14.0 Special Magic Squares, Prime Numbers
14.12 Magic Squares, Sum of Latin Squares
Prime Number Magic Squares and Prime Number Pan Magic Squares of odd order can also be constructed based on La Hirian Primaries, meaning based on suitable selected Latin Squares (ref. Attachment 14.8.1a).
For some early examples of La Hirian Primaries, reference is made to Charles D. Shuldham, The Monist, Vol. 24, No.4 (October 1914).
14.12.1 Magic Squares (4 x 4)
The elements of two Latin Squares A1 and B1, with latin main diagoanls, result in a Prime Number Magic Square C with elements
c_{i} = a_{i} + b_{i},
i = 1 ... 16 (ref. Attachment 14.8.1a).
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_{i} + b_{j} (i,j = 1 ... 4) are distinct prime numbers (16 ea).
Attachment 14.8.2 shows Correlated Magic Lines for a_{i} = 1 ... 61, b_{j} = 0 ... 60 (i,j = 1 ... 4) and the related Magic Sums of {a_{i}}, {b_{i}} and {a_{i} + b_{i}}, i = 1 ... 4.
Attachment 14.8.3 shows the resulting unique Prime Number Magic Squares (40 ea) with the related Magic Sums. Each square shown corresponds with 1152 Prime Number Magic Squares.
Note:
The supplementary conditions, required to construct Prime Number Pan Magic Squares based on two Latin Squares, are illustrated in Attachment 14.8.1b.
14.12.2 Pan Magic Squares (5 x 5)
The same method can be used to construct Prime Number (Pan) Magic Squares of order 5 (ref. Attachment 14.8.1a).
The key to possible solutions is to find Correlated Magic Lines {a_{i}, i = 1 ... 5} and {b_{j}, j = 1 ... 5} such that
c_{ij} = a_{i} + b_{j} (i,j = 1 ... 5) are distinct prime numbers (25 ea).
Attachment 14.8.4 shows Correlated Magic Lines for a_{i} = 1 ... 127, b_{j} = 0 ... 126 (i,j = 1 ... 5) and the related Magic Sums of {a_{i}}, {b_{i}} and {a_{i} + b_{i}}, i = 1 ... 5.
Attachment 14.8.5 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.
Note:
The supplementary conditions, required to construct Prime Number Ultra Magic Squares based on two Latin Squares, are illustrated in Attachment 14.8.1b.
14.12.3 Magic Squares (6 x 6)
Although not exactly Latin Square based, the same method can be used to construct Prime Number Magic Squares of order 6 (ref. Attachment 14.8.1b).
The key to possible solutions is to find Correlated Balanced Magic Lines {a_{i}, i = 1 ... 6} and {b_{j}, j = 1 ... 6} such that c_{ij} = a_{i} + b_{j} (i,j = 1 ... 6) are distinct prime numbers (36 ea).
Attachment 14.8.10 shows some Correlated Balanced Magic Lines for a_{i} = 1 ... 8389, b_{j} = 0 ... 12888 (i,j = 1 ... 6) and the related Magic Sums of {a_{i}}, {b_{i}} and {a_{i} + b_{i}}, i = 1 ... 6.
Attachment 14.8.11 shows the resulting unique Prime Number Magic Squares (48 ea) with the related Magic Sums.
14.12.4 Pan Magic Squares (7 x 7)
The same method can be used to construct Prime Number Pan Magic Squares of order 7 (ref. Attachment 14.8.1a).
The key to possible solutions is to find Correlated Magic Lines {a_{i}, i = 1 ... 7} and {b_{j}, j = 1 ... 7} such that
c_{ij} = a_{i} + b_{j} (i,j = 1 ... 7) are distinct prime numbers (49 ea).
Attachment 14.8.6 shows some Correlated Magic Lines for a_{i} = 1 ... 601, b_{j} = 0 ... 600 (i,j = 1 ... 7) and the related Magic Sums of {a_{i}}, {b_{i}} and {a_{i} + b_{i}}, i = 1 ... 7.
Attachment 14.8.7 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.
Note:
The supplementary conditions, required to construct Prime Number Ultra Magic Squares based on two Latin Squares, are illustrated in Attachment 14.8.1b.
14.12.5 Magic Squares (8 x 8)
The same method can be used to construct Prime Number Magic Squares of order 8 (ref. Attachment 14.8.1a).
