Office Applications and Entertainment, Magic Squares

       Special Magic Squares, Prime Numbers

16.3   Prime Numbers, Sum of Squares

16.3.1 Magic Squares (4 x 4)

In another article Christian Boyer mentions an interesting method to construct Prime Number Magic Squares, using sums of squares, as published by André Gérardin early 1916 (Sphinx-Oedipe).

The elements of two latin squares A1 (MC = 50) and B1 (MC = 120), with latin main diagoanls, result in a Prime Number Magic Square C (MC = 7588) with elements c_i = a_i² + b_i², i = 1 ... 16 (ref. Attachment 16.3.1).

The key to possible solutions is to find Correlated Magic Lines {a_i, i = 1 ... 4} and {b_j, j = 1 ... 4} such that c_ij = a_i² + b_j² (i,j = 1 ... 4) are distinct prime numbers (16 ea).

Attachment 16.3.2 shows Correlated Magic Lines for a_i, b_j = 1 ... 58 (i,j = 1 ... 4) and the related Magic Sums of {a_i}, {b_i}, {a_i²}, {b_i²} and {a_i² + b_i²}, i = 1 ... 4.

Attachment 16.3.3 shows the resulting unique Prime Number Magic Squares (68 ea) with the related Magic Sums. Each square shown corresponds with 1152 Prime Number Magic Squares.

16.3.2 Pan Magic Squares (5 x 5)

The same method can be used to construct Prime Number (Pan) Magic Squares of order 5 (ref. Attachment 16.3.1).

The key to possible solutions is now to find Correlated Magic Lines {a_i, i = 1 ... 5} and {b_j, j = 1 ... 5} such that c_ij = a_i² + b_j² (i,j = 1 ... 5) are distinct prime numbers (25 ea).

Attachment 16.3.5 shows Correlated Magic Lines for a_i, b_j = 1 ... 115 (i,j = 1 ... 5) and the related Magic Sums of {a_i}, {b_i}, {a_i²}, {b_i²} and {a_i² + b_i²}, i = 1 ... 5.

Attachment 16.3.6 shows the resulting unique Prime Number Pan Magic Squares (48 ea) with the related Magic Sums. Each square shown corresponds with 28800 Prime Number Pan Magic Squares.

16.3.3 Pan Magic Squares (7 x 7)

The same method can be used to construct Prime Number (Pan) Magic Squares of order 7 (ref. Attachment 16.3.1).

Correlated Magic Lines {a_i, i = 1 ... 7} and {b_j, j = 1 ... 7} should be found such that c_ij = a_i² + b_j² (i,j = 1 ... 7) are distinct prime numbers (49 ea).

Attachment 16.3.7 shows Correlated Magic Lines for a_i, b_j = 1 ... 550 (i,j = 1 ... 7) and the related Magic Sums of {a_i}, {b_i}, {a_i²}, {b_i²} and {a_i² + b_i²}, i = 1 ... 7.

Attachment 16.3.8 shows the resulting unique Prime Number Pan Magic Squares (48 ea) with the related Magic Sums. Each square shown corresponds with 304.819.200 Prime Number Pan Magic Squares.

16.3.4 Magic Squares (8 x 8)

The same method can be used to construct Prime Number Magic Squares of order 8 (ref. Attachment 16.3.1).

Correlated Magic Lines {a_i, i = 1 ... 8} and {b_j, j = 1 ... 8} should be found such that c_ij = a_i² + b_j² (i,j = 1 ... 8) are distinct prime numbers (64 ea).

Attachment 16.3.9 shows some Correlated Magic Lines for a_i, b_j = 1 ... 1500 (i,j = 1 ... 8) and the related Magic Sums of {a_i}, {b_i}, {a_i²}, {b_i²} and {a_i² + b_i²}, i = 1 ... 8.

Attachment 16.3.10 shows the resulting unique Prime Number Magic Squares (16 ea) with the related Magic Sums. Each square shown corresponds with numerous Prime Number Magic Squares.

16.3.5 Summary

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

A comparable method (La Hirian Primaries), which constructs Prime Number Magic Squares C based on the sum of the elements of two Latin Squares A1 and B1, has been discussed in Section 14.12.