Unsolved Problems “Primes Magic Squares”

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  • #123
    Natalia Makarova
    Partecipante

    I tell here about unsolved problems in the field of “Magic Squares”.

    Problem # 1
    “Pandiagonal Magic Squares of Prime Numbers”

    The competition was held on this issue (2013):
    http://www.azspcs.net/Contest/PandiagonalMagicSquares

    Lots of good solutions found winner Jarek Wroblewski.
    However the problem is not completely solved.

    For example, the optimal solution found for n = 7:

    3,7,173,223,17,197,113, 
    181,211,11,79,131,23,97, 
    43,41,149,89,137,191,83, 
    233,103,107,73,127,31,59, 
    29,167,101,19,199,67,151, 
    5,47,139,179,109,61,193, 
    239,157,53,71,13,163,37

    Magic constant S = 733.

    The solution by Jarek Wroblewski for n = 8:

    5,37,107,157,229,311,271,131, 
    73,239,397,173,197,13,113,43, 
    293,313,11,97,181,149,103,101, 
    223,83,151,71,53,241,233,193, 
    167,127,179,31,277,317,61,89, 
    67,59,139,281,269,109,17,307, 
    137,349,257,211,19,29,199,47, 
    283,41,7,227,23,79,251,337

    Magic constant S = 1248.

    It is unknown whether this solution smallest.
    And then for all n> 7 we have the same issue.

    Jarek Wroblewski found solutions also for n> 20:
    http://www.math.uni.wroc.pl/~p-k1g4/PMS/

    The best solution you can send to the website:
    http://www.azspcs.net/Contest/PandiagonalMagicSquares/FinalReport

    and here:
    http://www.primepuzzles.net/puzzles/puzz_663.htm

    Another similar problem:

    Pandiagonal magic squares of consecutive primes
    http://www.primepuzzles.net/puzzles/puzz_723.htm

    I invite everyone to take part in solving these problems.

    Links

    1. Discussion on scientific forum (in Russian)
    http://dxdy.ru/topic12959.html
    http://dxdy.ru/topic73817.html
    2. Natalia Makarova. Unconventional pandiagonal squares (series of articles). Part 1 here:
    http://www.natalimak1.narod.ru/pannetr.htm
    3. The algebraic theory of diabolic magic squares. By Barkley Rosser and R. J. Walker
    http://yadi.sk/d/tl-_Ab-o5AYhS

    #125
    Natalia Makarova
    Partecipante

    Problem # 2
    Most Perfect Magic Squares of Prime Numbers

    Definitions can be found here:
    http://www.primepuzzles.net/puzzles/puzz_671.htm

    My solutions for n = 6 and n = 8:

    149, 9161, 2309, 6701, 2609, 8861
    9067, 1483, 6907, 3943, 6607, 1783
    4139, 5171, 6299, 2711, 6599, 4871
    3229, 7321, 1069, 9781, 769, 7621
    5987, 3323, 8147, 863, 8447, 3023
    7219, 3331, 5059, 5791, 4759, 3631

    Magic constant S=29790.

    19, 5923, 1019, 4423, 4793, 1277, 3793, 2777
    4877, 1193, 3877, 2693, 103, 5839, 1103, 4339
    499, 5443, 1499, 3943, 5273, 797, 4273, 2297
    5297, 773, 4297, 2273, 523, 5419, 1523, 3919
    1213, 4729, 2213, 3229, 5987, 83, 4987, 1583
    5903, 167, 4903, 1667, 1129, 4813, 2129, 3313
    733, 5209, 1733, 3709, 5507, 563, 4507, 2063
    5483, 587, 4483, 2087, 709, 5233, 1709, 3733

    Magic constant S=24024.

    It is not known whether these solutions minimal.
    I have not found solutions for n>8.

    It is most perfect magic square of order 10 of different natural numbers:

    1, 448, 12, 441, 6, 435, 14, 446, 7, 440
    418, 33, 407, 40, 413, 46, 405, 35, 412, 41
    342, 107, 353, 100, 347, 94, 355, 105, 348, 99
    201, 250, 190, 257, 196, 263, 188, 252, 195, 258
    156, 293, 167, 286, 161, 280, 169, 291, 162, 285
    15, 436, 4, 443, 10, 449, 2, 438, 9, 444
    404, 45, 415, 38, 409, 32, 417, 43, 410, 37
    356, 95, 345, 102, 351, 108, 343, 97, 350, 103
    187, 262, 198, 255, 192, 249, 200, 260, 193, 254
    170, 281, 159, 288, 165, 294, 157, 283, 164, 289

    See my article:
    http://www.natalimak1.narod.ru/sovnetr.htm
    http://www.natalimak1.narod.ru/netradic1.htm

    #209
    Natalia Makarova
    Partecipante

    Published a new puzzle
    http://www.primepuzzles.net/puzzles/puzz_749.htm

    Maybe this will help us solve this problem.
    Dear colleagues, connect, please!

