Problems & Puzzles: Puzzles

Puzzle 37.-  Set of even numbers { ai } such that every ai + aj + 1 is prime ( i & j are different )

I have found the following set of 14 even numbers with the property described above:

{ 4, 6, 12, 24, 54, 186, 3246, 25926, 169314,  412026, 541524, 37949286, 124716066, 324532464 }

These are the 91 primes formed:

11  17  29  59  191  3251  25931  169319  412031  541529  37949291  124716071 324532469 / 19  31  61  193  3253  25933  169321  412033  541531  37949293  124716073  324532471/ 37  67  199  3259  25939  169327  412039  541537  37949299 124 716079  324532477 / 79  211  3271  25951  169339  412051  541549  37949311  124716091  324532489 / 241  3301  25981  169369  412081  541579  37949341  124716121 324532519 /
3433
26113  169501  412213  541711  37949473  124716253  324532651/ 29173  172561  415273  544771  37952533  124719313  324535711 /  95241  437953  567451  37975213  124741993  324558391/ 581341  710839  8118601  124885381 324701779 / 953551  38361313  125128093  324944491/ 38490811  125257591  325073989 / 162665353  362481751/
449248531

Questions:

a) Find another set with 14 members

b) Does the before mentioned set of 14 members accept another valid member?

c) Find a larger set

d)  Find the largest “ Set of even numbers {ai} such that every ai +aj - 1 is prime (i !j)

Jack Brennen (4/2/99) has found solution to questions a), c) & d). Here is his communicate:

"Carlos,  I have some results on your latest puzzle!!!

In response to puzzle 37, question (c):{2 56 194 236 254 446 464 506 716 854 4016 4226 39314 56476 128156}. This set of 15 even numbers meets the requirements. Note that the rules don't require that all the primes formed be distinct; in this sequence, 194+506+1 == 236+464+1 == 254+446+1 == 701, 254+716+1 == 464+506+1 == 971, 254+4226+1 == 464+4016+1 == 4481, and 506+4226+1 == 716+4016+1 == 4733.  Therefore, this sequence does not yield a full complement of 105 distinct primes [94/105, CBRF].

Also, in response to question (a) of the same puzzle: {14 56 134 176 224 254 322 686 806 926 2564 10046 15746 100136.}This set of 14 even numbers meets the requirements. As above, some prime numbers are formed more than once by this sequence, so fewer than 91 distinct primes are created [83/91, CBRF].

In response to question (d) of the same puzzle:{12 36 62 96 102 216 348 762 846 876 1266 79806 914766} This set of 13 even numbers meets the requirements -- the sum of any two is one MORE than a prime number. Once again, some prime numbers are formed more than once, so fewer than 78 distinct primes are created [76/78, CBRF]".

***

Well, this is a very nice work! By the way Jack has pointed out that my solution - { 4, 6, 12, 24, 54, 186, 3246, 25926, 169314,  412026, 541524, 37949286, 124716066, 324532464 } - contains a condition that I was not aware of it: all the primes produced by this solution are different!!...[91/91, CBRF]

Obviously, my next question - after the Brennen work - is this:

e) Find solution to a), b), c) & d) with the added condition that all the primes produced are different.

***

I have found a solution to d) that can be a good starting point to improve:

The 14 members set is this:{16 92 136 142 298 472 1186 1732 1996 17242 61546 392332 1155562 61853122}

Its 91 distinct primes are:{107 151 157 313 487 1201 1747 2011 17257 61561 392347 1155577 61853137 227 233 389 563 1277 1823 2087 17333 61637 392423 1155653 61853213 277 433 607 1321 1867 2131 17377 61681 392467 1155697 61853257 439 613 1327 1873 2137 17383 61687 392473 1155703 61853263 769 1483 2029 2293 17539 61843 392629 1155859 61853419 1657 2203 2467 17713 62017 392803 1156033 61853593 2917 3181 18427 62731 393517 1156747 61854307 3727 18973 63277 394063 1157293 61854853 19237 63541 394327 1157557 61855117 78787 409573 1172803 61870363 453877 1217107 61914667 1547893 62245453 63008683}(CBRF, 6/2/99)

