Problems & Puzzles: Puzzles

Puzzle 707 The smallest sum for a set... In the always interesting Prime Curios! site we have the following entry: 48 is the smallest possible sum for a set of four distinct primes such that the sum of any three is prime: {5, 7, 17, 19} In another Curio we find that: 108 is the smallest possible sum for a set of six distinct primes such that the sum of any five is prime: {5, 7, 11, 19, 29, 37}. I have computed the minimal sum set solution of distinct primes such that... for K = 8, 10, 12, 14 & 16 K MinSum Set Author 4 48 {5, 7, 17, 19} Unknown 6 108 {5, 7, 11, 19, 29, 37} Honaker, Noe, Porter 8 204 {5 7 13 23 31 37 41 47} {5 7 11 13 31 37 47 53} CR 10 324 {7 11 13 17 31 41 43 47 53 61} CR 12 630 {13 23 29 37 43 53 59 61 67 73 83 89} {11 23 31 37 43 53 59 61 67 73 83 89} CR 14 630 {11 13 17 23 29 31 37 43 53 61 67 73 83 89} CR 16 1050 {11 17 29 31 37 41 53 59 67 73 79 97 103 109 113 131} {11 17 19 31 37 41 53 59 67 79 83 97 103 109 113 131} CR Q. Find the corresponding set of K distinct primes such that the sum of any K-1 is prime, for K>16, ... (as large as you can).

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Contributions came from J. K. Andersen & Jud McCranie

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Andersen wrote:

http://oeis.org/A191837 shows MinSum for K up to 5000.
MinSum for K=12 is 624: {7 11 23 31 37 47 53 61 67 83 101 103}

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Jud wrote:

Solutions for Puzzle 707 I found, first a better one for K=12:

K   Sum   Primes
12  624  {  5   7  11  17  23  37  47  53  83 101 103 137}

18 1320  { 13  17  19  23  29  31  37  41  43  61  71  83  89  97 103 107 227 229

20 1590  {  7  11  19  23  31  37  41  47  59  67  79  97 101 107 109 131 137 139 167 181}

22 2040  {  3   5   7  19  23  29  31  41  43  47  53  61  67  89 107 109 127 139 163 229 317 331}

24 2052  {  3   5   7  11  13  17  19  31  41  43  47  53  59  73  79 101 103 139 151 163 173 229 241 251}

26 2130  {  3   5   7  11  13  17  19  23  29  31  37  41  43  79  83  89  97 109 113 131 137 151 157 197 251 257}

28 2712  {  3   5  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79  83  89  97 101 103 107 673 677}

30 2714  {  3   5   7  11  13  17  19  23  29  31  37  41  43  47  53  59  61  67  71  73  79  83  89  97 101 103 107 109 617 619}

32 3750  {  3   7  11  13  23  29  31  43  47  53  67  73  83  89 103 109 113 127 131 149 151 163 167 173 191 193 211 223 227 233 251 263}

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