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Order 4 set of primes

Home Forum Magic Cubes of Prime Numbers Order 4 set of primes

Questo argomento contiene 11 risposte, ha 2 partecipanti, ed è stato aggiornato da  Natalia Makarova 3 anni, 11 mesi fa.

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  • #107

    primesmagicgames
    Amministratore del forum

    Those here are sequences of 64 primes numbers that teoretically can be used to form a cube with the given MAGIC constant.
    You can use them as a starts for searching into cubes constructions:

    
    MAGIC=828
    191 7 241 331 53 193 367 101 379 127 359 349 419 173 239 389 257 443 19 277 41 5 107 71 97 103 29 89 67 439 47 449 337 109 307 211 409 269 17 251 59 79 227 199 233 149 61 283 23 457 421 373 11 229 137 83 383 313 31 73 139 401 223 293 
    
    MAGIC=780
    293 263 449 173 23 149 257 53 3 7 97 421 137 139 61 83 433 127 439 443 89 419 277 151 211 179 47 13 251 11 31 101 109 227 307 107 17 349 281 457 73 233 241 103 37 397 5 379 229 59 409 239 79 337 163 283 331 19 431 167 271 71 41 199 
    
    MAGIC=746
    307 13 281 271 97 181 151 379 53 191 19 293 109 73 257 167 157 197 5 139 37 311 383 71 107 89 79 251 137 11 389 229 223 59 179 3 113 163 283 17 41 233 239 127 317 349 409 367 67 439 347 401 103 47 457 331 313 241 131 31 211 7 61 193 
    
    MAGIC=740
    113 17 241 131 277 31 281 13 211 43 5 367 193 103 179 101 67 293 61 383 149 449 11 47 7 157 199 19 151 127 59 71 331 89 173 251 443 191 181 389 353 373 197 263 227 73 167 109 359 223 419 163 433 97 349 37 83 139 311 239 233 53 79 257 
    
    MAGIC=736
    317 211 67 13 131 311 127 47 199 109 37 353 101 59 97 263 79 293 139 29 7 191 179 19 449 163 5 31 269 197 71 307 229 241 397 113 421 251 149 83 271 419 373 43 277 73 41 283 3 61 359 11 433 257 409 383 193 103 233 181 89 281 23 223 
    
    MAGIC=712
    307 181 293 67 311 281 197 73 337 59 113 167 53 239 349 131 101 41 421 233 389 157 139 151 313 71 47 3 149 229 277 439 7 37 29 347 359 23 97 193 11 199 107 409 379 83 109 241 89 103 367 43 179 5 191 13 397 79 173 19 163 263 223 137 
    
    MAGIC=706
    421 17 193 199 229 11 3 389 31 443 7 131 239 331 167 71 41 277 37 223 283 29 67 281 137 23 337 97 109 397 383 83 157 181 179 103 113 127 101 59 317 107 13 233 367 163 89 53 47 151 43 79 149 173 439 241 73 373 61 293 227 449 311 139 
    
    MAGIC=696
    79 109 43 37 317 229 271 59 107 101 139 397 263 71 233 211 331 47 389 17 7 191 251 157 3 89 239 199 227 83 137 103 173 163 67 29 193 421 337 41 353 443 311 151 13 11 409 97 61 31 23 179 113 283 53 19 293 73 5 269 257 223 457 449 
    
    MAGIC=692
    103 53 151 107 5 23 307 281 139 137 173 239 29 277 3 227 43 181 83 113 61 271 101 109 229 31 257 131 179 421 353 359 431 79 373 233 127 37 17 263 163 47 157 251 443 59 191 13 379 67 197 71 41 241 11 89 419 7 149 337 313 283 389 19 
    
    MAGIC=688
    313 181 59 37 53 409 79 29 179 271 73 113 331 149 439 389 419 17 433 11 89 227 173 367 127 251 311 317 233 61 151 157 13 101 397 191 41 5 67 83 443 103 71 229 197 383 23 47 137 139 97 43 379 131 263 7 257 163 31 3 223 107 167 19 
    
    MAGIC=682
    137 47 83 197 59 23 283 67 163 103 73 37 7 239 173 419 5 233 29 61 89 353 409 193 31 431 227 3 439 149 43 13 127 109 167 97 101 359 271 151 293 311 113 107 223 241 389 79 281 397 53 11 263 257 277 41 211 17 421 191 19 181 157 179
    