***

Jack Brennen (8/2/99) has produced a larger solution to d): a set with 15 members and 105/105 distinct primes. This is the amazing set:

{2 6 66 126 192 378 906 5922 12036 969342 2850186 19283442     29129916 32536812 53878566}

***

Wilfred Whiteside has found better (larger) solutions to c) of this puzzle. At 3/05/99 he wrote:

"Here is the 16 member set and the 120 unique primes generated by ai+aj+1

{10 200 368 608 1058 1868 3170 5678 10058 23708 50588 56918 152228 1819640 3325508 5067098}   [120/120]

Here are the 120 primes:

211 379 619 1069 1879 3181 5689 10069 23719 50599 56929 152239 1819651 3325519 5067109 / 569 809 1259 2069 3371 5879 10259 23909 50789 57119 152429 1819841 3325709 5067299 / 977 1427 2237 3539 6047 10427 24077 50957 57287 152597 1820009 3325877 5067467 / 1667 2477 3779 6287 10667 24317 51197 57527 152837 1820249 3326117 5067707 / 2927 4229 6737 11117 24767 51647 57977 153287 1820699 3326567 5068157 / 5039 7547 11927 25577 52457 58787 154097 1821509 3327377 5068967 / 8849 13229 26879 53759 60089 155399 1822811 3328679 5070269 / 15737 29387 56267 62597 157907 1825319 3331187 5072777 / 33767 60647 66977 162287 1829699 3335567 5077157 / 74297
80627 175937 1843349 3349217 5090807 / 107507 202817 1870229 3376097 5117687 / 209147 1876559 3382427 5124017 / 1971869 3477737 5219327 / 5145149 6886739 / 8392607

For the same c) he also found 21 sets of 15 members that produce 105/105 disctinct primes each.

***

Four years later (Set. 2003) W. Whiteside, got new records. He also propose new puzzling questions tied to this puzzle.

2 perfect solutions were found containing 17 numbers (generating 136 unique primes).

Solution 1 with 17 numbers:

116 200 440 446 1910 3560 7922 9806 11386 38666 43826 67766 122030 123290 1269896 7361840 67471130 [136/136] - ***perfect

Which generates the unique primes listed below:

317 557 563 2027 3677 8039 9923 11503 38783 43943 67883 122147 123407 1270013 7361957 67471247 / 641 647 2111 3761 8123 10007 11587 38867 44027 67967 122231 123491 1270097 7362041 67471331 / 887 2351 4001 8363 10247 11827 39107 44267 68207 122471 123731 1270337 7362281 67471571 / 2357 4007 8369 10253 11833 39113 44273 68213 122477 123737 1270343 7362287 67471577 / 5471 9833 11717 13297 40577 45737 69677 123941 125201 1271807 7363751 67473041 / 11483 13367 14947 42227 47387 71327 125591 126851 1273457 7365401 67474691 / 17729 19309 46589 51749 75689 129953 131213 1277819 7369763 67479053 / 21193 48473 53633 77573 131837 133097 1279703 7371647 67480937 / 50053 55213 79153 133417 134677 1281283 7373227 67482517 / 82493 106433 160697 161957 1308563 7400507 67509797 / 111593 165857 167117 1313723 7405667 67514957 / 189797 191057 1337663 7429607 67538897 / 245321 1391927 7483871 67593161 / 1393187 7485131 67594421 / 8631737 68741027 / 74832971

Solution 2 with 17 numbers:

6 52 414 1164 1206 1854 2946 8184 15024 31326 48474 131424 150564 1928394 7353474 21732474 72960624 [136/136] - ***perfect

Which generates the unique primes listed below:

59 421 1171 1213 1861 2953 8191 15031 31333 48481 131431 150571 1928401 7353481 21732481 72960631 / 467 1217 1259 1907 2999 8237 15077 31379 48527 131477 150617 1928447 7353527 21732527 72960677 / 1579 1621 2269 3361 8599 15439 31741 48889 131839 150979 1928809 7353889 21732889 72961039 / 2371 3019 4111 9349 16189 32491 49639 132589 151729 1929559 7354639 21733639 72961789 / 3061 4153 9391 16231 32533 49681 132631 151771 1929601 7354681 21733681 72961831 / 4801 10039 16879 33181 50329 133279 152419 1930249 7355329 21734329 72962479 / 11131 17971 34273 51421 134371 153511 1931341 7356421 21735421 72963571 / 23209 39511 56659 139609 158749 1936579 7361659 21740659 72968809 / 46351 63499 146449 165589 1943419 7368499 21747499 72975649 / 79801 162751 181891 1959721 7384801 21763801 72991951 / 179899 199039 1976869 7401949 21780949 73009099 / 281989 2059819 7484899 21863899 73092049 / 2078959 7504039 21883039 73111189 / 9281869 23660869 74889019 / 29085949 80314099 / 94693099

