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14.0 Special Magic Squares, Prime Numbers
Prime Number Magic Twin Squares are paired Magic Squares (A , B) with the property {bi} = {ai + d} for i = 1 ... n and d >= 2. This section will consider Twin Squares for d = 2.
14.14.1 Magic Twin Squares (3 x 3)
The enumeration of order 3 Prime Number Magic Twin Squares,
which can be found within prime number range (2 ... 9923) for d = 2 ... 9184,
has been discussed in Attachment 14.2.
Attachment 14.14.1
shows for prime number range (17 ... 11779) and d = 2 the first occurring Prime Number Magic Twin Squares for miscellaneous Magic Sums.
14.14.2 Magic Twin Squares (4 x 4)
Attachment 14.14.2
shows for prime number range (7 ... 661) and d = 2 the first occurring Prime Number Magic Twin Squares for miscellaneous Magic Sums.
14.14.3 Magic Twin Squares (5 x 5)
Attachment 14.14.16
shows for prime number range (5 ... 1609) and d = 2 the first occurring Prime Number Simple Magic Twin Squares for miscellaneous Magic Sums.
Attachment 14.14.17
shows for prime number range (11 ... 9463) and d = 2 the first occurring Prime Number Pan Magic Twin Squares for miscellaneous Magic Sums,
based on La Hirian Primaries as discussed in Section 14.12.2.
Attachment 14.14.6
shows for prime number range (13 ... 14629) and d = 2 the first occurring Prime Number Associated Magic Twin Squares for miscellaneous Magic Sums.
Based on the order 3 Prime Number Simple Magic Twin Squares as discussed in Section 14.4.1 above, following order 5 Prime Number Magic Twin Squares can be constructed:
Occasionally order 5 Prime Number Magic Twin Squares can be constructed based on consecutive prime numbers
{bi} for i = 1 ... 25.
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MC5a = 2173
107 311 599 809 347 821 431 461 281 179 191 269 827 227 659 857 521 137 239 419 197 641 149 617 569 MC5b = 2183
109 313 601 811 349 823 433 463 283 181 193 271 829 229 661 859 523 139 241 421 199 643 151 619 571
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The right square (5b) has been constructed with the Generator Method as discussed in detail in Section 14.13.4
for order 6 squares.
14.14.4 Magic Twin Squares (6 x 6)
Attachment 14.14.76
shows for prime number range (13 ... 3769) and d = 2 the first occurring Prime Number Simple Magic Twin Squares
with Symmetric Main Diagonals for miscellaneous Magic Sums.
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MC6a = 18264
5741 569 1607 3119 5231 1997 431 1871 1667 3851 4787 5657 1451 2789 5849 809 2729 4637 2549 2237 1301 4217 4421 3539 4001 5279 3359 3299 239 2087 4091 5519 4481 2969 857 347 MC6b = 18276
5743 571 1609 3121 5233 1999 433 1873 1669 3853 4789 5659 1453 2791 5851 811 2731 4639 2551 2239 1303 4219 4423 3541 4003 5281 3361 3301 241 2089 4093 5521 4483 2971 859 349
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Attachment 14.14.71
shows for prime number range (13 ... 17837) and d = 2 the first occurring Prime Number Concentric Magic Twin Squares for miscellaneous Magic Sums.
Alternatively order 6 Prime Number Magic Twin Squares can be constructed based on consecutive prime numbers
{bi} for i = 1 ... 36.
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Mc6a = 5646
1451 659 1289 827 269 1151 1619 419 1427 1019 281 881 599 431 1031 857 1667 1061 347 1301 809 617 1481 1091 311 1229 521 1277 1487 821 1319 1607 569 1049 461 641 Mc6b = 5658
1453 661 1291 829 271 1153 1621 421 1429 1021 283 883 601 433 1033 859 1669 1063 349 1303 811 619 1483 1093 313 1231 523 1279 1489 823 1321 1609 571 1051 463 643
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The right square (6b) has been constructed with the Generator Method as discussed in detail in Section 14.13.4.
Each pair shown corresponds with numerous pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 36.
14.14.5 Magic Twin Squares (7 x 7)
Based on the order 5 Prime Number Magic Twin Squares found in Section 14.4.3 above, following order 7 Prime Number Magic Twin Squares can be constructed (ref. Section 14.5.1):
Based on the order 3 and 4 Prime Number Magic Twin Squares found in Section 14.4.1 and 2 above, following order 7 Prime Number Magic Twin Squares can be constructed (ref. Section 14.5.7):
Attachment 14.14.87
contains a few examples of sets of Prime Number Pan Magic Twin Squares,
based on La Hirian Primaries as discussed in Section 14.12.4.
