a set for k=17, 19, 21, and so on….

Question to B: Find an example for k=>14

Solution

Felice Russo (7/03/99) has found the least solution to k=17, 19 & 21 for Question A

For k=17 the consecutive primes are: 210097 210101 210109 210113 210127
210131 210139 210143 210157 210169 210173 210187 210191 210193 210209 210229 210233
For k=19 the consecutive primes are: 210071 210097 210101 210109 210113
210127 210131 210139 210143 210157 210169 210173 210187 210191 210193 210209 210229 210233 210241
For k=21 the consecutive primes are: 2614159 2614163 2614169 2614177
2614181 2614193 2614211 2614219 2614223 2614237 2614279 2614301 2614303 2614307 2614327 2614333 2614351 2614361 2614363 2614369 2614373

***

Felice Russo (16/04/99) wrote:
"Carlos,  about puzzle 44  I made a search for k=22 (part A) and for k=14 (part B). I didn't find any solution up to p=14.557.471 and p=14.469.817 respectively"

***

Sudipta Das has found (12/12/2002) more solutions to Question A and the first asked solutions to Question B:

For Question A:

For k=22 : 16697669  16697683  16697687  16697699  16697711  16697729  16697741  16697749  16697771  16697773  16697797  16697843  16697867  16697869  16697873  16697899  16697909  16697911  16697929  16697951  16697969  16697981 ( Sudipta Das - 6/12/02 )

For k=23 : 2614133  2614159  2614163  2614169  2614177  2614181  2614193  2614211  2614219  2614223  2614237  2614279  2614301  2614303  2614307  2614327  2614333  2614351  2614361  2614363  2614369  2614373  2614393 ( Sudipta Das - 6/12/02 )

For Question B:

For k=14 : 136450033  136450049  136450063  136450073  136450079  136450081  136450129  136450151  136450199  136450201  136450207  136450217  136450231  136450247 ( 272900281 ) ( Sudipta Das - 6/12/02 )

For k=15 : 1169151281  1169151293  1169151307  1169151311  1169151323  1169151337  1169151349  1169151359  1169151371  1169151383  1169151397  1169151409  1169151413  1169151427  1169151439 ( 2338302721 ) ( Sudipta Das - 6/12/02 )

***

Gennady Gusev wrote on Dec 12, 2025:

For Question A:
k=24, primes: 81694678441 81694678463 81694678469 81694678487 81694678493 81694678543 81694678553 81694678579 81694678589 81694678591 81694678633 81694678649 81694678717 81694678733 81694678739 81694678753 81694678757 81694678783 81694678799 81694678817 81694678859 81694678867 81694678883 81694678889
k=25, primes: 11774661169 11774661179 11774661187 11774661191 11774661203 11774661221 11774661253 11774661263 11774661277 11774661281 11774661307 11774661343 11774661373 11774661407 11774661443 11774661467 11774661473 11774661487 11774661497 11774661527 11774661553 11774661557 11774661593 11774661611 11774661623
k=26, primes: 2001623354051 2001623354089 2001623354141 2001623354201 2001623354249 2001623354273 2001623354317 2001623354323 2001623354327 2001623354371 2001623354383 2001623354387 2001623354429 2001623354431 2001623354443 2001623354459 2001623354477 2001623354503 2001623354519 2001623354531 2001623354587 2001623354593 2001623354639 2001623354719 2001623354771 2001623354779
k=27, primes: 234361988977 234361988989 234361989011 234361989073 234361989079 234361989139 234361989149 234361989161 234361989163 234361989181 234361989221 234361989223 234361989227 234361989247 234361989259 234361989263 234361989271 234361989311 234361989347 234361989349 234361989371 234361989413 234361989473 234361989479 234361989481 234361989503 234361989539
k=28, primes: 6568310280967 6568310280971 6568310280977 6568310280991 6568310281003 6568310281007 6568310281009 6568310281031 6568310281067 6568310281111 6568310281129 6568310281151 6568310281177 6568310281229 6568310281237 6568310281271 6568310281301 6568310281319 6568310281367 6568310281381 6568310281387 6568310281409 6568310281411 6568310281427 6568310281439 6568310281501 6568310281517 6568310281523
k=29, primes: 234361988951 234361988977 234361988989 234361989011 234361989073 234361989079 234361989139 234361989149 234361989161 234361989163 234361989181 234361989221 234361989223 234361989227 234361989247 234361989259 234361989263 234361989271 234361989311 234361989347 234361989349 234361989371 234361989413 234361989473 234361989479 234361989481 234361989503 234361989539 234361989541
k=31, primes: 1545440400599 1545440400641 1545440400659 1545440400689 1545440400719 1545440400733 1545440400761 1545440400817 1545440400827 1545440400853 1545440400883 1545440400913 1545440400937 1545440400967 1545440400971 1545440400979 1545440401007 1545440401031 1545440401061 1545440401097 1545440401133 1545440401157 1545440401163 1545440401223 1545440401237 1545440401277 1545440401279 1545440401289 1545440401319 1545440401429 1545440401441
k=33, primes: 1545440400593 1545440400599 1545440400641 1545440400659 1545440400689 1545440400719 1545440400733 1545440400761 1545440400817 1545440400827 1545440400853 1545440400883 1545440400913 1545440400937 1545440400967 1545440400971 1545440400979 1545440401007 1545440401031 1545440401061 1545440401097 1545440401133 1545440401157 1545440401163 1545440401223 1545440401237 1545440401277 1545440401279 1545440401289 1545440401319 1545440401429 1545440401441 1545440401447
k=35, primes: 1545440400589 1545440400593 1545440400599 1545440400641 1545440400659 1545440400689 1545440400719 1545440400733 1545440400761 1545440400817 1545440400827 1545440400853 1545440400883 1545440400913 1545440400937 1545440400967 1545440400971 1545440400979 1545440401007 1545440401031 1545440401061 1545440401097 1545440401133 1545440401157 1545440401163 1545440401223 1545440401237 1545440401277 1545440401279 1545440401289 1545440401319 1545440401429 1545440401441 1545440401447 1545440401469
 
For Question B:
k=16, primes: 74422046563 74422046579 74422046603 74422046623 74422046651 74422046663 74422046681 74422046683 74422046687 74422046689 74422046707 74422046719 74422046747 74422046767 74422046791 74422046807
k=17, primes: 13977870089 13977870091 13977870103 13977870139 13977870149 13977870157 13977870161 13977870191 13977870197 13977870199 13977870229 13977870233 13977870241 13977870251 13977870287 13977870299 13977870301
k=18, primes: 74422046551 74422046563 74422046579 74422046603 74422046623 74422046651 74422046663 74422046681 74422046683 74422046687 74422046689 74422046707 74422046719 74422046747 74422046767 74422046791 74422046807 74422046819
k=19, primes: 3300260211491 3300260211503 3300260211527 3300260211529 3300260211533 3300260211551 3300260211571 3300260211583 3300260211593 3300260211601 3300260211607 3300260211617 3300260211629 3300260211649 3300260211667 3300260211671 3300260211673 3300260211697 3300260211709
k=21, primes: 4145811882109 4145811882137 4145811882161 4145811882169 4145811882181 4145811882211 4145811882227 4145811882269 4145811882283 4145811882287 4145811882323 4145811882343 4145811882347 4145811882361 4145811882403 4145811882419 4145811882449 4145811882461 4145811882469 4145811882493 4145811882521
 
I didn't find any solutions up to 2*10^13.

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