Office Applications and Entertainment, Magic Squares

Vorige Pagina Volgende Pagina Index About the Author

14.0    Special Magic Squares, Prime Numbers

14.17   Sophie Germain Primes

Sophie Germain Primes {ai} are prime numbers for which the operation {bi} = 2 * {ai} + 1 results in prime numbers {bi} for i = 1 ... n.

Paired Prime Number Magic Squares (A , B), with the property {bi} = 2 * {ai} + 1 for i = 1 ... n, can be generated with comparable routines as described in previous sections.

14.17.1 Magic Squares (3 x 3)

Attachment 14.17.1 shows for the range {bi} = (23 ... 174299) the first occurring pairs of 3 x 3 Sophie Germain Magic 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.

14.17.2 Magic Squares (4 x 4)

Attachment 14.17.14 shows for the range {bi} = (11 ... 1319) the first occurring pairs of 4 x 4 Sophie Germain Simple Magic Squares for miscellaneous (smaller) Magic Sums (ref. Priem4a).

Attachment 14.17.2 shows for the range {bi} = (23 ... 43607) the first occurring pairs of 4 x 4 Sophie Germain Simple Magic Squares for miscellaneous (larger) Magic Sums (ref. Priem4b).

Each pair shown corresponds with 32 pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 16.

Occasionally pairs of 4 x 4 Sophie Germain Pan Magic Squares can be found, of which an example is shown below:

MC4a = 21944
3779 3623 6899 7643
3761 10781 641 6761
4073 3329 7193 7349
10331 4211 7211 191
MC4b = 43892
7559 7247 13799 15287
7523 21563 1283 13523
8147 6659 14387 14699
20663 8423 14423 383

The pair shown corresponds with 384 pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 16.

Attachment 14.17.4 shows for the range {ci} = (5639 ... 3827639) a few triplets of 4 x 4 Simple Magic Squares with Sophie Germain Primes {ai}, {bi} = 2 * {ai} + 1 and {ci} = 2 * {bi} + 1 for miscellaneous Magic Sums.

Each triplet shown corresponds with 32 triplets with the same Magic Sums and variable values {ai}/{bi}/{ci}, i = 1 ... 16.

14.17.3 Magic Squares (5 x 5)

Attachment 14.17.3 shows for the range {bi} = (23 ... 43607) the first occurring pairs of 5 x 5 Sophie Germain Simple Magic Squares for miscellaneous Magic Sums.

Attachment 14.17.4 shows for the range {bi} = (23 ... 43607) the first occurring pairs of 5 x 5 Sophie Germain Associated Magic Squares for miscellaneous Magic Sums.

Attachment 14.17.5 shows for the range {bi} = (23 ... 43607) the first occurring pairs of 5 x 5 Sophie Germain Pan Magic Squares for miscellaneous Magic Sums, based on La Hirian Primaries as discussed in Section 14.12.2.

Based on the pairs of 3 x 3 Sophie Germain Magic Squares as discussed in Section 14.17.1 above, following pairs of 5 x 5 Sophie Germain Magic Squares can be constructed:

Each pair shown corresponds with numerous pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 25.

14.17.4 Magic Squares (6 x 6)

Attachment 14.17.9 shows for the range {bi} = (23 ... 43607) the first occurring pairs of 6 x 6 Sophie Germain Simple Magic Squares, with symmetrical main diagonals, for miscellaneous Magic Sums.

Based on pairs of 4 x 4 Sophie Germain Simple Magic Squares as discussed in Section 14.17.2 above, pairs of 6 x 6 Sophie Germain Bordered Magic Squares can be constructed (ref. Priem6b2).

An example of a pair of 6 x 6 Sophie Germain Bordered Magic Squares, with Simple Magic Center Squares, is shown below:

MC6a = 20946
6173 719 2039 2063 6329 3623
419 6899 431 1583 5051 6563
1901 131 6323 5171 2339 5081
3593 5003 1811 659 6491 3389
5501 1931 5399 6551 83 1481
3359 6263 4943 4919 653 809
MC6b = 41898
12347 1439 4079 4127 12659 7247
839 13799 863 3167 10103 13127
3803 263 12647 10343 4679 10163
7187 10007 3623 1319 12983 6779
11003 3863 10799 13103 167 2963
6719 12527 9887 9839 1307 1619

Attachment 14.17.10 shows for the range {bi} = (23 ... 43607) the first occurring pairs of 6 x 6 Sophie Germain Bordered Magic Squares for miscellaneous Magic Sums.

Each pair shown corresponds with numerous pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 25.

14.17.5 Magic Squares (7 x 7)

Attachment 14.17.11 shows a few examples of pairs of 7 x 7 Sophie Germain Bordered Magic Squares, based on the pairs of 5 x 5 Sophie Germain Simple Magic Squares as discussed in Section 14.17.3 above.

Attachment 14.17.15 shows a few examples of pairs of 7 x 7 Sophie Germain Concentric Magic Squares, based on the pairs of 5 x 5 Sophie Germain Concentric Magic Squares as discussed in Section 14.17.3 above.

Each pair shown corresponds with numerous pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 49, (Order of magnitude 8 * (5!)2 = 115200 for the Same Center Square).