Order 7 Prime Number Magic Twin Squares with smaller Magic Sums can be constructed based on consecutive prime numbers
{bi} for i = 1 ... 49.
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MC7a = 9275
431 2267 2237 347 1487 2087 419 2129 2027 569 461 1427 2141 521 857 827 1931 1721 881 1607 1451 1289 821 809 1949 1871 659 1877 1091 1667 1787 1019 1061 1031 1619 1997 617 641 2081 1229 599 2111 1481 1049 1301 1697 1319 1151 1277 MC7b = 9289
433 2269 2239 349 1489 2089 421 2131 2029 571 463 1429 2143 523 859 829 1933 1723 883 1609 1453 1291 823 811 1951 1873 661 1879 1093 1669 1789 1021 1063 1033 1621 1999 619 643 2083 1231 601 2113 1483 1051 1303 1699 1321 1153 1279
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Each pair shown corresponds with numerous pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 49.
14.14.6 Magic Twin Squares, Composed (8 x 8)
In Section 14.6.1 was discussed how Prime Number Magic Squares of order 8 - with Magic Sum 2 * s1 - can be composed out of 4th order Prime Number Magic Squares with Magic Sum s1.
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MC8a = 155224
11057 7559 31247 27749 10709 1787 37019 28097 2339 29567 9239 36467 7487 32609 6197 31319 25469 1877 36929 13337 20747 4649 34157 18059 38747 38609 197 59 38669 38567 239 137 18917 10499 28307 19889 29387 23669 15137 9419 4049 25409 13397 34757 3299 13007 25799 35507 16187 5867 32939 22619 6689 4157 34649 32117 38459 35837 2969 347 38237 36779 2027 569 MC8b = 155240
11059 7561 31249 27751 10711 1789 37021 28099 2341 29569 9241 36469 7489 32611 6199 31321 25471 1879 36931 13339 20749 4651 34159 18061 38749 38611 199 61 38671 38569 241 139 18919 10501 28309 19891 29389 23671 15139 9421 4051 25411 13399 34759 3301 13009 25801 35509 16189 5869 32941 22621 6691 4159 34651 32119 38461 35839 2971 349 38239 36781 2029 571
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Attachment 14.14.4
contains a few more sets of Prime Number Magic Twin Squares, which can be used to construct Composed Magic Twin Squares of order 8
(ref. Priem4c2).
Order 8 Prime Number Magic Twin Squares with smaller Magic Sums can be constructed based on consecutive prime numbers
{bi} for i = 1 ... 64.
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MC8a = 7612
17 1619 1091 1697 1229 881 269 809 1319 179 857 101 347 1931 1451 1427 521 227 1289 2129 107 311 1997 1031 29 1301 1787 827 1061 71 1877 659 1607 1667 461 821 191 641 137 2087 1481 431 59 1019 2081 2111 149 281 1487 1949 197 599 569 1049 1721 41 1151 239 1871 419 2027 617 11 1277 MC8b = 7628
19 1621 1093 1699 1231 883 271 811 1321 181 859 103 349 1933 1453 1429 523 229 1291 2131 109 313 1999 1033 31 1303 1789 829 1063 73 1879 661 1609 1669 463 823 193 643 139 2089 1483 433 61 1021 2083 2113 151 283 1489 1951 199 601 571 1051 1723 43 1153 241 1873 421 2029 619 13 1279
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Each pair shown corresponds with numerous pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 64.
14.14.7 Magic Twin Squares (9 x 9)
Composed Magic Twin Squares of order 9 can be constructed based on a combination of
one order 3 Magic Twin Square with 8 Semi Magic Twin Squares (6 Magic Lines).
of which a few examples are shown in Attachment 14.14.10.
Order 9 Prime Number Magic Twin Squares with smaller Magic Sums can be constructed based on consecutive prime numbers
{bi} for i = 1 ... 81.
