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Magic cubes constructed using all prime numbers are relatively rare. The four I found all appeared in three articles in the Journal of Recreational Mathematics.
Prime number magic cubes may come in a variety of types, just as prime magic squares do. However, they can never be classed as normal because they cannot be constructed using consecutive numbers.
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This order-3 prime number magic cube was constructed by Akio Suzuki in 1977 [1]. Like all order 3 magic cubes, it is associated, but not pantriagonal. And like odd order associated magic cubes, the 3 central orthogonal planes are associated magic squares. Prime numbers used range from 107 to 2999. Each complement pair sums 3106. The constant is 4659 (sorry, not a prime). Also, like other associated magic cubes, this one is semi-pantriagonal. |
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Also constructed by Akio Suzuki in 1977
[1]. This
cube has exactly the same characteristics as the above cube except it uses a
smaller prime numbers. Prime numbers used range from 53 to 2153. |
Addendum: As a result of a computer search, Allen Wm, Johnson, Jr. [2] confirmed that this cube has the smallest possible sum for an order 3 prime magic cube using distinct digits.
Constructed by Gakuho Abe in 1977 [1], this magic cube is
not associated.
It is simple magic with no extra features (except that it uses prime numbers).
The magic constant, S = 4020. Prime numbers used range from 7 to 2003.
|
7 |
1999 |
17 |
1997 |
|
1873 |
37 |
1979 |
131 |
|
233 |
1013 |
991 |
1783 |
|
1907 |
971 |
1033 |
109 |
|
1753 |
733 |
1283 |
251 |
|
311 |
1549 |
467 |
1693 |
|
1069 |
557 |
1447 |
947 |
|
887 |
1181 |
823 |
1129 |
|
257 |
1277 |
727 |
1759 |
|
1699 |
461 |
1543 |
317 |
|
941 |
1453 |
563 |
1063 |
|
1123 |
829 |
1187 |
881 |
|
2003 |
11 |
1993 |
13 |
|
137 |
1973 |
31 |
1879 |
|
1777 |
997 |
1019 |
227 |
|
103 |
1039 |
977 |
1901 |
Note that none of the magic cubes shown on this page use consecutive prime numbers. Is such a cube possible?
This cube, constructed in 1985 [3], consists of 4 digit
primes. Johnson calls this cube ‘pandiagonal’, a common name for a cube where
all the oblique square pandiagonals are correct. We now call it
pantriagonal because all the pan-3-agonals are correct. S = 19740.
(A pandiagonal magic cube is one where all orthogonal planes are pandiagonal
magic squares.)
The difference in the sums of the two pairs of numbers in each pantriagonal that are spaced m/2 apart is 660. If the difference was 0, this cube would be called complete (the two sets of numbers would be complement pairs). That is a feature that is fairly common in order 4 pantriagonal magic cubes.
Prime numbers used range from 283 to 9587. This cube is
associated because two numbers symmetrically located across the center
point of the cube sum to the total of the first and last numbers in the series.
I assume it is only possible for an order 4 pantriagonal magic cube to be
associated if the cube is not normal (i.e. the numbers used are not consecutive
and start at 1).
In the same article, Johnson showed several other order-4 prime cubes that were
not associated.
|
5851 |
5743 |
6143 |
2003 |
|
8243 |
4877 |
6007 |
613 |
|
3209 |
5573 |
2281 |
8677 |
|
2437 |
3547 |
5309 |
8447 |
|
4547 |
8573 |
283 |
6337 |
|
6073 |
5521 |
2333 |
5813 |
|
3217 |
2767 |
8117 |
5639 |
|
5903 |
2879 |
9007 |
1951 |
|
7919 |
863 |
6991 |
3967 |
|
4231 |
1753 |
7103 |
6653 |
|
4057 |
7537 |
4349 |
3797 |
|
3533 |
9587 |
1297 |
5323 |
|
1423 |
4561 |
6323 |
7433 |
|
1193 |
7589 |
4297 |
6661 |
|
9257 |
3863 |
4993 |
1627 |
|
7867 |
3727 |
4127 |
4019 |
This order-6 cube and the order-8 following are both firsts
to the best of my knowledge. hh (see The Cube
TimeLine)
I received this cube in an email from Zhong Ming of Dazhou city in Sichuan
province, China, and Peng Baowang of Qinghe county in Hebei, China on September
7, 2009. The cube was dated August 31, 2009.
