Problems & Puzzles: Puzzles

 Puzzle 534. The sum of any two terms a prime or... JM Bergot sent the following puzzle. Q1. How large a set can you find such that the sum of any two terms is a prime or twice a prime? An example is {1, 3, 10, 16}.

Contributions came from J. K. Andersen, Luis Rodríguez, Farideh Firoozbakht & Emmanuel Vantieghem & J-C Colin.

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Andersen wrote:

In puzzle 37 I found a set of 21 PRIMES such that the sum of any two
is twice a prime and all primes are distinct:
{66772273, 96208141, 4124025181, 5489743873, 117302261653, 163892796481,
181976431633, 273114794041, 577134609901, 708575135581, 2004867173041,
643966478961193, 1074902910136753, 2430714408803833, 2787742815203461,
3497947627590601, 5684113455430441, 6596939759874313, 16426063554792193,
29473090317646981, 37033804397792473}

If each number ai in my reply to c) in puzzle 37 is replaced by 2*ai+1 then
we get a set of 22 numbers (not all of them prime) such that the sum of any
two is twice a prime and all primes are distinct:
{661, 173713, 266533, 644341, 769501, 858133, 1091641, 1144993, 1450993, 1501921, 1540141, 9892661107453, 2621281732760113, 5419859456649361, 5692882369219921, 6224924154629293, 17601163839742633, 25743460863024481,
38970846380610541, 51828696468918601, 52304220066744493, 65821391040787933}

The same happens if each number ai in my reply to d) is replaced by 2*ai-1:
{2879, 86519, 198347, 820679, 850667, 920807, 948287, 956687, 1306619,
1331327, 1493879, 6443237109467, 28066985530007, 251630208854747, 353662203592847, 486906122452199, 3247021265830799, 4936365955623407,
8907480103468379, 16287793741915667, 17709809937155879, 22847423576295887}

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Luis wrote:

The largest set that I have found such that the sum of of any two terms is a
prime or twice a prime, have 5 terms. For 5 terms and a value maximum of 25 there are 12 solutions . The smallest is 1,4,9,10,13

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Farideh wrote:

Each of such sets has at most two even terms. Because sum of two even numbers which
they are congruent mod 4 is a multiple of 4 so they cannot be in the set.

The following set of 14 numbers has the mentioned property.

A = {1, 4, 6997, 54517, 66697, 1285057, 1310797, 3000225, 3533377, 8210437, 78047197, 79515517, 128073082, 40214715517}.

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Emanuel wrote:

After some computations I was able to find following solutions :

{1,13,25,61,133,193,733,4561,36853,1517506,11289961}
{1,61,301,361,2041,19261,42348,54121,73081,779521,807661,1480630}
{61,241,301,850,1021,1081,3421,8161,9232,38701,2569741,4662481,25913221,124804081}

I am convinced that there may exist solutions of arbitrary length. But I'm also convinced that it may take much more time to find a length 15 solution.

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Colin wrote:

I found a 14 numbers set {25, 28, 33, 46, 61, 421, 673, 1453, 2833, 11653, 22201, 115273, 122316493, 219929221} for which all sums of two any terms are a prime or twice a prime. I also found several 12 numbers sets which can may be prolonged :

{1, 10, 16, 37, 57, 121, 157, 34597, 51121, 419917, 974857, 6650317}
{1, 16, 21, 22, 25, 37, 841, 67477, 163837, 219661, 2038117, 6848461}
{5, 6, 17, 41, 77, 101, 257, 461, 992, 46637, 154481, 5594417}
{5, 6, 17, 41, 56, 101, 257, 881, 92357, 193457, 810437, 2757341}
{16, 21, 22, 25, 37, 841, 2881, 4237, 4657, 62617, 4467037, 7819981}
{23, 24, 35, 38, 59, 143, 239, 20303, 57263, 846539, 1029443, 3624659}
{39, 43, 58, 79, 88, 175, 799, 3823, 5419, 38803, 375223, 2359939}

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