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  • #176
    Natalia Makarova
    Partecipante

    System decided colleague in Russia Dmitry Ezhov.
    Thank you, Dmitry!

    You see this solution:

    x67 = x69 - x7 + x9
    x62 = -x63 + x69 + x9
    x61 = -x65 + x69 + x9
    x5 = -x65 + x69 + x9 
    x47 = -x49 + x69 + x9 
    x43 = x45 - x49 + x53 - x63 - 3 x65 + 2 x69 - x7 + x73 + 2 x9 
    x42 = x45 + x49 + x53 + x63 + x65 - 2 x69 + x73 - 3 x9 
    x41 = -x45 + 2 x49 - 2 x53 + 2 x65 + x7 - 2 x73 + x9 
    x33 = 2 x49 - x53 + x7 - 2 x73 + x9 
    x3 = 2 x49 + x63 + 2 x65 - 2 x69 + x7 - 3 x9 
    x29 = -x49 + x69 + x9
    x27 = x49
    x25 = -x45 + x69 + x9 
    x23 = -x45 - x49 - x53 - x63 + x65 + 2 x69 - x73 + 3 x9
    x22 = -x45 + x49 - x53 + x63 + x65 + x7 - x73 
    x21 = x45 - 2 x49 + 2 x53 - 2 x65 + x69 - x7 + 2 x73 
    x2 = -2 x49 - x63 - 2 x65 + 3 x69 - x7 + 4 x9 
    x13 = -2 x49 + 2 x69 - x7 + x73 + x9
    x1 = x65 
    s = 2 x69 + 2 x9

    I reviewed the solution.

    Hypothesis
    in a nontraditional perfect cube of order 4 is bound to have the same numbers.

    This perfect cube of arbitrary natural numbers I made manually (without the program):

    203 275 207 279
    317 209 211 227
    165 195 349 255
    279 285 197 203
    
    461 151 85 267
    117 251 231 365
    151 231 251 331
    235 331 397 1
    
    21 321 407 215
    255 231 251 227
    441 251 231 41
    247 161 75 481
    
    279 217 265 203
    275 273 271 145
    207 287 133 337
    203 187 295 279

    There are a lot of identical numbers. With a program you can reduce their number, but they still will be.
    Pay attention to the following formula:

    x61 = -x65 + x69 + x9
    x5 = -x65 + x69 + x9
    x47 = -x49 + x69 + x9
    x29 = -x49 + x69 + x9
    x27 = x49
    x1 = x65

    These elements will inevitably be the same.

    I could be wrong. Colleagues, there are other opinions?

    #177
    Natalia Makarova
    Partecipante

    In the illustration you can see a scheme of unconventional perfect cube of order 4.

    When you create a formula that perfect cube I used the general formula of the magic square of order 4 by Max Alekseev.
    See http://dxdy.ru/post291286.html#p291286
    as well as in the illustration.

    I found a solution – a perfect cube of order 4 primes:

    229  257  283  311 
    383  101  109  487 
    157  349  521  53 
    311  373  167  229 
    
    131  521  349  79 
    311  283  257  229 
    151  257  283  389 
    487  19  191  383 
    
    409  109  101  461 
    307  257  283  233 
    311  283  257  229 
    53  431  439  157 
    
    311  193  347  229 
    79  439  431  131 
    461  191  19  409 
    229  257  283  311 

    Магическая константа этого куба S = 1080.

    Here many of the same elements.
    You can find the best solution (less identical elements)?

    #178
    Natalia Makarova
    Partecipante

    Let me show you two perfect cube of order 4 primes.

    Magic constant S = 660

    97  47  283  233 
    191  149  13  307 
    139  181  317  23 
    233  283  47  97 
    
    167  293  37  163 
    173  223  107  157 
    37  107  223  293 
    283  37  293  47 
    
    163  157  173  167 
    157  107  223  173 
    293  223  107  37 
    47  173  157  283 
    
    233  163  167  97 
    139  181  317  23 
    191  149  13  307 
    97  167  163  233 

    Magic constant S = 420

    101  43  167  109 
    193  19  11  197 
    17  191  199  13 
    109  167  43  101 
    
    103  199  71  47 
    79  107  103  131 
    71  103  107  139 
    167  11  139  103 
    
    107  131  19  163 
    131  103  107  79 
    139  107  103  71 
    43  79  191  107 
    
    109  47  163  101 
    17  191  199  13 
    193  19  11  197 
    101  163  47  109

    Unfortunately, not all elements are unique.
    As I said, a perfect cube of order 4 can not be made up of different numbers.

    Is there a solution with less magic constant?

    #179
    Natalia Makarova
    Partecipante

    For many years believed that classic perfect cube of order 5 does not exist.

    And in 2003 a cube was found!
    The authors of this solution: Walter Trump and Christian Boyer.
    See http://www.trump.de/magic-squares/magic-cubes/cubes-1.html

    25 16 80 104 90
    115 98 4 1 97
    42 111 85 2 75
    66 72 27 102 48
    67 18 119 106 5
    
    91 77 71 6 70
    52 64 117 69 13
    30 118 21 123 23
    26 39 92 44 114
    116 17 14 73 95
    
    47 61 45 76 86
    107 43 38 33 94
    89 68 63 58 37
    32 93 88 83 19
    40 50 81 65 79
    
    31 53 112 109 10
    12 82 34 87 100
    103 3 105 8 96
    113 57 9 62 74
    56 120 55 49 35
    
    121 108 7 20 59
    29 28 122 125 11
    51 15 41 124 84
    78 54 99 24 60
    36 110 46 22 101

    Magic constant S = 315.