Because all rows, columns and pillars as well as the 4 tragonals sum to the same
constant, the cube is magic. It contains no magic squares (in fact, no diagonals
sum correct), and no additional features, so is a Simple magic cube. It consists
of 216 unique numbers (i.e. no duplicates), all of which are prime.
Top Plane Top - 1 Top - 2 4831 4783 67 9811 4639 5479 131 761 379 9403 9497 9439 337 8849 8821 1409 1307 8887 191 241 193 9473 9769 9743 8951 2437 3547 5309 8447 919 7013 5903 2879 9007 1951 2857 331 577 5009 4751 9619 9323 9643 3209 5573 2281 8677 227 8009 3217 2767 8117 5639 1861 8273 9719 8933 1123 829 733 2143 8243 4877 6007 613 7727 9049 6073 5521 2333 5813 821 8423 7499 8287 1789 1801 1811 8311 5851 5743 6143 2003 1559 4219 4547 8573 283 6337 5651 7561 6791 7121 2663 2953 2521 431 9109 9491 467 373 9739 983 1021 1049 8461 8563 9533 Bottom + 2 Bottom + 1 Bottom Plane 8543 8839 9277 173 1831 947 8419 3299 8317 1607 5419 2549 7349 3079 2749 7207 6917 2309 4177 3533 9587 1297 5323 5693 9151 7867 3727 4127 4019 719 127 9629 9677 397 101 9679 7487 4057 7537 4349 3797 2383 3593 9257 3863 4993 1627 6277 547 9293 4861 5119 251 9539 31 4231 1753 7103 6653 9839 977 1193 7589 4297 6661 8893 9137 151 937 8747 9041 1597 449 7919 863 6991 3967 9421 149 1423 4561 6323 7433 9721 8059 2371 1583 8081 8069 1447 8923 1031 593 9697 8039 1327 7321 6571 1553 8263 4451 1451 4391 5087 9803 59 5231 5039
On September 8, 2009, I received another email from Zhong
Ming and Peng Baowang. Attached was an order-8 prime number magic cube
containing an order-6 and an order-4 cube.
All cubes are magic because all orthogogonal lines and the 4 triagonals sum
correctly to the constant.
No planar diagonals sum correctly so there are no magic squares in any of the 3
cubes. However, all broken triagonals sum correctly in the order-4 cube, so it
is pantriagonal magic. (The other 2 are simple magic.). All numbers are unique
primes.
The longest consecutive run of primes is right at the high end , with 9 adjacent
primes.
The order-4 cube uses primes from 283 to 9587. S = 19740
The order-6 cube uses primes from 31 to 9839. S = 29610 The green numbers
indicate the outside planes of this cube.
The order-4 cube uses primes from 11 to 9857. S = 39480
Because the high and low numbers are scattered throughout the 3 cubes, this cube
is considered concentric rather then bordered.