Also, attached are the 495 sets of 16 [not shown here]. My favorite two are shown below. I like them because the numbers are so small.

2 128 464 518 854 1094 3388 6788 8108 11804 14588 16088 21374 74888 90398 212504 [118/120] 66 246 480 540 690 1306 3366 4110 15126 16500 27210 99756 146850 148182 166116 251460 [120/120] - ***perfect

PPS. Have you ever tried this same puzzle, but where Ai+Aj+1 and Ai+Aj-1 must both be prime (i.e.. a twin prime generator)? Or where Ai+Aj+Ak+1 must be prime for all triplets?

***

J. K. Andersen wrote (Dec. 08):

b) The original set of 14 members accepts many new members. Infinitely many
according to the k-tuple conjecture. The smallest is 130240985189886.
The set accepts these 5 numbers together to give a set with 19 members:
{15350434870543764, 23356976320119804, 55328077307324106, 75101700483381696, 379945206648412494}
Another case of 5 accepted numbers:
{16568293072203786, 83106156195399504, 115618940968754466, 273778741223561004, 369396131124243024}

The 171 produced primes are distinct in both cases.

c) A set with 22 members:
{330, 86856, 133266, 322170, 384750, 429066, 545820, 572496, 725496, 750960,
770070, 4946330553726, 1310640866380056, 2709929728324680, 2846441184609960,
3112462077314646, 8800581919871316, 12871730431512240, 19485423190305270,
25914348234459300, 26152110033372246, 32910695520393966}
The 231 produced primes are distinct.
Around 15000 sets with 21 members and one other with 22 were found.

d) A set with 22 members:
{1440, 43260, 99174, 410340, 425334, 460404, 474144, 478344, 653310, 665664,
746940, 3221618554734, 14033492765004, 125815104427374, 176831101796424,
243453061226100, 1623510632915400, 2468182977811704, 4453740051734190,
8143896870957834, 8854904968577940, 11423711788147944}
The 231 produced primes are distinct.
Around 13000 sets with 21 members and one other with 22 were found.

Whiteside suggested a variant with twin primes.
A set with 11 members where ai + aj +/- 1 is prime for i and j different:
{71556, 2330736, 4008096, 5486196, 5747316, 2780309060586, 2561228504876826,
16981826091904116, 45374308604816676, 49582677199072956, 58745659496092626}
The 55 produced twin prime pairs are distinct.
Around 500000 sets with 10 members and 86 other with 11 were found.

Primality proofs were only peformed for the sets listed above.

***

I noticed a connection to a pattern in "Prime number patterns" by Andrew
Granville: http://www.dms.umontreal.ca/~andrew/PDF/PrimePatterns.pdf.
Among other things, Granville studies sets of n primes whose pairwise
averages are all distinct primes.

If ai+aj+1 in puzzle 37 is also prime for i=j, meaning that 2*ai+1 is prime
for each i, then the set of 2*ai+1 satisfies Granville's average pattern:
((2*ai+1) + (2*aj+1))/2 = ai+aj+1.
Here is such a set with 21 members where 2*ai+1 is listed:
{66772273, 96208141, 4124025181, 5489743873, 117302261653, 163892796481,
181976431633, 273114794041, 577134609901, 708575135581, 2004867173041,
643966478961193, 1074902910136753, 2430714408803833, 2787742815203461,
3497947627590601, 5684113455430441, 6596939759874313, 16426063554792193,
29473090317646981, 37033804397792473}

The 21 primes and 210 pairwise averages are distinct.
Granville lists the sets with the smallest largest element for n up to 12.
37033804397792473 is far from the smallest for n=21.

***

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