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 14.0    Special Magic Squares, Prime Numbers 14.14   Magic Twin Squares 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. Prime Number Magic Twin Squares can be generated with comparable routines as described in previous sections, however based on those prime variable values {bi} for which also {bi - 2} is prime for i = 1 ... n. 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. Each pair shown corresponds with 8 pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 9. 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. Each pair corresponds with 32 pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 16. Attachment 14.14.3 shows for prime number range (13 ... 50551) and d = 2 the first occurring Prime Number Pan Magic Twin Squares for miscellaneous Magic Sums. Each pair shown corresponds with 384 pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 16. 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: Attachment 14.14.9  Prime Number Concentric Magic Twin Squares         (ref. Priem5c2) Attachment 14.14.13 Prime Number Magic Twin Squares with Square  Inlay (ref. Priem5g3) Attachment 14.14.14 Prime Number Magic Twin Squares with Diamond Inlay (ref. Priem5g2) Each pair shown corresponds with numerous pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 25. 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. Attachment 14.14.72 shows for prime number range (19 ... 49921) and d = 2 the first occurring Prime Number Associated Magic Twin Squares for miscellaneous Magic Sums. Attachment 14.14.75 shows the corresponding Prime Number Pan Magic and Complete Twin Squares (Eulers Transformation). Based on order 4 Prime Number (Pan) Magic Twin Squares, order 6 Prime Number Concentric Magic Twin Squares can be constructed (ref. Priem6b2). An example of a pair of Prime Number Concentric Magic Twin Squares, with Pan Magic Center Squares, is shown below:
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
 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. Following order 6 Prime Number Simple Magic Twin Squares can be constructed, based on combinations of order 3 Prime Number Semi Magic Twin Squares (6 Magic Lines): Attachment 14.14.73 Prime Number Simple Magic Twin Squares, four order 3 Semi Magic Sub Squares Attachment 14.14.74 Prime Number Simple Magic Twin Squares, two order 3 Semi Magic Bottom Squares 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): Attachment 14.14.81 Prime Number Concentric Magic Twin Squares Attachment 14.14.82 Prime Number Bordered   Magic Twin Squares, Center Square with Square  Inlay Attachment 14.14.83 Prime Number Bordered   Magic Twin Squares, Center Square with Diamond Inlay Attachment 14.14.84 Prime Number Bordered   Magic Twin Squares, Associated Center Square 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 14.14.85 Prime Number Composed   Magic Twin Squares, Simple Magic Corner Squares Attachment 14.14.86 Prime Number Composed   Magic Twin Squares, Simple Embedded Magic Squares 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. 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. An example of a Magic Sum for which a set of 4 Prime Number Magic Twin Squares can be found is s1 = 77620:
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
 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). Attachment 14.14.7 shows for prime number range (59 ... 98929) and d = 2 the first occurring Prime Number Composed Magic Twin Squares of order 8 for a few Magic Sums. Attachment 14.14.5 shows for prime number range (13 ... 50593) and d = 2 the first occurring Prime Number Concentric Magic Twin Squares for a few Magic Sums. 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). Attachment 14.14.20 shows for prime number range (5 ... 45343) and d = 2 the first occurring Prime Number Composed Magic Twin Squares for a few Magic Sums (ref. Priem9b). Based on the order 7 Prime Number Magic Twin Squares found in Section 14.4.5 above, following order 9 Prime Number Magic Twin Squares can be constructed (ref. Section 14.7.4): Prime Number Concentric Magic Twin Squares Prime Number Bordered   Magic Twin Squares, Center Square with Square Inlay Prime Number Bordered   Magic Twin Squares, Center Square with Diamond Inlay Prime Number Bordered   Magic Twin Squares, Associated Center Square of which a few examples are shown in Attachment 14.14.10. Order 9 Associated Magic Squares with order 4 and 5 Square Inlays, can be obtained by means of a transformation of order 9 Composed Magic Squares (ref. Section 14.7.11). An Example, based on order 4 and 5 Prime Number Magic Twin Squares found in Section 14.4.2 and 3 above, is enclosed in Attachment 14.14.95 which contains: Prime Number Composed   Magic Twin Squares with Associated Corner Squares and Rectangles Prime Number Associated Magic Twin Squares with Associated Embedded Magic Squares Each pair shown corresponds with numerous pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 81. 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 14.14.12 Prime Number Concentric Magic Twin Squares Attachment 14.14.18 Prime Number Bordered   Magic Twin Squares, Composed Center Square 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. Each pair corresponds with numerous pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 100. 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): Prime Number Concentric Magic Twin Squares Prime Number Bordered   Magic Twin Squares, Center Square with Square Inlay Prime Number Bordered   Magic Twin Squares, Center Square with Diamond Inlay of which a few examples are shown in Attachment 14.14.11. Each pair corresponds with numerous pairs with the same Magic Sum and variable values {ai}/{bi}, i = 1 ... 121. 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). Attachment 14.14.15 shows for prime number range (13 ... 98809) and d = 2 the first occurring Prime Number Concentric Magic Twin Squares for a few Magic Sums. 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): Composed, 16 Semi Magic Sub Squares (3 x 3) Composed, Associated Center Square  (6 x 6), 12 Semi Magic Border Squares (3 x 3) Composed, Concentric Center Square  (6 x 6), 12 Semi Magic Border Squares (3 x 3) of which a few examples are shown in Attachment 14.14.22. Prime Number Magic Squares of order 12 - with Magic Sum 3 * s1 - can be composed out of 4th order Prime Number Magic Squares with Magic Sum s1. Attachment 14.14.23 shows for prime number range (19 ... 49939) and d = 2 the first occurring Prime Number Composed Magic Twin Squares of order 12, for a few Magic Sums. Each pair corresponds with numerous pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 144. 14.14.11 Summary 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 - - - -
 Following sections will provide miscellaneous construction methods for paired squares based on Sophie Germain Primes.