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MC9a = 13845
107 101 137 197 3359 3329 3299 3167 149 3257 3251 2969 2711 191 179 239 821 227 269 347 311 281 3119 1031 2801 2687 2999 1481 1931 1619 1277 1607 1319 1427 1487 1697 857 1877 2309 659 2267 2129 809 827 2111 1667 1949 1997 2141 881 1019 2081 1049 1061 1787 1091 1229 1451 1301 2027 2087 1151 1721 1871 569 617 2339 599 2381 641 2237 2591 2549 2729 2657 2789 521 431 461 419 1289 MC9b = 13863
109 103 139 199 3361 3331 3301 3169 151 3259 3253 2971 2713 193 181 241 823 229 271 349 313 283 3121 1033 2803 2689 3001 1483 1933 1621 1279 1609 1321 1429 1489 1699 859 1879 2311 661 2269 2131 811 829 2113 1669 1951 1999 2143 883 1021 2083 1051 1063 1789 1093 1231 1453 1303 2029 2089 1153 1723 1873 571 619 2341 601 2383 643 2239 2593 2551 2731 2659 2791 523 433 463 421 1291
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The right square (9b) has been constructed with the Generator Method as discussed in detail in Section 14.13.10
for order 10 squares.
14.14.8 Magic Twin Squares (10 x 10)
Based on the order 8 Prime Number Concentric Magic Twin Squares found in Section 14.4.6 above, order 10 Prime Number Magic Twin Squares can be constructed (ref. Section 14.8.3):
Attachment 14.14.30
shows for prime number range (29 ... 42703) and d = 2 the first occurring Prime Number Composed Magic Twin Squares, with order 4 Associated Center Square, for a few Magic Sums.
Order 10 Prime Number Magic Twin Squares with smaller Magic Sums can be constructed based on consecutive prime numbers
{bi} for i = 1 ... 100.
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MC10a = 16784
197 3527 179 1877 149 227 3389 191 3581 3467 3257 347 3251 311 3359 3329 281 2141 239 269 1289 2129 1931 1319 1151 1277 1301 2381 1667 2339 461 2999 1949 3167 419 431 2969 569 3299 521 5 1871 29 41 17 11 3371 3767 3851 3821 3671 107 3461 2081 3539 3557 101 59 71 137 2549 1049 2267 1061 2591 2309 1229 1607 1031 1091 2711 617 809 2789 3119 2729 659 2111 599 641 1787 1451 2027 1481 1619 2087 1487 1721 1427 1697 857 2687 881 2657 821 827 1997 2237 1019 2801 MC10b = 16804
199 3529 181 1879 151 229 3391 193 3583 3469 3259 349 3253 313 3361 3331 283 2143 241 271 1291 2131 1933 1321 1153 1279 1303 2383 1669 2341 463 3001 1951 3169 421 433 2971 571 3301 523 7 1873 31 43 19 13 3373 3769 3853 3823 3673 109 3463 2083 3541 3559 103 61 73 139 2551 1051 2269 1063 2593 2311 1231 1609 1033 1093 2713 619 811 2791 3121 2731 661 2113 601 643 1789 1453 2029 1483 1621 2089 1489 1723 1429 1699 859 2689 883 2659 823 829 1999 2239 1021 2803
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The right square (10b) has been constructed with the Generator Method as discussed in detail in Section 14.13.10.
14.14.9 Magic Twin Squares (11 x 11)
Based on the order 9 Prime Number Magic Twin Squares found in Section 14.4.7 above, following order 11 Prime Number Magic Twin Squares can be constructed (ref. Section 14.9.1):
of which a few examples are shown in Attachment 14.14.11.
Order 11 Prime Number Magic Twin Squares with smaller Magic Sums can be constructed based on consecutive prime numbers
{bi} for i = 1 ... 121.
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MC11a = 25189, MC4a = 8420
1931 2711 1229 2549 4931 107 149 4967 137 4481 1997 3539 1289 1721 1871 2237 4799 197 4649 191 4517 179 2129 3329 2081 881 269 2267 4787 4721 239 4259 227 821 1091 3389 3119 281 2309 347 311 4547 4337 4637 4157 4217 521 4421 1481 461 4271 431 4241 419 569 1607 1619 1667 1697 3557 3581 1949 1877 3257 1787 2591 3917 4049 4229 641 809 599 4127 617 1451 659 4091 71 2789 101 2729 3527 3299 2999 2801 3461 41 3371 1061 1151 3851 1319 1487 3767 2687 1427 3671 1301 3467 3929 857 4019 3821 3359 1031 1019 1277 827 1049 4001 2027 2087 2381 2141 3251 2969 2657 2111 3167 2339 59 MC11b = 25211, MC4b = 8428
1933 2713 1231 2551 4933 109 151 4969 139 4483 1999 3541 1291 1723 1873 2239 4801 199 4651 193 4519 181 2131 3331 2083 883 271 2269 4789 4723 241 4261 229 823 1093 3391 3121 283 2311 349 313 4549 4339 4639 4159 4219 523 4423 1483 463 4273 433 4243 421 571 1609 1621 1669 1699 3559 3583 1951 1879 3259 1789 2593 3919 4051 4231 643 811 601 4129 619 1453 661 4093 73 2791 103 2731 3529 3301 3001 2803 3463 43 3373 1063 1153 3853 1321 1489 3769 2689 1429 3673 1303 3469 3931 859 4021 3823 3361 1033 1021 1279 829 1051 4003 2029 2089 2383 2143 3253 2971 2659 2113 3169 2341 61
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The right square (11b) has been constructed with the Generator Method as discussed in detail in Section 14.13.21 and
Section 14.13.23.