    MAGIC=674
    199 23 3 211 79 181 139 347 269 137 163 283 281 271 179 409 191 89 193 67 197 7 29 43 73 257 397 107 101 127 41 5 83 293 419 227 173 103 239 233 59 11 251 109 17 97 223 313 61 53 131 19 331 47 149 389 113 229 241 13 421 31 151 457 
    
    MAGIC=668
    383 19 181 367 89 283 43 197 67 157 101 97 263 127 131 137 29 233 359 317 149 139 113 239 251 11 163 41 53 79 241 229 337 223 23 73 269 13 3 401 277 47 293 109 17 349 281 37 103 5 379 313 59 227 179 83 151 71 7 311 61 419 173 107 
    
    MAGIC=658
    223 197 13 367 5 149 211 59 157 167 349 71 3 317 151 37 191 67 97 17 181 31 199 127 47 107 103 29 73 19 139 389 137 373 257 353 43 11 311 233 7 83 101 53 383 173 89 229 239 293 449 401 163 307 277 23 227 113 41 193 79 61 263 271 
    
    MAGIC=652
    79 109 5 179 211 181 71 61 41 311 157 379 151 97 439 89 229 241 23 113 31 293 307 127 131 257 389 173 167 29 191 239 7 101 103 37 317 199 383 149 19 83 13 107 139 59 67 269 193 457 283 233 401 251 197 271 263 137 43 47 11 3 17 73 
    
    MAGIC=648
    31 17 127 89 197 367 251 359 149 263 47 61 3 37 191 13 101 397 223 67 71 43 109 457 11 331 7 139 97 83 313 53 113 283 317 131 241 5 227 59 103 229 421 311 271 193 23 151 157 73 41 29 277 181 79 107 167 409 281 19 199 173 257 137 
    
    MAGIC=646
    167 5 419 181 449 3 41 103 257 131 107 113 191 157 67 241 397 199 139 101 83 229 97 47 37 281 277 383 227 23 179 197 31 353 349 17 71 173 373 233 239 151 317 379 193 53 211 7 61 59 109 29 89 43 19 127 283 11 149 13 163 293 137 73 
    
    MAGIC=642
    137 83 373 241 229 13 379 127 17 73 41 173 293 337 59 151 193 457 167 197 113 89 7 3 283 131 179 353 257 331 47 107 163 31 139 61 97 251 359 67 199 11 349 53 29 23 157 101 211 79 223 109 5 233 191 269 271 277 103 263 71 181 37 19 
    
    MAGIC=638
    181 179 191 257 239 3 151 113 37 67 109 23 61 227 29 83 7 13 211 283 233 269 313 443 107 41 383 317 17 397 137 11 229 173 149 73 97 31 373 103 127 263 271 131 199 5 449 89 347 43 71 139 251 59 163 101 53 197 223 47 281 79 19 241
    
    MAGIC=634
    197 281 331 233 401 167 379 11 149 257 83 89 37 23 359 179 199 103 59 139 61 3 151 53 229 239 173 211 349 113 101 227 317 7 191 97 71 79 353 421 73 29 13 41 47 43 5 181 17 263 137 19 307 277 107 67 241 271 157 131 163 293 31 109
    
    MAGIC=632
    149 157 101 197 199 131 127 311 137 11 113 347 47 293 53 3 23 41 167 31 401 89 251 389 83 13 73 239 7 163 191 5 409 241 43 257 181 313 97 173 179 269 139 67 37 79 233 59 29 317 151 19 263 17 193 107 271 367 211 61 223 229 109 227
    
    MAGIC=626
    71 127 307 83 373 359 353 149 281 101 31 131 239 137 379 421 61 73 53 37 47 17 211 223 41 79 269 251 103 227 89 151 11 43 7 139 257 107 337 97 271 277 167 3 29 67 5 191 311 199 23 13 59 179 163 193 113 197 109 283 263 19 181 229
    
    MAGIC=622
    347 103 367 109 211 97 379 239 89 199 17 373 191 23 433 41 233 163 179 137 317 13 139 31 37 29 197 43 71 293 227 193 11 151 59 61 277 73 79 167 107 83 251 257 7 181 157 281 131 113 223 269 263 389 5 47 127 271 19 149 3 101 67 53
    
    MAGIC=620
    53 3 181 97 233 197 37 19 173 113 11 13 229 281 109 199 31 43 239 101 331 17 337 59 79 409 131 107 179 103 419 41 167 293 373 257 283 47 23 7 151 89 163 149 73 347 137 67 5 271 383 211 191 127 223 61 193 29 263 139 157 83 313 71
    
