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Magic squares of twin primes

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Questo argomento contiene 6 risposte, ha 1 partecipante, ed è stato aggiornato da  Natalia Makarova 3 anni, 11 mesi fa.

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  • #263

    Natalia Makarova
    Partecipante

    Magic squares of twin primes

    See
    http://www.primepuzzles.net/puzzles/puzz_080.htm

    n=3 (minimal, author Radko Nachev)

    239 17  191
    101 149 197
    107 281 59

    S=447

    Magic square of the second number-twins (+2):

    241 19  193
    103 151 199
    109 283 61

    S=453

    I found the following solutions:

    n=4 (minimal)

    17  11  419 137
    269 227 59  29
    107 197 101 179
    191 149 5   239

    S=584

    n=5 (minimal)

    5   11  617 179 281
    71  311 101 191 419
    239 461 149 17  227
    347 41  29  569 107
    431 269 197 137 59

    S=1093

    n = 6 (minimal)

    5    17  1049 11  857 419 
    29   881 41   59  521 827 
    149  659 569  227 137 617 
    347  431 281  641 461 197 
    1019 101 239  821 71  107 
    809  269 179  599 311 191

    S=2358

    I have not found a solution for n = 7.
    Next, I already have a solution for n = 8 (minimal), n = 9 (not minimal ?), n = 10 (minimal).

    Anybody can find a solution for n = 7?

    I found pandiagonal square 7th order of primes twins.
    See http://www.primepuzzles.net/puzzles/puzz_689.htm

    But this magic square has a very large magic constant.

    #265

    Natalia Makarova
    Partecipante

    My solutions to the puzzle # 80

    n=8 (minimal)

    179, 419, 1277, 239, 1721, 1451, 71, 1997,
    1229, 821, 599, 1667, 191, 1319, 1301, 227,
    107, 2141, 41, 2027, 281, 809, 1931, 17,
    29, 1289, 1487, 101, 641, 1787, 149, 1871,
    1949, 461, 1619, 857, 1091, 311, 5, 1061,
    2111, 827, 431, 569, 521, 1607, 1151, 137,
    1481, 1049, 881, 197, 1031, 11, 2087, 617,
    269, 347, 1019, 1697, 1877, 59, 659, 1427

    S=7354

    n=9 (not minimal ?)

    1319, 5, 107, 71, 3389, 2027, 2081, 3251, 29,
    11, 2129, 1229, 521, 3119, 1289, 881, 311, 2789,
    101, 17, 1049, 2999, 179, 3371, 2339, 2087, 137,
    1787, 617, 431, 1931, 239, 197, 2729, 1019, 3329,
    1277, 2381, 1481, 191, 2267, 1667, 1301, 1487, 227,
    3167, 461, 3299, 1451, 281, 1061, 821, 1697, 41,
    3257, 2711, 1877, 809, 659, 641, 347, 827, 1151,
    1091, 3539, 857, 1619, 149, 1427, 59, 569, 2969,
    269, 419, 1949, 2687, 1997, 599, 1721, 1031, 1607

    S=12279

    n=10 (minimal)

    41, 2657, 2129, 149, 1997, 3539, 827, 2381, 1787, 1277,
    1721, 3371, 3851, 179, 3359, 599, 1451, 137, 2087, 29,
    1619, 1667, 3671, 71, 191, 2729, 2267, 2081, 1871, 617,
    3461, 227, 1301, 461, 17, 1877, 3581, 11, 3299, 2549,
    1151, 2801, 1487, 2111, 1031, 431, 101, 2969, 881, 3821,
    197, 659, 59, 3119, 3257, 2339, 3557, 2789, 569, 239,
    3389, 1481, 2309, 1949, 1319, 311, 1061, 1931, 1427, 1607,
    419, 2999, 821, 2591, 857, 2711, 1697, 269, 1091, 3329,
    3767, 281, 347, 2687, 1229, 2141, 5, 1049, 3251, 2027,
    1019, 641, 809, 3467, 3527, 107, 2237, 3167, 521, 1289

    S=16784

    See
    http://www.primepuzzles.net/puzzles/puzz_080.htm

    The non-minimal pandiagonal 7×7 square of twin primes:

    17, 5279, 7589, 37361, 3371, 44069087, 17189,
    34031, 44066819, 6197, 13679, 4019, 1319, 13829,
    17681, 59, 7559, 10499, 44097479, 3929, 2687,
    44073947, 34589, 419, 6689, 13721, 6299, 4229,
    2729, 19961, 2969, 44067677, 11057, 31079, 4421,
    4787, 7547, 35081, 461, 8969, 16631, 44066417,
    6701, 5639, 44080079, 3527, 1277, 11549, 31121

    S=44139893.