    #210
    Natalia Makarova
    Partecipante

    This is the minimal pandiagonal square of order 4 of consecutive primes:

    170693941183817 170693941183933 170693941183949 170693941183981
    170693941183979 170693941183951 170693941183847 170693941183903
    170693941183891 170693941183859 170693941184023 170693941183907
    170693941183993 170693941183937 170693941183861 170693941183889

    S=2731103058942720

    Author Max Alekseyev.

    See post on the forum in Russia
    http://dxdy.ru/post891839.html#p891839
    This problem is solved!

    I invite all to solve the following problem: find the minimal pandiagonal square of order 5 of consecutive primes.

    #211
    Natalia Makarova
    Partecipante

    S=2731103058942720

    Sorry, this is a mistake.

    correct:
    S=682775764735680

    #216
    Natalia Makarova
    Partecipante

    Hello, dear colleagues!

    I bring to your attention an interesting report by my colleague Max Alekseyev on conference MathFest 2014

    “An efficient backtracking method for solving a system of
    linear equations over a finite set with application for construction of magic squares”

    http://home.gwu.edu/~maxal/MathFest2014_slides.pdf

    #221
    Natalia Makarova
    Partecipante

    Hello, dear colleagues!

    We know two pandiagonal squares of order 4 of consecutive primes.
    This is a minimal solution by Max Alekseyev:

    170693941183817 170693941183933 170693941183949 170693941183981
    170693941183979 170693941183951 170693941183847 170693941183903
    170693941183891 170693941183859 170693941184023 170693941183907
    170693941183993 170693941183937 170693941183861 170693941183889

    S = 682775764735680

    See http://oeis.org/A245721

    The second solution is from J. Wroblewski and J. К. Andersen:

    320572022166380833 320572022166380921 320572022166380849 320572022166380917
    320572022166380909 320572022166380857 320572022166380893 320572022166380861
    320572022166380911 320572022166380843 320572022166380927 320572022166380839
    320572022166380867 320572022166380899 320572022166380851 320572022166380903

    S = 1282288088665523520

    See
    http://dxdy.ru/post751928.html#p751928
    http://www.primepuzzles.net/conjectures/conj_042.htm

    Required to find pandiagonal square of order 4 of consecutive primes with a magic constant 682775764735680 < S < 1282288088665523520.

    #222
    Natalia Makarova
    Partecipante

    Pandiagonal magic squares of consecutive primes

    n=4 (minimal, author Max Alekseyev)

    {170693941183817: 0, 30, 42, 44, 72, 74, 86, 90, 116, 120, 132, 134, 162, 164, 176, 206}
    
    170693941183817 170693941183933 170693941183949 170693941183981
    170693941183979 170693941183951 170693941183847 170693941183903
    170693941183891 170693941183859 170693941184023 170693941183907
    170693941183993 170693941183937 170693941183861 170693941183889
    
    S=682775764735680

    See http://oeis.org/A245721

    n=5, solution is unknown.

    This is my solution with 5 errors:

    {13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113}
    
    13 47 111* 89 53
    79 107 29 43 55*
    59 51* 23 97 83
    41 73 113 67 19
    121* 35* 37 17 103
    
    S=313

    n=6 (minimal)

    {67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251}
    
    67 193 71 251 109 239
    139 233 113 181 157 107 
    241 97 191 89 163 149
    73 167 131 229 151 179
    199 103 227 101 127 173
    211 137 197 79 223 83
    
    S=930

    See http://oeis.org/A073523

    n=7, solution is unknown.

    I tried to solve this problem for the next array of consecutive primes:

    {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, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239}

    This is my solution with 5 errors:

    97 167 233 179 11 103 7
    59 19 71 163 101 211 173
    157 137 89 181 23 83 127
    113 131 139 109 121* 123* 61
    191 67 53 17 229 47 193
    -17* 197 227 107 239 31 13
    197* 79 -15* 41 73 199 223
    
    S=797

    I think that such a solution exists.

    Dear Colleagues!
    I ask you to take part in solving this problem.

    See more information:

    http://dxdy.ru/post751921.html#p751921
    http://dxdy.ru/topic87170.html
    http://www.primepuzzles.net/puzzles/puzz_723.htm
    http://www.primepuzzles.net/puzzles/puzz_731.htm
    http://www.primepuzzles.net/puzzles/puzz_736.htm

    #363
    Natalia Makarova
    Partecipante

    Dear colleagues!