14.14.10 Magic Twin Squares (12 x 12)
Based on the order 10 Prime Number Concentric Magic Twin Squares found in Section 14.4.8 above, order 12 Prime Number Concentric Magic Twin Squares can be constructed (ref. Priem12a).
Based on the order 6 Prime Number Magic Twin Squares as discussed in Section 14.4.4 above, following order 12 Composed Prime Number Magic Twin Squares can be constructed (ref. Priem12b):
of which a few examples are shown in Attachment 14.14.22.
Order 12 Prime Number Magic Twin Squares with smaller Magic Sums can be constructed based on consecutive prime numbers
{bi} for i = 1 ... 144.
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Mc12a = 36138
881 2309 6197 4649 281 2591 599 5657 347 269 6269 6089 2141 809 5867 5519 5501 1049 2381 419 5741 521 5879 311 5477 3917 659 5417 641 5849 5639 617 821 5441 569 1091 5099 857 1061 5021 1151 5279 5231 1301 4271 5009 827 1031 1871 4241 4229 2027 3929 1877 4421 3299 1997 4217 2081 1949 1427 2801 6659 191 6569 1721 431 6131 3329 149 179 6551 1931 461 239 4337 6449 1019 2969 5651 197 227 6359 6299 4517 4637 1607 1787 1619 4547 1481 4049 4259 4481 1487 1667 3359 3467 3389 137 3461 2789 3539 2999 3371 3251 3257 3119 4799 4931 1277 1229 1697 4967 1289 3671 4721 4787 1319 1451 2087 4127 2267 2657 2129 2111 4157 2237 3527 4019 4091 2729 2549 3581 2687 3167 2711 2339 4001 107 3557 3767 3821 3851 Mc12b = 36162
883 2311 6199 4651 283 2593 601 5659 349 271 6271 6091 2143 811 5869 5521 5503 1051 2383 421 5743 523 5881 313 5479 3919 661 5419 643 5851 5641 619 823 5443 571 1093 5101 859 1063 5023 1153 5281 5233 1303 4273 5011 829 1033 1873 4243 4231 2029 3931 1879 4423 3301 1999 4219 2083 1951 1429 2803 6661 193 6571 1723 433 6133 3331 151 181 6553 1933 463 241 4339 6451 1021 2971 5653 199 229 6361 6301 4519 4639 1609 1789 1621 4549 1483 4051 4261 4483 1489 1669 3361 3469 3391 139 3463 2791 3541 3001 3373 3253 3259 3121 4801 4933 1279 1231 1699 4969 1291 3673 4723 4789 1321 1453 2089 4129 2269 2659 2131 2113 4159 2239 3529 4021 4093 2731 2551 3583 2689 3169 2713 2341 4003 109 3559 3769 3823 3853
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The right square (12b) has been constructed with the Generator Method as discussed in detail in Section 14.13.31.
The obtained results regarding the miscellaneous types of Prime Number Magic Twin Squares as deducted and discussed in previous sections are summarized in following table: |
Order
Main Characteristics
Subroutine
Results
3
Simple Magic
4
Simple Magic
Pan Magic
5
Simple Magic
Pan Magic
Associated
Concentric
Magic, Square Inlay
Magic, Diamond Inlay
6
Concentric
Associated
Pan Magic and Complete
Euler
Simple, Symmetric Main Diagonals
Simple, Four Semi Magic Sub Squares
Simple, Two Semi Magic Bottom Squares
7
Bordered
Composed, Simple Magic Corner Squares
Pan Magic
8
Composed, Simple Magic Sub Squares
Concentric
9
Composed, Semi Magic Sub Squares
Concentric
10
Concentric
Bordered, Composed Center Square
Composed, Associated Center Square
11
Concentric
12
Concentric
Composed, Miscellaneous Types
Composed, Simple Magic Sub Squares
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-
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Following sections will provide miscellaneous construction methods for paired squares based on Sophie Germain Primes.
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