    On the basis of this cube I compose the classic perfect cube of order 25.
    You can view the decision here:
    http://yadi.sk/d/i-bNylp1HHsjS

    Unconventional perfect cube of order 5 different natural numbers make easily.
    For example:

    123 78 398 518 448
    573 488 18 3 483
    208 553 423 8 373
    328 358 133 508 238
    333 88 593 528 23
    
    453 383 353 28 348
    258 318 583 343 63
    148 588 103 613 113
    128 193 458 218 568
    578 83 68 363 473
    
    233 303 223 378 428
    533 213 188 163 468
    443 338 313 288 183
    158 463 438 413 93
    198 248 403 323 393
    
    153 263 558 543 48
    58 408 168 433 498
    513 13 523 38 478
    563 283 43 308 368
    278 598 273 243 173
    
    603 538 33 98 293
    143 138 608 623 53
    253 73 203 618 418
    388 268 493 118 298
    178 548 228 108 503

    Magic constant S = 1565.

    I want to try to find a perfect cube of order 5 different primes.

    It’s an interesting challenge!
    I invite everyone to participate in finding solutions.

    #187
    Natalia Makarova
    Partecipante

    My program has found a perfect cube of order 4 with magic constant S = 336.
    This is a minimal solution.

    97  41  127  71 
    107  101  31  97 
    61  163  41  71 
    71  31  137  97 
    
    149  71  97  19 
    11  79  89  157 
    37  89  79  131 
    139  97  71  29 
    
    19  97  71  149 
    157  89  79  11 
    131  79  89  37 
    29  71  97  139 
    
    71  127  41  97 
    61  67  137  71 
    107  5  127  97 
    97  137  31  71 
    #188
    Natalia Makarova
    Partecipante

    I propose a scheme for perfect cube of order 5

    Variables Xi – the first level variables (free and dependent), variables Yi – the dependent variables of the second level.
    In drawing up the system of equations I use only the first level variables.

    Each layer in this cube is a magic square of order 5.
    I use the general formula of the magic square of order 5 by Max Alekseev.
    See http://dxdy.ru/post291405.html#p291405

    Now I want to write a system of equations describing a perfect cube of order 5.

    You can offer other search algorithms solutions.

    #195
    Natalia Makarova
    Partecipante

    I wrote a system of equations describing a perfect cube of order 5 with a magic constant s:

    X1+X41+X81+X121+X161=s
    X2+X42+X82+X122+X162=s
    X3+X43+X83+X123+X163=s
    X4+X44+X84+X124+X164=s
    X20+X60+X100+X140+X180=s
    X6+X46+X86+X126+X166=s
    X18+X58+X98+X138+X178=s
    X10+X50+X90+X130+X170=s
    X22+X62+X102+X142+X182=s
    X7+X47+X87+X127+X167=s
    X11+X51+X91+X131+X171=s
    X15+X55+X95+X135+X175=s
    X8+X48+X88+X128+X168=s
    X13+X53+X93+X133+X173=s 
    X1+X46+X91+X133-X161-X166-X171-X173=0
    -X1-X2-X3-X4+X50+X91+X128+X161+X162+X163+X164-X168-X170-X171=0
    X1+X2+X3+X4-X8-X10-X11+X48+X91+X130-X161-X162-X163-X164=0
    -X1-X6-X11-X13+X53+X91+X126+X161=0
    X1+X2+X3+X4-X8-X10-X11-X42-X46-X47-X48-X83+X87+X95+2*(X86+X88+X90+X91+X93)-X124-X130-X133-X135-X161-X166-X171-X173=0
    -X1-X6-X11-X13-X44-X50-X53-X55-X83+X87+X95+2*(X86+X88+X90+X91+X93)-X122-X126-X127-X128+X161+X162+X163+X164-X168-X170-X171=0
    -2*X1-X20-X22-X2-X3-X4+X8+X10+X11+X48-X98-X91-X87-X95-2*(X86+X88+X90+X91+X93)+X133+2*X161+X162+X163+X164+2*X166+X180+X178+X170+2*X171+X182+X167+X175+X173=0
    2*X1+X2+X3+X4+2*X6+2*X11+X13+X10+X18+X20+X7+X15+X22+X53-X98-X91-X87-X95+X128-2*(X86+X88+X90+X91+X93)-2*X161-X162-X163-X164+X168+X170+X171-X180-X182=0
    X22+X47+X91+X135-X167-X171-X175-X182=0
    -X7-X11-X15-X22+X55+X91+X127+X182=0
    X20+X46+X98+X130-X166-X170-X178-X180=0
    -X6-X10-X18-X20+X50+X98+X126+X180=0
    X1+X42+X83+X124-X161-X162-X163-X164=0
    -X1-X2-X3-X4+X44+X83+X122+X161=0
    X1+X60+X102-2*X121-X140-X142-X122-X123-X124+X128+X130+X131+X161+X162+X163+X164-X168-X170-X171=0
    X1+X2+X3+X4-X8-X10-X11-2*X41-X60-X62-X42-X43-X44+X48+X50+X51+X102+X140+X161=0
    X2+X46+X87+X128-X162-X166-X167-X168=0
    -X2-X6-X7-X8+X48+X87+X126+X162=0
    X3+X58+X91-X138-X131-2*X126-X127-2*X128-2*X130-2*X131-2*X133-X135-X163+2*X166+X167+2*X168+2*X170+2*X171+2*X173+X175=0
    -X3+2*X6+X7+2*X8+2*X10+2*X11+2*X13+X15-X43-3*X51-X58+X43-2*X46-X47-2*X48-2*X50-2*X53-X55+X91+X138+X163=0
    X4+X50+X95+X133-X164-X170-X173-X175=0
    -X4-X10-X13-X15+X53+X95+X130+X164=0
    -X1-X2-X3-X4-X46-X50-X58-X60-X87-X91-X95-X102-X161-X166-X171-X173+2*X121+X122+X123+X124+2*X126+2*X131+X133+X130+X127+X135+X138+X140+X142=0
    -X1-X6-X11-X13-X87-X91-X95-X102-X126-X130-X138-X140-X161-X162-X163-X164+2*X41+X42+X43+X44+2*X46+X50+X58+X60+X47+2*X51+X55+X62+X53=0

    I ask colleagues to help me solve the system.
    Thanks in advance.

