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- Questo topic ha 42 risposte, 1 partecipante ed è stato aggiornato l'ultima volta 9 anni, 9 mesi fa da Natalia Makarova.
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Giugno 18, 2014 alle 11:25 am #176Natalia MakarovaPartecipante
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?
Giugno 19, 2014 alle 5:32 pm #177Natalia MakarovaPartecipanteIn 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)?Giugno 21, 2014 alle 11:57 am #178Natalia MakarovaPartecipanteLet 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?
Giugno 21, 2014 alle 3:11 pm #179Natalia MakarovaPartecipanteFor 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.html25 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-bNylp1HHsjSUnconventional 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.Giugno 22, 2014 alle 3:55 am #187Natalia MakarovaPartecipanteMy 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
Giugno 22, 2014 alle 4:45 am #188Natalia MakarovaPartecipanteI 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#p291405Now I want to write a system of equations describing a perfect cube of order 5.
You can offer other search algorithms solutions.
- Questa risposta è stata modificata 9 anni, 9 mesi fa da Natalia Makarova.
Giugno 24, 2014 alle 4:05 am #195Natalia MakarovaPartecipanteI 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.Giugno 25, 2014 alle 11:50 am #196Natalia MakarovaPartecipanteSystem 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?
Giugno 26, 2014 alle 11:04 am #197Natalia MakarovaPartecipanteMy 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 = 5040This concentric magic cube of order 6.
I think there is a right solution.- Questa risposta è stata modificata 9 anni, 9 mesi fa da Natalia Makarova.
Luglio 1, 2014 alle 6:58 am #199Natalia MakarovaPartecipanteI 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.
Luglio 2, 2014 alle 6:53 am #200Natalia MakarovaPartecipanteSystem of equations in integers solved at a forum in Russia
http://forum.exponenta.ru/viewtopic.php?t=9466&postdays=0&postorder=asc&start=30The 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.
Luglio 2, 2014 alle 8:42 am #201Natalia MakarovaPartecipanteI 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.Luglio 2, 2014 alle 10:08 am #202Natalia MakarovaPartecipanteThis cube looks like
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