    MAGIC=588
    227 43 277 71 29 197 293 137 79 109 317 269 223 263 257 283 5 307 241 281 7 67 173 13 59 19 101 229 139 3 251 179 167 127 89 73 211 103 53 41 239 233 97 151 149 31 311 37 83 107 163 61 11 17 199 157 271 193 313 191 131 23 181 47
    
    MAGIC=580
    179 19 139 173 137 103 227 229 11 131 233 47 193 223 151 73 181 263 83 191 197 67 307 251 281 7 293 53 199 149 59 211 5 269 17 79 277 317 37 157 31 89 271 107 71 163 311 23 113 313 97 61 41 3 109 167 283 101 257 13 43 29 127 239
    
    MAGIC=592
    137 5 173 233 41 331 257 67 283 83 241 311 281 109 157 229 167 163 13 179 23 3 71 43 59 61 17 53 307 199 29 317 211 181 191 107 89 151 11 313 269 37 47 193 113 139 149 101 251 131 7 293 271 97 197 19 73 223 79 227 263 31 127 239
    
    MAGIC=582
    23 233 307 29 199 7 103 277 113 43 97 61 227 241 223 281 283 239 101 173 37 137 67 211 127 53 109 181 157 313 41 293 107 251 59 257 131 179 19 13 269 89 79 47 139 31 11 317 5 167 71 163 197 149 193 17 3 263 191 271 73 331 151 83
    
    MAGIC=578
    277 107 199 43 7 5 61 173 193 103 313 101 331 229 41 29 139 13 167 191 53 127 113 137 19 31 151 149 67 211 97 163 197 271 11 23 307 263 233 37 181 3 47 227 89 83 317 239 17 241 283 281 223 157 71 251 179 131 257 59 73 269 79 109
    
    
    #109

    Natalia Makarova
    Partecipante

    I suggest this pattern for magic cube of order 4:

    2 1 1 2
    1 1 2 2
    1 2 2 1
    2 2 1 1
    
    1 1 2 2
    2 1 2 1
    1 2 1 2
    2 2 1 1
    
    1 2 2 1
    1 2 1 2
    0 1 1 1
    1 1 2 2
    
    2 2 1 1
    2 2 1 1
    1 1 2 2
    1 1 2 2

    This pattern is composed of residues modulo 3.

    #118

    Natalia Makarova
    Partecipante

    I found a solution for n = 4, S = 750:

    11  7  283  449 
    193  457  71  29 
    439  59  89  163 
    107  227  307  109 
    
    31  61  389  269 
    443  67  101  139 
    79  383  157  131 
    197  239  103  211 
    
    277  419  41  13 
    97  53  241  359 
    3  127  421  199 
    373  151  47  179 
    
    431  263  37  19 
    17  173  337  223 
    229  181  83  257 
    73  133*  293  251

    There is one element of the wrong – 133 is not a prime number.
    This solution corresponds to the pattern of residues modulo 5:

    1  2  3  4 
    3  2  1  4 
    4  4  4  3 
    2  2  2  4
     
    1  1  4  4 
    3  2  1  4 
    4  3  2  1 
    2  4  3  1
     
    2  4  1  3 
    2  3  1  4 
    3  2  1  4 
    3  1  2  4
     
    1  3  2  4 
    2  3  2  3 
    4  1  3  2 
    3  3  3  1

    I suggest looking for a solution to this pattern.

    Array of prime numbers:

    group # 1
    11 31 41 61 71 101 131 151 181 191 211 241 251 271 281 311 331 401 421 431 461 491 521 541 571 601 631 641 661 691 701

    group # 2
    7 17 37 47 67 97 107 127 137 157 167 197 227 257 277 307 317 337 347 367 397 457 467 487 547 557 577 587 607 617 647 677 727

    group # 3
    3 13 23 43 53 73 83 103 113 163 173 193 223 233 263 283 293 313 353 373 383 433 443 463 503 523 563 593 613 643 653 673 683

    group # 4
    19 29 59 79 89 109 139 149 179 199 229 239 269 349 359 379 389 409 419 439 449 479 499 509 569 599 619 659 709 719

    Try it all!
    I do not know whether there is such a solution.

    #135

    Natalia Makarova
    Partecipante

    The general formula of the magic cube of order 4

    This scheme magic cube of order 4 with magic constant s:

    Free variables are green, the dependent variables are blue.