    Note. The scheme of cube shows above.
    I checked the system of equations for the data from the classical perfect cube of order 5 W.Trump and C.Boyer, all equations are satisfied for these data.

    #196
    Natalia Makarova
    Partecipante

    System of equations decided colleague Dmitry Ezhov.
    Thank you, Dmitry!

    So, we have the general formula perfect cube of order 5:

    X86 = -X87 - X88 - X90 + 6 X91 - X93 - X95
    X47 = -4 X48 + 8 X50 + 3 X51 + 4 X53 + 3 X55 + 2 X58 - 4 X87 - X88 + 2 X90 - 18 X91 + X93 + 2 X95 + 3 X98
    X46 = X48 - 3 X50 - (3 X51)/2 - X53 - X55 - X58 + (3 X87)/2 + X88/2 - X90/2 + (15 X91)/2 - (3 X98)/2
    X173 = X175/4 + X178/4 + X18/4 + X182/2 - 2 X2 - X22/2 - X3/2 - X42 + X44 + 3 X48 - 4 X50 - (3 X51)/2 - 3 X53 - X55 - X58 - (5 X7)/4 - X8 - (3 X82)/4 + X83/2 + X84/4 + (5 X87)/2 + X88/2 - X90 + 14 X91 - X93 - X95/2 - 2 X98
    X171 = -X175 - X182 + X22 - 4 X48 + 8 X50 + 3 X51 + 4 X53 + 2 X55 + 2 X58 + X7 - 4 X87 - X88 + 2 X90 - 16 X91 + X93 + X95 + 3 X98
    X170 = -(X178/2) + X18/2 - X180 + X20 + X48 - 4 X50 - (3 X51)/2 - X53 - X55 - X58 + X6 + X87 - (3 X90)/2 + (21 X91)/2 - X93/2 - X95/2 - (3 X98)/2
    X168 = (3 X175)/4 + X178/4 + X18/4 + X182/2 - X2 - X22/2 - X3/2 - X4 + 2 X48 - 4 X50 - (3 X51)/2 - 2 X53 - X55 - X58 - (3 X7)/4 - X8 - X82/4 + X83/2 - X84/4 + (5 X87)/2 - X90 + 13 X91 - X93/2 - X95/2 - 2 X98
    X167 = -3 X175 - 2 X182 + 4 X2 + 2 X22 + 2 X3 + 4 X4 - 8 X48 + 16 X50 + 6 X51 + 8 X53 + 4 X55 + 4 X58 + 2 X7 + X82 - 2 X83 + X84 - 7 X87 - 2 X88 + 4 X90 - 44 X91 + 2 X93 + 3 X95 + 6 X98
    X166 = (3 X175)/4 - X178/4 - X18/4 + X182/2 - X2 - X22/2 - X3/2 - X4 + 2 X48 - 4 X50 - (3 X51)/2 - 2 X53 - X55 - X58 - X6 - (3 X7)/4 - X82/4 + X83/2 - X84/4 + (5 X87)/2 + X88 - X90/2 + 11 X91 - X95/2 - X98
    X164 = -(X175/2) - X18 + X180 + X2 - X20 + X42 - X44 - 2 X48 + 5 X50 + (3 X51)/2 + X53 + X55 + X58 - X6 + X7/2 + X8 + X82/2 - X84/2 - 2 X87 + (3 X90)/2 - (19 X91)/2 + X93/2 + X95/2 + (5 X98)/2
    X163 = -(X175/2) + X178/2 + (5 X18)/2 - 2 X180 - X182 - 4 X2 + 2 X20 + X22 - 2 X42 + 2 X44 + 4 X50 + 3 X51 + 4 X53 + 2 X55 + 2 X58 - (3 X7)/2 - 4 X8 - (3 X82)/2 + X83 + X84/2 - X88 + X90 - 10 X91 + X93 + 2 X95
    X162 = (3 X175)/4 - X178/4 - X18/4 + X182/2 - X22/2 - X3/2 - X4 + 2 X48 - 7 X50 - 3 X51 - 3 X53 - 2 X55 - 2 X58 + X6 + X7/4 + X8 - X82/4 + X83/2 - X84/4 + 2 X87 + X88/2 - 2 X90 + (39 X91)/2 - X93 - (3 X95)/2 - (5 X98)/2
    X15 = 2 X175 + 2 X182 - 4 X2 - 2 X22 - 2 X3 - 4 X4 + 8 X48 - 16 X50 - 6 X51 - 8 X53 - 4 X55 - 4 X58 - 3 X7 - X82 + 2 X83 - X84 + 7 X87 + 2 X88 - 4 X90 + 48 X91 - 2 X93 - 3 X95 - 6 X98
    X142 = (8 X161)/3 - (7 X175)/6 + X178/2 + (7 X18)/6 - (2 X180)/3 - (5 X182)/3 - 4 X2 + (2 X20)/3 + X22/3 - X3/3 - 2 X42 + 2 X44 - (8 X48)/3 + (32 X50)/3 + 5 X51 + (16 X53)/3 + (10 X55)/3 + (10 X58)/3 - (4 X6)/3 - X62 - (5 X7)/6 - (8 X8)/3 + (2 X81)/3 - (3 X82)/2 + 2 X83 + X84/2 - (7 X87)/3 - X88 + 3 X90 - 25 X91 + (5 X93)/3 + 3 X95 + (10 X98)/3
    X140 = (2 X161)/3 - X175/6 + X178/2 + X18/6 + X180/3 + X182/3 - 2 X2 - X20/3 + X22/3 - X3/3 + 2 X41 + X43 + 2 X44 - (2 X48)/3 + (8 X50)/3 + X51 + (4 X53)/3 + (4 X55)/3 + (4 X58)/3 - (4 X6)/3 + X60 + X62 - (5 X7)/6 - (2 X8)/3 + (2 X81)/3 - X82/2 + X83 + X84/2 - X87/3 + X90 - 15 X91 + (2 X93)/3 + X95 + (4 X98)/3
    X138 = -X178 - X18 - X58 + 5 X91 - X98
    X135 = -3 X175 - 2 X182 + 4 X2 + 2 X22 + 2 X3 + 4 X4 - 8 X48 + 16 X50 + 6 X51 + 8 X53 + 3 X55 + 4 X58 + 3 X7 + X82 - 2 X83 + X84 - 7 X87 - 2 X88 + 4 X90 - 43 X91 + 2 X93 + 2 X95 + 6 X98