    See
    http://www.primepuzzles.net/puzzles/puzz_689.htm

    Is required for puzzle # 80:

    1. find a solution for n = 7 with a magic constant S < 44139893;
    2. find a solution for n = 9 with a magic constant S < 12279.

    #267

    Natalia Makarova
    Partecipante

    I found a solution!

    n = 7 (not minimal ?)

    821, 281, 599, 347, 1451, 827, 1667, 
    29, 659, 5, 227, 1949, 1427, 1697, 
    881, 1721, 101, 1607, 311, 1301, 71, 
    1787, 1061, 1229, 239, 857, 179, 641, 
    1049, 521, 569, 1877, 1091, 269, 617, 
    107, 461, 1871, 1277, 197, 1931, 149, 
    1319, 1289, 1619, 419, 137, 59, 1151

    S = 5993

    My algorithm

    Here you can see an scheme of the magic square of order 7:

    ai (i = 1, 2, 3, …, 34) – independent variables, xk (k = 1, 2, 3, …, 15) – dependent variables.
    Magic constant S is given.

    The general formula of the magic square of order 7:

    x1=S-a1-a2-a3-a4-a5-a6
    x2=S-a7-a8-a9-a10-a11-a12
    x3=S-a6-a12-a16-a18-a21-a26
    x4=S-a1-a8-a14-a18-a22-a30
    x5=S-x3-a31-a32-a33-a34-x4
    x6=S-a25-a26-a27-a28-a29-a30
    x7=S-a6-x2-a17-a24-x6-x4
    x8=S-a4-a10-a15-a18-a28-a32
    x9=S-a5-a11-a16-a22-a29-a33
    x10=S-a3-a9-a14-a21-a27-a31
    x11=S-x1-a12-a19-a23-a30-a34
    x12=S-a13-a14-a15-a16-x11-a17
    x13=S-a1-a7-x12-a20-a25-x3
    x14=S-x13-x10-a18-x9-a19-x7
    x15=S-a2-a8-a13-x14-a26-x5

    For details see the article:
    http://www.natalimak1.narod.ru/formul2.htm

    Now we need to find a solution for n = 7 with a magic constant S < 5993.

    See also
    http://dxdy.ru/post942305.html#p942305

    #268

    Natalia Makarova
    Partecipante

    I found a minimal solution for n = 7.

    419, 1061, 881, 71, 569, 107, 1301,
    17, 641, 821, 179, 1031, 1289, 431,
    1427, 41, 269, 1151, 191, 521, 809,
    1229, 857, 461, 659, 827, 137, 239,
    599, 1607, 347, 1319, 281, 29, 227,
    101, 5, 1481, 11, 1451, 1049, 311,
    617, 197, 149, 1019, 59, 1277, 1091

    S=4409

    Now we need to find a solution for n = 9 with a magic constant S < 12279.
    There is a solution?

    #269

    Natalia Makarova
    Partecipante

    Progress!

    1619, 1487, 179, 2027, 617, 827, 1949, 2657, 11,
    1151, 2549, 191, 1061, 2687, 599, 1697, 1289, 149,
    821, 269, 239, 2339, 857, 29, 1319, 2789, 2711,
    1931, 1277, 2801, 1091, 641, 2111, 1019, 461, 41,
    2267, 1427, 809, 431, 1301, 1871, 659, 521, 2087,
    1787, 1229, 1877, 2309, 311, 2081, 17, 281, 1481,
    1031, 59, 2729, 101, 227, 1997, 881, 3299, 1049,
    569, 107, 2129, 1667, 2141, 137, 2381, 5, 2237,
    197, 2969, 419, 347, 2591, 1721, 1451, 71, 1607

    S = 11373

    I guess it is a minimal solution.

    #272

    Natalia Makarova
    Partecipante

    Published a new puzzle:

    Magic squares and consecutive twin primes

    http://www.primepuzzles.net/puzzles/puzz_769.htm

    I invite all to solve this problem for n = 3, 4, 7.

    #273

    Natalia Makarova
    Partecipante

    My colleague Serg Zorkin found minimal solution for n = 7

    431  2267 2237 347  1487 2087 419
    2129 2027 569  461  1427 2141 521
    857  827  1931 1721 881  1607 1451
    1289 821  809  1949 1871 659  1877
    1091 1667 1787 1019 1061 1031 1619
    1997 617  641  2081 1229 599  2111
    1481 1049 1301 1697 1319 1151 1277

    S=9275

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