    The general formula:

    x23 = 2 s + x10 - x18 + x19 - x2 - x3 - x4 - x5 - x6 - x7 - x8 - x9,
    x24 = 2 s - x10 - x11 - x12 - x17 - x19 - x2 - x20 - x21 - x3 + x4 + x7,
    x26 = 2 s - x10 - x13 - x15 - x16 - x19 + x2 - x22 - x25 + x3 - x4 - x7,
    x28 = s - x1 - x10 - x19,
    x29 = s - x11 - x2 - x20,
    x30 = s - x12 - x21 - x3,
    x31 = s - x13 - x22 - x4,
    x32 = -s - x10 - x14 + x18 - x19 + x2 + x3 + x4 + x6 + x7 + x8 + x9,
    x33 = -s + x10 + x11 + x12 - x15 + x17 + x19 + x2 + x20 + x21 + x3 - x4 - x6 - x7, 
    x34 = s - x16 - x25 - x7,
    x35 = -s + x10 + x13 + x15 + x16 - x17 + x19 - x2 + x22 + x25 - x3 + x4 + x7 - x8, 
    x36 = s - x18 - x27 - x9

    Calculated variables yi obvious.
    For example:
    y1 = s – x1 – x2 – x3

    #154

    Natalia Makarova
    Partecipante

    This is my best solution of a task # 1 for n = 4:

    17  7  439  317 
    139  487  107  47 
    331  59  167  223 
    293  227  67  193 
    
    19  61  281  419 
    191  199  179  211 
    337  347  43  53 
    233  173  277  97 
    
    283  443  23  31 
    421  83  127  149 
    3  103  433  241 
    73  151  197  359 
    
    461  269  37  13 
    29  11  367  373 
    109  271  137  263 
    181  229  239  131

    Magic constant of the cube S = 780.
    I think that this is not an optimal solution.

    I found an approximate solution for S = 750 with one error:

    11  7  283  449 
    193  457  71  29 
    439  59  89  163 
    107  227  307  109 
    
    31  61  389  269 
    443  67  101  139 
    79  383  157  131 
    197  239  103  211 
    
    277  419  41  13 
    97  53  241  359 
    3  127  421  199 
    373  151  47  179 
    
    431  263  37  19 
    17  173  337  223 
    229  181  83  257 
    73  133*  293  251

    Theoretically the minimum magic constant is 578.
    I propose to continue a address this problem.

    #155

    Natalia Makarova
    Partecipante

    It is an associative cube of order 4 with a magic constant S = 1260:

    23  521  433  283 
    373  29  457  401 
    587  139  11  523 
    277  571  359  53 
    
    263  379  557  61 
    613  13  131  503 
    317  449  31  463 
    67  419  541  233 
    
    397  89  211  563 
    167  599  181  313 
    127  499  617  17 
    569  73  251  367 
    
    577  271  59  353 
    107  619  491  43 
    229  173  601  257 
    347  197  109  607 

    Constant associativity of the cube K = 630.

    Optimal solution!

    #172

    Natalia Makarova
    Partecipante

    On this web page
    http://www.magic-squares.net/c-t-htm/c_prime.htm

    we see:
    «Note that none of the magic cubes shown on this page use consecutive prime numbers».

    This is a very good challenge!

    I tried to find a magic cube of order 4 consecutive primes
    First potential array:

    79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439

    From this set of consecutive primes, possibly amounted magic cube of order 4 with magic constant S = 1008.
    My program has quickly found a solution with 9 errors:

    79  127  419  383 
    277  421  179  131 
    263  107  199  439 
    389  353  211  55* 
    
    97  313  239  359 
    311  229  83  385* 
    373  137  349  149 
    227  329*  337  115* 
    
    401  251  193  163 
    223  173  379  233 
    101  433  167  307 
    283  151  269  305* 
    
    431  317  157  103 
    197  185*  367  259* 
    271  331  293  113 
    109  175*  191  533*

    Bad solution, but it is difficult to look further – the program runs long.
    Is there a solution?

    A few more potential arrays for this problem:

    S= 1122
    103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463

    S=1168
    109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479

    S= 2232
    359 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761

    S= 2388
    397 401 409 419 421 431 433 439 443 449 457 461 463 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811

    S= 2976
    541 547 557 563 569 571 577 587 593 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953 967

    Magic squares of order 4 consecutive primes are found very well.
    See here:
    http://oeis.org/A173981

    #173

    Natalia Makarova
    Partecipante

    On this web page
    http://www.multimagie.com/

    we see
    «Nearly perfect magic cube of order 4, by Fermat, 1640».