    X133 = (3 X175)/4 - X178/4 - X18/4 + X182/2 - X2 - X22/2 - X3/2 - X4 + 2 X48 - 4 X50 - (3 X51)/2 - 3 X53 - X55 - X58 - (3 X7)/4 - X82/4 + X83/2 - X84/4 + (3 X87)/2 + X88/2 - X90 + 15 X91 - X93 - (3 X95)/2 - X98
    X131 = -X51 + X87 + X95
    X130 = (3 X175)/4 + X178/4 + X18/4 + X182/2 - X2 - X22/2 - X3/2 - X4 + 2 X48 - 5 X50 - (3 X51)/2 - 2 X53 - X55 - X58 - (3 X7)/4 - X82/4 + X83/2 - X84/4 + 2 X87 + X88/2 - (3 X90)/2 + 14 X91 - X93/2 - X95 - 2 X98
    X13 = -X175 - X182 + 3 X2 + X22 + X3 + X4 + X42 - X44 - 5 X48 + 8 X50 + 3 X51 + 5 X53 + 2 X55 + 2 X58 + 2 X7 + X8 + X82 - X83 - 4 X87 - X88 + 2 X90 - 24 X91 + X93 + 2 X95 + 3 X98
    X128 = -((3 X175)/4) - X178/4 - X18/4 - X182/2 + X2 + X22/2 + X3/2 + X4 - 3 X48 + 4 X50 + (3 X51)/2 + 2 X53 + X55 + X58 + (3 X7)/4 + X82/4 - X83/2 + X84/4 - (5 X87)/2 - X88 + X90 - 8 X91 + X93/2 + X95/2 + 2 X98
    X127 = 3 X175 + 2 X182 - 4 X2 - 2 X22 - 2 X3 - 4 X4 + 12 X48 - 24 X50 - 9 X51 - 12 X53 - 7 X55 - 6 X58 - 3 X7 - X82 + 2 X83 - X84 + 10 X87 + 3 X88 - 6 X90 + 67 X91 - 3 X93 - 5 X95 - 9 X98
    X126 = -((3 X175)/4) + X178/4 + X18/4 - X182/2 + X2 + X22/2 + X3/2 + X4 - 3 X48 + 7 X50 + 3 X51 + 3 X53 + 2 X55 + 2 X58 + (3 X7)/4 + X82/4 - X83/2 + X84/4 - 3 X87 - X88/2 + 2 X90 - (39 X91)/2 + X93 + (3 X95)/2 + (5 X98)/2
    X124 = X175/2 + X18 - X180 - X2 + X20 - X4 - X42 + 2 X48 - 5 X50 - (3 X51)/2 - X53 - X55 - X58 + X6 - X7/2 - X8 - X82/2 - X84/2 + 2 X87 - (3 X90)/2 + (29 X91)/2 - X93/2 - X95/2 - (5 X98)/2
    X123 = X175/2 - X178/2 - (5 X18)/2 + 2 X180 + X182 + 4 X2 - 2 X20 - X22 - X3 + 2 X42 - X43 - 2 X44 - 4 X50 - 3 X51 - 4 X53 - 2 X55 - 2 X58 + (3 X7)/2 + 4 X8 + (3 X82)/2 - 2 X83 - X84/2 + X88 - X90 + 15 X91 - X93 - 2 X95
    X122 = -((3 X175)/4) + X178/4 + X18/4 - X182/2 - X2 + X22/2 + X3/2 + X4 - X42 - 2 X48 + 7 X50 + 3 X51 + 3 X53 + 2 X55 + 2 X58 - X6 - X7/4 - X8 - (3 X82)/4 - X83/2 + X84/4 - 2 X87 - X88/2 + 2 X90 - (29 X91)/2 + X93 + (3 X95)/2 + (5 X98)/2
    X121 = -2 X161 + (3 X175)/4 - X178/4 - X18/4 + X182/2 + 2 X2 - X22/2 + X3/2 - X41 + X42 - X44 + 2 X48 - 7 X50 - 3 X51 - 3 X53 - 2 X55 - 2 X58 + X6 + X7/4 + X8 - X81 + (3 X82)/4 - X83/2 - X84/4 + 2 X87 + X88/2 - 2 X90 + (39 X91)/2 - X93 - (3 X95)/2 - (5 X98)/2
    X11 = X175 + X182 - X22 + 4 X48 - 8 X50 - 3 X51 - 4 X53 - 2 X55 - 2 X58 - X7 + 3 X87 + X88 - 2 X90 + 20 X91 - X93 - 2 X95 - 3 X98
    X102 = -((8 X161)/3) + (7 X175)/6 - X178/2 - (7 X18)/6 + (2 X180)/3 + (2 X182)/3 + 4 X2 - (2 X20)/3 - (4 X22)/3 + X3/3 + 2 X42 - 2 X44 + (8 X48)/3 - (32 X50)/3 - 5 X51 - (16 X53)/3 - (10 X55)/3 - (10 X58)/3 + (4 X6)/3 + (5 X7)/6 + (8 X8)/3 - (2 X81)/3 + (3 X82)/2 - 2 X83 - X84/2 + (7 X87)/3 + X88 - 3 X90 + 30 X91 - (5 X93)/3 - 3 X95 - (10 X98)/3
    X100 = -((2 X161)/3) + X175/6 - X178/2 - X18/6 - (4 X180)/3 - X182/3 + 2 X2 - (2 X20)/3 - X22/3 + X3/3 - 2 X41 - X43 - 2 X44 + (2 X48)/3 - (8 X50)/3 - X51 - (4 X53)/3 - (4 X55)/3 - (4 X58)/3 + (4 X6)/3 - 2 X60 - X62 + (5 X7)/6 + (2 X8)/3 - (2 X81)/3 + X82/2 - X83 - X84/2 + X87/3 - X90 + 20 X91 - (2 X93)/3 - X95 - (4 X98)/3
    X10 = -((3 X175)/4) + X178/4 - (3 X18)/4 + X180 - X182/2 + X2 - X20 + X22/2 + X3/2 + X4 - 3 X48 + 8 X50 + 3 X51 + 3 X53 + 2 X55 + 2 X58 - X6 + (3 X7)/4 + X82/4 - X83/2 + X84/4 - 3 X87 - X88/2 + 2 X90 - (39 X91)/2 + X93 + (3 X95)/2 + (7 X98)/2
    X1 = X161 - (3 X175)/4 + X178/4 + X18/4 - X182/2 - 2 X2 + X22/2 - X3/2 - X42 + X44 - 2 X48 + 7 X50 + 3 X51 + 3 X53 + 2 X55 + 2 X58 - X6 - X7/4 - X8 - (3 X82)/4 + X83/2 + X84/4 - 2 X87 - X88/2 + 2 X90 - (29 X91)/2 + X93 + (3 X95)/2 + (5 X98)/2
    s = 5 X91