    Once again I remind interesting problem:

    https://primesmagicgames.altervista.org/wp/forums/topic/unsolved-problems-primes-magic-squares/#post-125

    Published sequence in the OEIS
    “Smallest magic constant of most-perfect magic squares of order 2n composed of distinct prime numbers”
    https://oeis.org/A258082

    Now we need to find a solution for n > 8.

    I invite all to solve this complex problem.

    #364
    Natalia Makarova
    Partecipante

    The scheme for most perfect square of order 10

    Here k – complementarity constant, S = 5k, if S – magic constant of a square.

    The general formula of most perfect square of order 10

    K = (2*X(36)+ X(20)- X(15)-2*X(45)+2*X(10)+2*X(26))/2 
    X(1) = (6*X(36)+5*X(20)- X(15)-2*X(45)+2*X(10)-4*X(47)-4*X(48)-4*X(49)+6*X(26))/4 
    X(11) = (2*X(36)-5*X(20)-3*X(15)-6*X(45)+2*X(10)+4*X(47)+4*X(48)+4*X(49)+2*X(26))/4 
    X(12) = X(20)+ X(45)- X(47) 
    X(13) = 2*X(36)- X(15)-3*X(45)+2*X(10)- X(48)+2*X(26) 
    X(14) = X(20)+ X(45)- X(49) 
    X(16) = (2*X(36)- X(20)-7*X(15)-6*X(45)+2*X(10)+4*X(47)+4*X(48)+4*X(49)+2*X(26))/4 
    X(17) = X(15)+ X(45)- X(47) 
    X(18) = 2*X(36)+ X(20)-2*X(15)-3*X(45)+2*X(10)- X(48)+2*X(26) 
    X(19) = X(15)+ X(45)- X(49) 
    X(2) = - X(45)+ X(10)+ X(47)
    X(21) = -(-2*X(36)-5*X(20)-3*X(15)-2*X(45)-2*X(10)+4*X(47)+4*X(48)+4*X(49))/2 
    X(22) = (2*X(36)-5*X(20)-7*X(15)-10*X(45)+2*X(10)+8*X(47)+4*X(48)+4*X(49)+6*X(26))/4 
    X(23) = -(2*X(36)-5*X(20)-7*X(15)-10*X(45)+2*X(10)+4*X(47)+4*X(49)+6*X(26))/4 
    X(24) = (2*X(36)-5*X(20)-7*X(15)-10*X(45)+2*X(10)+4*X(47)+4*X(48)+8*X(49)+6*X(26))/4 
    X(25) = (6*X(36)+5*X(20)- X(15)-2*X(45)+6*X(10)-4*X(47)-4*X(48)-4*X(49)+2*X(26))/4 
    X(27) = (6*X(36)+5*X(20)- X(15)-6*X(45)+6*X(10)-4*X(48)-4*X(49)+2*X(26))/4 
    X(28) = -(6*X(36)+5*X(20)- X(15)-6*X(45)+6*X(10)-4*X(47)-8*X(48)-4*X(49)+2*X(26))/4 
    X(29) = (6*X(36)+5*X(20)- X(15)-6*X(45)+6*X(10)-4*X(47)-4*X(48)+2*X(26))/4 
    X(3) = X(45)- X(10)+ X(48) 
    X(30) = (2*X(36)-5*X(20)-7*X(15)-6*X(45)+2*X(10)+4*X(47)+4*X(48)+4*X(49)+6*X(26))/4 
    X(31) = (-3*X(20)-5*X(15)-6*X(45)+2*X(10)+4*X(47)+4*X(48)+4*X(49)+2*X(26))/2 
    X(32) = -(-2*X(36)-5*X(20)-7*X(15)-10*X(45)+2*X(10)+8*X(47)+4*X(48)+4*X(49)+2*X(26))/4 
    X(33) = (6*X(36)- X(20)-11*X(15)-18*X(45)+10*X(10)+4*X(47)+4*X(49)+10*X(26))/4 
    X(34) = -(-2*X(36)-5*X(20)-7*X(15)-10*X(45)+2*X(10)+4*X(47)+4*X(48)+8*X(49)+2*X(26))/4 
    X(35) = (-2*X(36)- X(20)-3*X(15)-6*X(45)+2*X(10)+4*X(47)+4*X(48)+4*X(49)+2*X(26))/4 
    X(37) = (-2*X(36)- X(20)-3*X(15)-2*X(45)+2*X(10)+4*X(48)+4*X(49)+2*X(26))/4 
    X(38) = (10*X(36)+5*X(20)- X(15)-6*X(45)+6*X(10)-4*X(47)-8*X(48)-4*X(49)+6*X(26))/4 
    X(39) = (-2*X(36)- X(20)-3*X(15)-2*X(45)+2*X(10)+4*X(47)+4*X(48)+2*X(26))/4 
    X(4) = - X(45)+ X(10)+ X(49) 
    X(40) = -(-2*X(36)-5*X(20)-7*X(15)-6*X(45)+2*X(10)+4*X(47)+4*X(48)+4*X(49)+2*X(26))/4 
    X(41) = -(2*X(36)-5*X(20)-7*X(15)-10*X(45)+2*X(10)+4*X(47)+4*X(48)+4*X(49)+2*X(26))/4 
    X(42) = 2*X(36)-2*X(15)-4*X(45)+2*X(10)+ X(47)+2*X(26) 
    X(43) = -2*X(36)+2*X(15)+4*X(45)-2*X(10)+ X(48)-2*X(26) 
    X(44) = 2*X(36)-2*X(15)-4*X(45)+2*X(10)+ X(49)+2*X(26) 
    X(46) = (6*X(36)+5*X(20)- X(15)-6*X(45)+6*X(10)-4*X(47)-4*X(48)-4*X(49)+6*X(26))/4 
    X(5) = 2*X(36)-2*X(15)-2*X(45)+ X(10)+2*X(26) 
    X(50) = 2*X(36)-2*X(15)-3*X(45)+2*X(10)+2*X(26) 
    X(6) = -(2*X(36)-5*X(20)-7*X(15)-6*X(45)-2*X(10)+4*X(47)+4*X(48)+4*X(49)+2*X(26))/4 
    X(7) = 2*X(36)-2*X(15)-3*X(45)+ X(10)+ X(47)+2*X(26) 
    X(8) = -2*X(36)+2*X(15)+3*X(45)- X(10)+ X(48)-2*X(26) 
    X(9) = 2*X(36)-2*X(15)-3*X(45)+ X(10)+ X(49)+2*X(26)