14.17.6 Magic 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 suitable Sophie Germain Simple Magic Squares can be found is shown below:

MC8a = 43888
10883 911 2939 7211 8273 2819 7883 2969
1559 9371 7151 3863 683 10799 2393 8069
5741 3821 1601 10781 2459 6053 2699 10733
3761 7841 10253 89 10529 2273 8969 173
7349 281 6491 7823 3851 2549 6323 9221
1031 7643 4211 9059 8741 7121 1289 4793
6983 3329 6761 4871 4019 6101 7433 4391
6581 10691 4481 191 5333 6173 6899 3539
MC8b = 87784
21767 1823 5879 14423 16547 5639 15767 5939
3119 18743 14303 7727 1367 21599 4787 16139
11483 7643 3203 21563 4919 12107 5399 21467
7523 15683 20507 179 21059 4547 17939 347
14699 563 12983 15647 7703 5099 12647 18443
2063 15287 8423 18119 17483 14243 2579 9587
13967 6659 13523 9743 8039 12203 14867 8783
13163 21383 8963 383 10667 12347 13799 7079

Attachment 14.17.12 contains a few more sets of Sophie Germain Simple Magic Squares, which can be used to construct pairs of Composed Sophie Germain Magic Squares of order 8 (ref. Priem4c2).

Attachment 14.17.13 shows for prime number range (23 ... 43607) the first occurring pairs of Composed Sophie Germain Magic Squares of order 8 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, (Order of magnitude 4! * 324 = 25165824).

Attachment 14.17.16 shows a few examples of pairs of 8 x 8 Sophie Germain Concentric Magic Squares, based on the pairs of 6 x 6 Sophie Germain Bordered Magic Squares as discussed in Section 14.17.4 above.

Each pair shown corresponds with numerous pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 64, (Order of magnitude 8 * (6!)2 = 4147200 for the Same Center Square).

14.17.7 Magic Squares, Composed (9 x 9)

Pairs of Composed Sophie Germain Magic Squares of order 9 can be constructed based on a combination of order 3 Magic Center Squares with 8 Semi Magic Squares (6 Magic Lines).

Attachment 14.17.17 shows for prime number range (23 ... 174299) the first occurring pairs of Composed Sophie Germain Magic Squares of order 9 for a few Magic Sums (ref. Priem9b1).

Attachment 14.17.18 shows a few examples of pairs of 9 x 9 Sophie Germain Concentric Magic Squares, based on the pairs of 7 x 7 Sophie Germain Concentric Magic Squares as discussed in Section 14.7.5 above.

Each pair shown corresponds with numerous pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 81.

14.17.8 Magic Squares, Composed (10 x 10)

Pairs of Composed Sophie Germain Magic Squares of order 10 can be constructed based on a combination of order 4 Magic Center Squares with 4 Semi Magic Corner Squares (6 Magic Lines).

Attachment 14.17.19 shows for prime number range (23 ... 174299) the first occurring pairs of Composed Sophie Germain Magic Squares of order 10 for a few Magic Sums (ref. Priem10b1).

Attachment 14.17.20 shows a few examples of pairs of 10 x 10 Sophie Germain Concentric Magic Squares, based on the pairs of 8 x 8 Sophie Germain Concentric Magic Squares as discussed in Section 14.17.6 above.

Each pair shown corresponds with numerous pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 100.

14.17.9 Magic Squares, Composed (12 x 12)

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 14.17.23 shows for prime number range (23 ... 174299) the first occurring pairs of Composed Sophie Germain Magic Squares of order 12 for a few Magic Sums.

Each pair shown corresponds with numerous pairs with the same Magic Sums and variable values {ai}/{bi}, i = 1 ... 100.

14.17.10 Summary

The obtained results regarding the miscellaneous types of Prime Number Magic Squares as deducted and discussed in previous sections are summarized in following table:

Order

Main Characteristics

Subroutine

Results

3

Simple Magic

Priem3

Attachment 14.17.1

4

Simple Magic, Smaller Magic Sums

Priem4a

Attachment 14.17.14

Simple Magic, Larger  Magic Sums

Priem4b

Attachment 14.17.2

Simple Magic, Triplets

Attachment 14.17.24

5

Simple Magic

Priem5a

Attachment 14.17.3

Associated

Priem5e

Attachment 14.17.4

Pan Magic

La Hirian

Attachment 14.17.5

Concentric

Priem5c2

Attachment 14.17.6

Magic, Square  Inlay

Priem5g3

Attachment 14.17.7

Magic, Diamond Inlay

Priem5g2

Attachment 14.17.8

6

Symmetric Main Diagonals

Priem6a2

Attachment 14.17.9

Bordered

Priem6b2

Attachment 14.17.10

7

Bordered

Priem7a2

Attachment 14.17.11

Concentric

Attachment 14.17.15

8

Composed of Simple Magic Sub Squares

Priem4c2

Attachment 14.17.13

Concentric

Priem8a2

Attachment 14.17.16

9

Magic Center Square, Semi Magic Sub Squares

Priem9b1

Attachment 14.17.17

Concentric

Priem9a1

Attachment 14.17.18

10

Magic Center Square, Semi Magic Corner Squares

Priem10b1

Attachment 14.17.19

Concentric

Priem10c

Attachment 14.17.20

12

Composed of Simple Magic Sub Squares

Priem4c2

Attachment 14.17.23

Following sections will provide miscellaneous construction methods for Two and Four Way, V type Zig Zag Magic Squares.


Vorige Pagina Volgende Pagina Index About the Author