    I checked this formula for the perfect classic cube of order 5.

    Now I want to try this formula to make a perfect cube of order 5 primes. I think that a solution exists. How difficult is it to find?

    #197
    Natalia Makarova
    Partecipante

    My program has found the solution to task # 1, n = 6 for the magic constant
    S = 5040 with three errors:

    1327  739  479  821  1663  11 
    1453  149  13  1367  1087  971 
    1307  1093  1637  647  127  229 
    181  1291  1283  769  257  1259 
    641  1301  307  1237  383  1171 
    131  467  1321  199  1523  1399 
    
    23  53  1567  1021  1429  947 
    883  1987  7  863  503  797 
    857  613  2221  107  419  823 
    1483  239  101  2053  967  197 
    1061  521  1031  337  1471  619 
    733  1627  113  659  251  1657 
    
    191  109  1063  853  1223  1601 
    937  103  73  1997  1187  743 
    631  1847  757  317  439  1049 
    1619  601  461  1009  1289  61 
    1583  809  2069  37  445  97 
    79  1571  617  827  457  1489 
    
    1597  1439  449  463  431  661 
    571  701  2099  349  211  1109 
    67  409  311  919  1721  1613 
    1103  1973  787  269  331  577 
    683  277  163  1823  1097  997 
    1019  241  1231  1217  1249  83 
    
    1621  1487  1123  401  137  271 
    487  569  1181  151  1459  1193 
    727  491  71  2017  781  953 
    233  547  2011  29  773  1447 
    563  1753  97  1163  347  1117 
    1409  193  557  1279  1543  59 
    
    281  1213  359  1481  157  1549 
    709  1531  1667  313  593  227 
    1451  587  43  1033  1553  373 
    421  389  397  911  1423  1499 
    509  379  1373  443  1297  1039 
    1669  941  1201  859  17  353

    type 1
    size 6
    445 is not prime
    781 is not prime
    97 is not unique
    All Sums = 5040

    This concentric magic cube of order 6.
    I think there is a right solution.

    #199
    Natalia Makarova
    Partecipante

    I tried to find a solution on the general formula perfect cube of order 5 (see post # 196).
    Unfortunately, this solution is composed of rational numbers:

    -6586.5  1663  3  997  58438.5 
     613  79  313 -85892.5  139402.5 
     619  10477  92215  296541 -345337 
     90834.5  20719 -9100 -140571  92632.5 
    -30965  21577 -28916 -16559.5  109378.5 
    
     787  7  10369  21613  21739 
     21433  44425  21283  277 -32903 
     20899 -74093  11833  8839  87037 
     11239  20509 -1199  21397  2569 
     157  63667  12229  2389 -23927 
    
     8803  877  18919  12757  13159 
    -8039.33333333333  20353  1783  739  39679.3333333333 
     33948.6666666667  3169  10903  18637 -12142.6666666667 
     11155.6666666667  21067  20023  1453  816.333333333333 
     8647  9049  2887  20929  13003 
    
     50868.5 -45098.5 -38860  84301  3304 
     39649.3333333333 -87806.5  9643  96193.5 -3164.33333333333 
    -1870.66666666667  379191  9973 -270325 -62453.3333333333 
    -52544.1666666667 -73367.5  13885  65596.5  100945.166666667 
     18412 -118403.5  59874  78749  15883.5 
    
     643  97066.5  64084 -65153 -42125.5 
     859  77464.5  21493  43198 -88499.5 
     919 -264229 -70409  823  387411 
    -6170  65587.5  30906  106639.5 -142448 
     58264  78625.5  8441 -30992.5 -59823 
    
    S=54515

    Now I ask colleagues to help me, must solve the system of equations (see post # 195) in integers.

    Thank you in advance.