    Classic perfect magic cube of order 4 does not exist.

    Question:
    there is a perfect magic cube of order 4 of prime numbers?

    #212

    Natalia Makarova
    Partecipante

    Magic constants of associative 4 x 4 x 4 cubes composed of distinct prime numbers
    http://oeis.org/A240922

    1260, 1320, 1380, 1428, 1440, 1500

    I have the following potential magic constants:
    1512, 1548, 1560, 1584, 1596, 1608, 1620, 1632

    Solution with a magic constant S = 1548 does not exist.
    Magic constants 1512, 1584, 1608, 1632 need to be checked.
    This solutions for the magic constants 1560, 1596, 1620.

    53  599  631  277 
    719  19  281  541 
    751  233  7  569 
    37  709  641  173 
    
    223  643  587  107 
    467  11  349  733 
    523  397  23  617 
    347  509  601  103 
    
    677  179  271  433 
    163  757  383  257 
    47  431  769  313 
    673  193  137  557 
    
    607  139  71  743 
    211  773  547  29 
    239  499  761  61 
    503  149  181  727 
    S = 1560
    
    197  449  619  331 
    521  79  409  587 
    739  359  11  487 
    139  709  557  191 
    
    251  569  397  379 
    727  29  641  199 
    47  691  97  761 
    571  307  461  257 
    
    541  337  491  227 
    37  701  107  751 
    599  157  769  71 
    419  401  229  547 
    
    607  241  89  659 
    311  787  439  59 
    211  389  719  277 
    467  179  349  601 
    S = 1596 
    
    137  547  587  349 
    401  157  443  619 
    661  347  13  599 
    421  569  577  53 
    
    457  101  751  311 
    307  23  631  659 
    617  727  167  109 
    239  769  71  541 
    
    269  739  41  571 
    701  643  83  193 
    151  179  787  503 
    499  59  709  353 
    
    757  233  241  389 
    211  797  463  149 
    191  367  653  409 
    461  223  263  673 
    S = 1620 
    #215

    Natalia Makarova
    Partecipante

    I started to check the potential constant associativity K = 756.
    This solution with two errors found in 3 minutes:

    449  419  607  37 
    683  29  643  157 
    367  487  5  653 
    13  577  257  665* 
    
    313  277  563  359 
    617  23  463  409 
    199  619  47  647 
    383  593  439  97 
    
    659  317  163  373 
    109  709  137  557 
    347  293  733  139 
    397  193  479  443 
    
    91*  499  179  743 
    103  751  269  389 
    599  113  727  73 
    719  149  337  307
    
    K=756, S=1512

    Full check, unfortunately, runs long.
    Maybe my program is not optimal.

    This is an array of prime numbers, which should take part in drawing up the required associative cube:

    5 13 17 23 29 37 47 73 79 83 97 103 109 113 137 139 149 157 163 179 193 199 233 257 269 277 293 307 313 317 337 347 359 367 373 383 389 397 409 419 439 443 449 463 479 487 499 523 557 563 577 593 599 607 617 619 643 647 653 659 673 677 683 709 719 727 733 739 743 751

    #218

    Natalia Makarova
    Partecipante

    Hello, dear colleagues!

    Bring to your attention my article “Magic Cubes fourth order. Part 2
    http://natalimak1.narod.ru/Cube4Part2.pdf

    Are you interested in translating this article into English?

    #220

    Natalia Makarova
    Partecipante

    Hello, dear colleagues!

    My program does not find prime associative cube of order 4 with a magic constant S = 1512.
    I began checking the magic constant S = 1584.

    This solution with two errors:

    563  683  199  139 
    271  313  641  359 
    601  569  5  409 
    149  19  739  677* 
    
    733  349  419  83 
    269  23  541  751 
    509  613  331  131 
    73  599  293  619 
    
    173  499  193  719 
    661  461  179  283 
    41  251  769  523 
    709  373  443  59 
    
    115*  53  773  643 
    383  787  223  191 
    433  151  479  521 
    653  593  109  229

    Here is the wrong elements 677 and 115.
    Associative cube of order 4 with a magic constant S = 1584 should make of the numbers next array of primes:

    5 19 23 31 41 53 59 73 83 101 109 131 139 149 151 173 179 191 193 199 223 229 251 269 271 283 293 313 331 349 353 359 373 383 409 419 433 439 443 461 479 499 509 521 523 541 563 569 593 599 601 613 619 641 643 653 661 683 691 709 719 733 739 751 761 769 773 787

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