Top layer Top - 1 13 9859 6679 9829 2129 53 6869 4049 811 9127 7841 5867 7211 2909 3931 1783 1637 9781 103 8171 181 7577 9733 2297 6781 4831 4783 67 9811 4639 5479 3089 9511 349 3623 269 433 9787 7691 7817 4229 191 241 193 9473 9769 9743 5641 9631 257 7331 2477 9371 9413 521 479 409 331 577 5009 4751 9619 9323 9461 9283 1039 941 631 8837 661 8861 9227 7177 8273 9719 8933 1123 829 733 2693 6803 709 3613 8443 9187 3541 2617 4567 4967 8423 7499 8287 1789 1801 1811 4903 1493 8707 9043 907 8291 6701 1171 3167 7019 7561 6791 7121 2663 2953 2521 2851 1109 8779 8147 8753 1051 1747 2017 7877 8087 743 2029 4003 2659 6961 5939 9059 Top - 2 top - 3 8431 1289 4951 4933 1063 8941 9013 859 8783 1181 8093 1759 1933 6379 2633 8719 5717 131 761 379 9403 9497 9439 4153 4201 337 8849 8821 1409 1307 8887 5669 6151 8951 2437 3547 5309 8447 919 3719 3671 7013 5903 2879 9007 1951 2857 6199 19 9643 3209 5573 2281 8677 227 9851 8641 8009 3217 2767 8117 5639 1861 1229 2711 2143 8243 4877 6007 613 7727 7159 7523 9049 6073 5521 2333 5813 821 2347 3307 8311 5851 5743 6143 2003 1559 6563 2719 4219 4547 8573 283 6337 5651 7151 4133 431 9109 9491 467 373 9739 5737 2791 983 1021 1049 8461 8563 9533 7079 9011 8581 4919 4937 8807 929 857 1439 1151 8689 1777 8111 7937 3491 7237 1087 Bottom + 3 Bottom + 2 8669 1223 1483 7583 2267 7477 5197 5581 7529 7993 2111 3041 7789 3889 3947 3181 3779 8543 8839 9277 173 1831 947 6091 3449 8419 3299 8317 1607 5419 2549 6421 3917 4177 3533 9587 1297 5323 5693 5953 4021 9151 7867 3727 4127 4019 719 5849 5881 7487 4057 7537 4349 3797 2383 3989 1481 3593 9257 3863 4993 1627 6277 8389 2381 31 4231 1753 7103 6653 9839 7489 2113 977 1193 7589 4297 6661 8893 7757 3803 449 7919 863 6991 3967 9421 6067 7129 149 1423 4561 6323 7433 9721 2741 6761 8923 1031 593 9697 8039 1327 3109 7069 7321 6571 1553 8263 4451 1451 2801 4289 8647 8387 2287 7603 2393 4673 1201 6689 1877 7759 6829 2081 5981 5923 2341 Bottom + 1 Bottom layer 3251 7717 6599 5351 8269 1709 37 6547 1993 1091 1723 1117 8819 8123 7853 8761 6343 7349 3079 2749 7207 6917 2309 3527 7573 89 9767 1699 9689 2293 137 8233 5927 127 9629 9677 397 101 9679 3943 2053 9521 6247 9601 9437 83 2179 359 4027 547 9293 4861 5119 251 9539 5843 9391 9613 2539 7393 499 457 9349 239 7649 9137 151 937 8747 9041 1597 2221 643 8831 8929 9239 1033 9209 1009 587 5449 8059 2371 1583 8081 8069 1447 4421 5303 9161 6257 1427 683 6329 7253 3067 3511 4391 5087 9803 59 5231 5039 6359 6703 1163 827 8963 1579 3169 8699 8377 3323 2153 3271 4519 1601 8161 9833 6619 5821 11 3191 41 7741 9817 3001 9857
What is the smallest possible prime number magic cube? Now answered for order 3.
What is the
smallest possible consecutive prime numbers magic cube?
If such a cube is possible, it would use astronomically large
numbers. Nelson [4] constructed the first order-3 magic
square using 9 consecutive prime numbers starting with
1,480,028,129.
An order-3 prime cube would require a string of 27
suitable consecutive prime numbers!
[1] Gakuho Abe, Related Magic Squares with Prime
Elements, JRM 10:2 1977-78, pp.96-97.
[2] A. W. Johnson, Jr., Solution to Problem 2617, JRM 32:4, 2003-2004,
pp. 338-339
[3] A. W. Johnson, Jr., An Order 4 Prime Magic Cube, JRM 18:1, 1985-86,
pp 5-7
[4] H. L. Nelson, A Consecutive Prime 3 x 3 Magic Square, JRM, 1988, vol.
20:3, pp 214-216
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This page last updated
October 15, 2009
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