    We have 9 free variables and 42 dependent variables.

    #365
    Natalia Makarova
    Partecipante

    I tried to create a most perfect square of order 10 with magic constant S = 240240
    (K = 48048), using the general formula.
    We have a large array of primes, which consists of 972 complementary pairs:

    19 31 67 71 79 97 101 109 131 137 167 179 191 211 229 239 241 251 257 269 271 307 311 331 337 347 349 367 389 409 419 439 449 457 467 479 521 
    . . . . . . . . . . . . . . . . . . . . . . . . . 
    47527 47569 47581 47591 47599 47609 47629 47639 47659 47681 47699 47701 47711 47717 47737 47741 47777 47779 47791 47797 47807 47809 47819 
    47837 47857 47869 47881 47911 47917 47939 47947 47951 47969 47977 47981 48017 48029

    However, the result was bad:

     9712 28661 13627 45341 10477 34316  4057 38231 20737 35081
    20744 36979 16829 20299 19979 31324 26399 27409  9719 30559
    33851  4522 37766 21202 34616 10177 28196 14092 44876 10942
    37589 20134 33674  3454 36824 14479 43244 10564 26564 13714
    16214 22159 20129 38839 16979 27814 10559 31729 27239 28579
    13732 43991  9817 27311 12967 38336 19387 34421  2707 37571
    16724 21649 20639 38329 17489 27304 11069 31219 27749 28069
    37871 19852 33956  3172 37106 14197 43526 10282 26846 13432
    33569  4804 37484 21484 34334 10459 27914 14374 44594 11224
    20234 37489 16319 20809 19469 31834 25889 27919  9209 31069

    K = 48048, S = 240240

    We have in this solution 53 primes and 47 are not prime numbers.

    #367
    Natalia Makarova
    Partecipante

    Progress!

     8755 28661 13627 46298 10477 33359  4057 38231 21694 35081
    21701 36979 16829 19342 19979 32281 26399 27409  8762 30559
    31937  5479 36809 23116 33659 10177 27239 15049 44876 11899
    39503 19177 34631  1540 37781 14479 44201  9607 26564 12757
    15257 22159 20129 39796 16979 26857 10559 31729 28196 28579
    14689 43991  9817 26354 12967 39293 19387 34421  1750 37571
    15767 21649 20639 39286 17489 26347 11069 31219 28706 28069
    37871 20809 32999  3172 36149 16111 42569 11239 24932 14389
    33569  3847 38441 21484 35291  8545 28871 13417 46508 10267
    21191 37489 16319 19852 19469 32791 25889 27919  8252 31069

    K = 48048, S = 240240

    This solution has 65 primes and 35 are not prime numbers.

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