    #200
    Natalia Makarova
    Partecipante

    System of equations in integers solved at a forum in Russia
    http://forum.exponenta.ru/viewtopic.php?t=9466&postdays=0&postorder=asc&start=30

    The solution is obtained in Maple.
    You see this solution:

    {X1 = 3*_Z2-5*_Z3-3*_Z1+6*_Z14+3*_Z15+3*_Z16+3*_Z17-23*_Z18+11*_Z19-6*_Z20-6*_Z21+10*_Z22+4*_Z23+4*_Z24-30*_Z25+3*_Z26+3*_Z27+21*_Z28+2*_Z29+3*_Z31-9*_Z33+14*_Z34+65*_Z35-3*_Z36+18*_Z37-3*_Z10+2*_Z11-_Z12-3*_Z13+20*_Z4+20*_Z5+4*_Z6-3*_Z7+21*_Z8-3*_Z9,
    X10 = _Z3+2*_Z4+2*_Z5+_Z7+3*_Z8+_Z10-_Z18+2*_Z19+_Z20+_Z21+2*_Z22+2*_Z23+2*_Z24-2*_Z25+2*_Z28+_Z33+2*_Z34+6*_Z35+_Z36+4*_Z37,
    X100 = -_Z1-_Z8-_Z10-_Z26+5*_Z35,
    X102 = -6*_Z2+9*_Z3+5*_Z1-10*_Z14-5*_Z15-5*_Z16-5*_Z17+40*_Z18-19*_Z19+10*_Z20+10*_Z21-18*_Z22-7*_Z23-7*_Z24+52*_Z25-5*_Z26-5*_Z27-36*_Z28-4*_Z29-6*_Z31+16*_Z33-24*_Z34-105*_Z35+5*_Z36-31*_Z37+5*_Z10-4*_Z11+_Z12+5*_Z13-35*_Z4-35*_Z5-7*_Z6+5*_Z7-36*_Z8+5*_Z9,
    X11 = -2*_Z2+3*_Z3+3*_Z1-6*_Z14-3*_Z15-3*_Z16-3*_Z17+18*_Z18-9*_Z19+4*_Z20+4*_Z21-10*_Z22-5*_Z23-5*_Z24+24*_Z25-3*_Z26-3*_Z27-18*_Z28-2*_Z29-3*_Z31+6*_Z33-12*_Z34-46*_Z35+_Z36-17*_Z37+_Z10-2*_Z11+_Z12+3*_Z13-17*_Z4-17*_Z5-3*_Z6+_Z7-18*_Z8+3*_Z9,
    X121 = -4*_Z2+5*_Z3+3*_Z1-7*_Z14-3*_Z15-3*_Z16-3*_Z17+23*_Z18-11*_Z19+6*_Z20+6*_Z21-10*_Z22-4*_Z23-4*_Z24+30*_Z25-3*_Z26-3*_Z27-21*_Z28-3*_Z29-3*_Z31+9*_Z33-14*_Z34-60*_Z35+3*_Z36-18*_Z37+3*_Z10-2*_Z11+_Z12+3*_Z13-20*_Z4-20*_Z5-4*_Z6+3*_Z7-21*_Z8+3*_Z9,
    X122 = 2*_Z2-5*_Z3-3*_Z1+6*_Z14+3*_Z15+3*_Z16+2*_Z17-23*_Z18+11*_Z19-6*_Z20-6*_Z21+10*_Z22+4*_Z23+4*_Z24-30*_Z25+3*_Z26+3*_Z27+21*_Z28+2*_Z29+2*_Z31-9*_Z33+14*_Z34+65*_Z35-3*_Z36+18*_Z37-3*_Z10+2*_Z11-2*_Z13+20*_Z4+20*_Z5+4*_Z6-3*_Z7+21*_Z8-2*_Z9,
    X123 = -8*_Z2+20*_Z3+12*_Z1-24*_Z14-12*_Z15-13*_Z16-12*_Z17+86*_Z18-42*_Z19+26*_Z20+24*_Z21-36*_Z22-14*_Z23-14*_Z24+112*_Z25-12*_Z26-12*_Z27-80*_Z28-8*_Z29-_Z30-11*_Z31-_Z32+34*_Z33-54*_Z34-219*_Z35+12*_Z36-66*_Z37+12*_Z10-8*_Z11+8*_Z13-78*_Z4-76*_Z5-14*_Z6+12*_Z7-78*_Z8+8*_Z9,
    X124 = 2*_Z2-5*_Z3-3*_Z1+6*_Z14+2*_Z15+3*_Z16+3*_Z17-20*_Z18+10*_Z19-7*_Z20-6*_Z21+8*_Z22+3*_Z23+3*_Z24-26*_Z25+3*_Z26+3*_Z27+19*_Z28+2*_Z29+2*_Z31-8*_Z33+13*_Z34+52*_Z35-3*_Z36+15*_Z37-3*_Z10+2*_Z11-2*_Z13+19*_Z4+18*_Z5+3*_Z6-3*_Z7+18*_Z8-2*_Z9,
    X126 = -_Z3-_Z18-_Z25-_Z33+5*_Z35,
    X127 = 2*_Z4-2*_Z7-2*_Z10+2*_Z19-4*_Z20-2*_Z21-4*_Z22-3*_Z23-2*_Z24+2*_Z28+2*_Z34-2*_Z36-2*_Z37,
    X128 = -_Z4-_Z19-_Z28-_Z34+5*_Z35,
    X13 = _Z3-3*_Z4-3*_Z5-_Z6+2*_Z7-3*_Z8+2*_Z10+4*_Z18-2*_Z19+2*_Z20+2*_Z21+_Z22+_Z23+_Z24+4*_Z25-3*_Z28+2*_Z33-2*_Z34-13*_Z35+2*_Z36-_Z37,
    X130 = -_Z3+4*_Z4+3*_Z5-_Z7+3*_Z8-_Z10-3*_Z18+2*_Z19-2*_Z20-_Z21-4*_Z25+4*_Z28-_Z33+3*_Z34+11*_Z35-_Z36+2*_Z37,
    X131 = 2*_Z3+2*_Z7+2*_Z10+2*_Z18+2*_Z20+_Z21+2*_Z22+2*_Z23+2*_Z24+2*_Z25+2*_Z33-8*_Z35+2*_Z36+2*_Z37,
    X133 = _Z3-3*_Z4-3*_Z5-3*_Z8+2*_Z18-2*_Z19-2*_Z22-_Z23-_Z24+4*_Z25-3*_Z28+_Z33-2*_Z34-8*_Z35-3*_Z37,
    X135 = 2*_Z3-2*_Z4+4*_Z7+4*_Z10+4*_Z18+4*_Z20+4*_Z21+4*_Z22+3*_Z23+4*_Z24+2*_Z25-2*_Z28+3*_Z33-_Z34-5*_Z35+_Z36+4*_Z37,
    X138 = -2*_Z3-8*_Z4-6*_Z5-4*_Z7-6*_Z8-4*_Z10+2*_Z18-6*_Z19+2*_Z20-_Z21-4*_Z22-4*_Z23-5*_Z24+4*_Z25-8*_Z28-3*_Z33-7*_Z34-8*_Z35-_Z36-10*_Z37,
    X140 = _Z1,
    X142 = 12*_Z2-20*_Z3-14*_Z1+28*_Z14+14*_Z15+14*_Z16+14*_Z17-96*_Z18+48*_Z19-28*_Z20-26*_Z21+42*_Z22+18*_Z23+18*_Z24-126*_Z25+14*_Z26+13*_Z27+92*_Z28+10*_Z29+_Z30+13*_Z31+_Z32-36*_Z33+62*_Z34+249*_Z35-12*_Z36+78*_Z37-12*_Z10+8*_Z11-2*_Z12-10*_Z13+88*_Z4+86*_Z5+16*_Z6-12*_Z7+90*_Z8-10*_Z9,
