Office Applications and Entertainment, Magic Squares  
Index  About the Author 
14.0 Special Magic Squares, Prime Numbers
Prime Number Magic Twin Squares are paired Magic Squares (A , B) with the property {b_{i}} = {a_{i} + 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:
Each pair shown corresponds with numerous pairs with the same Magic Sums and variable values {a_{i}}/{b_{i}}, 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.

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.
Each pair shown corresponds with numerous pairs with the same Magic Sums and variable values {a_{i}}/{b_{i}}, 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.
Each pair shown corresponds with numerous pairs with the same Magic Sums and variable values {a_{i}}/{b_{i}}, 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 4^{th} order Prime Number Magic Squares with Magic Sum s1.

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).
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.
Each pair shown corresponds with numerous pairs with the same Magic Sums and variable values {a_{i}}/{b_{i}}, 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 14.14.30
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.
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.
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.
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.

Index  About the Author 