Office Applications and Entertainment, Magic Squares

14.0    Special Magic Squares, Prime Numbers

14.17   Sophie Germain Primes

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

Paired Prime Number Magic Squares (A , B), with the property {b_i} = 2 * {a_i} + 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 {b_i} = (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 {a_i}/{b_i}, i = 1 ... 9.

14.17.2 Magic Squares (4 x 4)

Attachment 14.17.14 shows for the range {b_i} = (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 {b_i} = (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 {a_i}/{b_i}, i = 1 ... 16.

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

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

Attachment 14.17.24 shows for the range {c_i} = (5639 ... 3827639) a few triplets of 4 x 4 Simple Magic Squares with Sophie Germain Primes {a_i}, {b_i} = 2 * {a_i} + 1 and {c_i} = 2 * {b_i} + 1 for miscellaneous Magic Sums.

Each triplet shown corresponds with 32 triplets with the same Magic Sums and variable values.

14.17.3 Magic Squares (5 x 5)

Attachment 14.17.3 shows for the range {b_i} = (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 {b_i} = (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 {b_i} = (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:

• Attachment 14.17.6 Concentric Magic Squares         (ref. Priem5c2)
  
  
• Attachment 14.17.7 Magic Squares with Square  Inlay (ref. Priem5g3)
  
  
• Attachment 14.17.8 Magic Squares with Diamond Inlay (ref. Priem5g2)

Occasionally order 5 pairs of Sophie Germain Magic Squares can be constructed based on consecutive Sophie Germain Primes {b_i} for i = 1 ... 25.

An example of a pair of Sophie Germain Magic Squares, based on consecutive Sophie Germain Primes, is shown below:

The right square (5b) has been constructed with the Generator Method as discussed in detail in Section 14.13.4 for order 6 squares.

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

Attachment 14.17.25 shows for the range {c_i} = (2039 ... 16379999) a few triplets of 5 x 5 Simple Magic Squares with Sophie Germain Primes {a_i}, {b_i} = 2 * {a_i} + 1 and {c_i} = 2 * {b_i} + 1 for miscellaneous Magic Sums.

Each triplet shown corresponds with 32 triplets with the same Magic Sums and variable values.

14.17.4 Magic Squares (6 x 6)

Attachment 14.17.9 shows for the range {b_i} = (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:

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

Alternatively order 6 pairs of Sophie Germain Magic Squares can be constructed based on consecutive Sophie Germain Primes {b_i} for i = 1 ... 36.

An example of a pair of Sophie Germain Magic Squares, based on consecutive Sophie Germain Primes, is shown below:

The right square (6b) has been constructed with the Generator Method as discussed in detail in Section 14.13.4.

Attachment 14.17.41 shows a few pairs of Sophie Germain Simple Magic Squares, based on consecutive Sophie Germain Primes and the related Magic Sums.

Each pair shown corresponds with numerous pairs with the same Magic Sums and variable values {a_i}/{b_i}, i = 1 ... 36.

14.17.5 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 {a_i}/{b_i}, i = 1 ... 49, (Order of magnitude 8 * (5!)² = 115200 for the Same Center Square).

Order 7 pairs of Sophie Germain Magic Squares with smaller Magic Sums can be constructed based on consecutive Sophie Germain Primes {b_i} for i = 1 ... 49.

An example of a pair of Sophie Germain Magic Squares, based on consecutive Sophie Germain Primes, is shown below:

The right square (7b) has been constructed with the Generator Method as discussed in detail in Section 14.13.5a.

The pair shown above corresponds with numerous pairs with the same Magic Sums and variable values {a_i}/{b_i}, i = 1 ... 49,

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 4^th 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:

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 {a_i}/{b_i}, 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 {a_i}/{b_i}, 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 {a_i}/{b_i}, 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 {a_i}/{b_i}, 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 4^th 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 {a_i}/{b_i}, i = 1 ... 144.

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:

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