    X15 = -2*_Z2+2*_Z3+3*_Z1-6*_Z14-2*_Z15-3*_Z16-4*_Z17+14*_Z18-8*_Z19+2*_Z20+2*_Z21-10*_Z22-6*_Z23-6*_Z24+20*_Z25-3*_Z26-3*_Z27-14*_Z28-2*_Z29-2*_Z31-_Z32+4*_Z33-10*_Z34-29*_Z35-16*_Z37-_Z10-2*_Z11-14*_Z4-14*_Z5-2*_Z6-_Z7-15*_Z8+2*_Z9,
    X161 = _Z2,
    X162 = -2*_Z2+5*_Z3+3*_Z1-6*_Z14-4*_Z15-3*_Z16-2*_Z17+23*_Z18-11*_Z19+6*_Z20+6*_Z21-10*_Z22-4*_Z23-4*_Z24+30*_Z25-3*_Z26-3*_Z27-21*_Z28-2*_Z29-_Z30-2*_Z31+9*_Z33-14*_Z34-60*_Z35+3*_Z36-18*_Z37+3*_Z10-2*_Z11+2*_Z13-20*_Z4-20*_Z5-4*_Z6+3*_Z7-21*_Z8+_Z9,
    X163 = 8*_Z2-20*_Z3-12*_Z1+24*_Z14+12*_Z15+12*_Z16+12*_Z17-86*_Z18+42*_Z19-26*_Z20-24*_Z21+36*_Z22+14*_Z23+14*_Z24-112*_Z25+12*_Z26+12*_Z27+80*_Z28+8*_Z29+_Z30+10*_Z31+_Z32-34*_Z33+54*_Z34+224*_Z35-12*_Z36+66*_Z37-12*_Z10+8*_Z11-_Z12-8*_Z13+78*_Z4+76*_Z5+14*_Z6-12*_Z7+78*_Z8-8*_Z9,
    X164 = -2*_Z2+5*_Z3+3*_Z1-6*_Z14-2*_Z15-3*_Z16-4*_Z17+20*_Z18-10*_Z19+7*_Z20+6*_Z21-8*_Z22-3*_Z23-3*_Z24+26*_Z25-3*_Z26-3*_Z27-19*_Z28-2*_Z29-2*_Z31-_Z32+8*_Z33-13*_Z34-47*_Z35+3*_Z36-15*_Z37+3*_Z10-2*_Z11+_Z13-19*_Z4-18*_Z5-3*_Z6+3*_Z7-18*_Z8+2*_Z9,
    X166 = _Z3,
    X167 = 2*_Z2-4*_Z3-3*_Z1+6*_Z14+4*_Z15+3*_Z16+2*_Z17-20*_Z18+10*_Z19-4*_Z20-4*_Z21+12*_Z22+6*_Z23+6*_Z24-28*_Z25+3*_Z26+3*_Z27+20*_Z28+2*_Z29+_Z30+2*_Z31-7*_Z33+13*_Z34+57*_Z35-2*_Z36+20*_Z37-_Z10+2*_Z11-2*_Z13+18*_Z4+20*_Z5+4*_Z6-_Z7+21*_Z8,
    X168 = _Z4,
    X170 = _Z5,
    X171 = 2*_Z2-5*_Z3-3*_Z1+6*_Z14+3*_Z15+3*_Z16+3*_Z17-20*_Z18+9*_Z19-6*_Z20-6*_Z21+8*_Z22+3*_Z23+3*_Z24-26*_Z25+3*_Z26+3*_Z27+18*_Z28+2*_Z29+3*_Z31-8*_Z33+12*_Z34+58*_Z35-3*_Z36+15*_Z37-3*_Z10+2*_Z11-_Z12-3*_Z13+17*_Z4+17*_Z5+3*_Z6-3*_Z7+18*_Z8-3*_Z9,
    X173 = _Z6,
    X175 = 2*_Z2-4*_Z3-3*_Z1+6*_Z14+2*_Z15+3*_Z16+4*_Z17-18*_Z18+8*_Z19-6*_Z20-6*_Z21+6*_Z22+2*_Z23+2*_Z24-22*_Z25+3*_Z26+3*_Z27+16*_Z28+2*_Z29+2*_Z31+_Z32-7*_Z33+11*_Z34+39*_Z35-2*_Z36+12*_Z37-3*_Z10+2*_Z11+16*_Z4+14*_Z5+2*_Z6-3*_Z7+15*_Z8-2*_Z9,
    X178 = 6*_Z4+4*_Z5+_Z7+4*_Z8+2*_Z10-2*_Z18+4*_Z19-2*_Z20+2*_Z22+2*_Z23+2*_Z24-4*_Z25+6*_Z28+_Z33+5*_Z34+12*_Z35+6*_Z37,
    X18 = _Z7,
    X180 = _Z8,
    X182 = -6*_Z2+11*_Z3+9*_Z1-18*_Z14-9*_Z15-9*_Z16-9*_Z17+56*_Z18-29*_Z19+18*_Z20+16*_Z21-24*_Z22-11*_Z23-11*_Z24+74*_Z25-9*_Z26-9*_Z27-56*_Z28-6*_Z29-_Z30-7*_Z31-_Z32+20*_Z33-38*_Z34-139*_Z35+7*_Z36-47*_Z37+7*_Z10-5*_Z11+_Z12+5*_Z13-53*_Z4-51*_Z5-9*_Z6+7*_Z7-54*_Z8+5*_Z9,
    X2 = _Z9,
    X20 = _Z10,
    X22 = _Z11,
    X3 = _Z12,
    X4 = _Z13,
    X41 = _Z14,
    X42 = _Z15,
    X43 = _Z16,
    X44 = _Z17,
    X46 = _Z18,
    X47 = -4*_Z3-4*_Z7-4*_Z10-6*_Z18-2*_Z19-2*_Z20-4*_Z21-2*_Z22-3*_Z23-4*_Z24-4*_Z25-5*_Z33-_Z34+19*_Z35-_Z36-4*_Z37,
    X48 = _Z19,
    X50 = _Z20,
    X51 = _Z21,
    X53 = _Z22,
    X55 = _Z23,
    X58 = _Z24,
    X6 = _Z25,
    X60 = _Z26,
    X62 = _Z27,
    X7 = -2*_Z2+6*_Z3+3*_Z1-6*_Z14-4*_Z15-3*_Z16-2*_Z17+24*_Z18-10*_Z19+8*_Z20+8*_Z21-8*_Z22-2*_Z23-2*_Z24+30*_Z25-3*_Z26-3*_Z27-22*_Z28-2*_Z29-_Z30-2*_Z31+10*_Z33-14*_Z34-63*_Z35+4*_Z36-16*_Z37+5*_Z10-2*_Z11+2*_Z13-20*_Z4-20*_Z5-4*_Z6+5*_Z7-21*_Z8,
    X8 = _Z28,
    X81 = _Z29,
    X82 = _Z30,
    X83 = _Z31,
    X84 = _Z32,
    X86 = _Z33,
    X87 = 2*_Z3+2*_Z7+2*_Z10+2*_Z18+2*_Z20+2*_Z21+2*_Z22+2*_Z23+2*_Z24+2*_Z25+2*_Z33-8*_Z35+_Z36+2*_Z37,
    X88 = _Z34,
    X90 = -6*_Z4-6*_Z5-6*_Z8+4*_Z18-4*_Z19-2*_Z22-2*_Z23-2*_Z24+6*_Z25-6*_Z28-5*_Z34-12*_Z35-6*_Z37,
    X91 = _Z35,
    X93 = -2*_Z3+6*_Z4+6*_Z5-2*_Z7+6*_Z8-2*_Z10-6*_Z18+4*_Z19-2*_Z20-2*_Z21-8*_Z25+6*_Z28-3*_Z33+4*_Z34+26*_Z35-2*_Z36+4*_Z37,
    X95 = _Z36,
    X98 = 2*_Z3+2*_Z4+2*_Z5+2*_Z7+2*_Z8+2*_Z10+2*_Z19+_Z21+2*_Z22+2*_Z23+2*_Z24+2*_Z28+2*_Z33+2*_Z34+_Z35+_Z36+4*_Z37,
    s = 5*_Z35}

    So, we have the general formula to perfect magic cube of order 5, composed of integers.

    #201
    Natalia Makarova
    Partecipante

    I found on this formula perfect magic cube of order 5 of distinct natural numbers:

    3664  2989  3053  3077  2397 
    3088  3071  2977  3122  2922 
    3015  2492  2429  2383  4861 
    1226  3045  5849  3038  2022 
    4187  3583  872  3560  2978 
    
    3064  3050  3044  2979  3043 
    3025  3037  3090  3042  2986 
    3003  2943  2994  3096  3144 
    2999  3012  3213  3017  2939 
    3089  3138  2839  3046  3068 
    
    3020  3034  3018  3049  3059 
    3080  3016  3159  2784  3141 
    1915  3115  3036  3031  4083 
    3930  3066  2987  3204  1993 
    3235  2949  2980  3112  2904 
    
    2338  3692  643  3637  4870 
    2985  3055  2637  3134  3369 
    5962  2902  3152  3129  35 
    2901  3030  3130  2924  3195 
    994  2501  5618  2356  3711 
    
    3094  2415  5422  2438  1811 
    3002  3001  3317  3098  2762 
    1285  3728  3569  3541  3057 
    4124  3027  1  2997  5031 
    3675  3009  2871  3106  2519 
    
    S=15180

    I think the perfect magic cube of order 5 of primes exist.
    I want to try to find the solution.

    #202
    Natalia Makarova
    Partecipante

    This cube looks like

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