The key to possible solutions is to find Correlated Magic Lines {a_{i}, i = 1 ... 8} and {b_{j}, j = 1 ... 8} such that
c_{ij} = a_{i} + b_{j} (i,j = 1 ... 8) are distinct prime numbers (64 ea).
Attachment 14.8.8 shows some Correlated Magic Lines for a_{i} = 1 ... 2341, b_{j} = 0 ... 2128 (i,j = 1 ... 8) and the related Magic Sums of {a_{i}}, {b_{i}} and {a_{i} + b_{i}}, i = 1 ... 8.
Attachment 14.8.9 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.
Note:
The supplementary conditions, required to construct Prime Number Most Perfect Pan Magic Squares based on Correlated Magic Lines (balanced), are illustrated in Attachment 14.8.1b.
14.12.6 Magic Squares (9 x 9)
The same method can be used to construct Prime Number Magic Squares of order 9 (ref. Attachment 14.8.1a).
The key to possible solutions is to find Correlated Magic Lines {a_{i}, i = 1 ... 9} and {b_{j}, j = 1 ... 9} such that
c_{ij} = a_{i} + b_{j} (i,j = 1 ... 9) are distinct prime numbers (81 ea).
Attachment 14.8.12 shows some Correlated Magic Lines for a_{i} = 1 ... 5647, b_{j} = 0 ... 2646 (i,j = 1 ... 9) and the related Magic Sums of {a_{i}}, {b_{i}} and {a_{i} + b_{i}}, i = 1 ... 9.
Attachment 14.8.13 shows the resulting unique Prime Number Magic Squares (24 ea) with the related Magic Sums. Each square shown corresponds with numerous Prime Number Magic Squares.
14.12.7 Magic Squares (10 x 10)
Although Euler postulated that sets of suitable Latin Squares did not exist for oddly even orders n ≡ 2 (mod 4), R. C. Bose and S. Shrinkhade proved the contrary for all n >= 10 (1959/1960).
Consequently the same method can be used to construct Prime Number Magic Squares of order 10 (ref. Attachment 14.8.1a)
The key to possible solutions is to find Correlated Magic Lines {a_{i}, i = 1 ... 10} and {b_{j}, j = 1 ... 10} such that
c_{ij} = a_{i} + b_{j} (i,j = 1 ... 10) are distinct prime numbers (100 ea).
Attachment 14.8.20 shows some Correlated Magic Lines for a_{i} = 1 ... 4775, b_{j} = 12 ... 3942 (i,j = 1 ... 10) and the related Magic Sums of {a_{i}}, {b_{i}} and {a_{i} + b_{i}}, i = 1 ... 10.
Attachment 14.8.21 shows the resulting unique Prime Number Magic Squares (8 ea) with the related Magic Sums. Each square shown corresponds with numerous Prime Number Magic Squares.
14.12.8 Pan Magic Squares (11 x 11)
The same method can be used to construct Prime Number Pan Magic Squares of order 11 (ref. Attachment 14.8.1a).
The key to possible solutions is to find Correlated Magic Lines {a_{i}, i = 1 ... 11} and {b_{j}, j = 1 ... 11} such that
c_{ij} = a_{i} + b_{j} (i,j = 1 ... 11) are distinct prime numbers (121 ea).
Attachment 14.8.22 shows some Correlated Magic Lines for a_{i} = 1 ... 8351, b_{j} = 0 ... 14418 (i,j = 1 ... 11) and the related Magic Sums.
Attachment 14.8.23 shows the resulting unique Prime Number Pan Magic Squares (8 ea) with the related Magic Sums. Each square shown corresponds with numerous Prime Number Pan Magic Squares.
14.12.9 Magic Squares (12 x 12)
The same method can be used to construct Prime Number Magic Squares of order 12 (ref. Attachment 14.8.1a).
The key to possible solutions is to find Correlated Magic Lines {a_{i}, i = 1 ... 12} and {b_{j}, j = 1 ... 12} such that
c_{ij} = a_{i} + b_{j} (i,j = 1 ... 12) are distinct prime numbers (144 ea).
Attachment 14.8.24 shows some Correlated Magic Lines for a_{i} = 1 ... 49359, b_{j} = 4 ... 56920 (i,j = 1 ... 12) and the related Magic Sums.
Attachment 14.8.25 shows the resulting unique Prime Number Magic Squares (4 ea) with the related Magic Sums. Each square shown corresponds with numerous Prime Number Magic Squares.
14.12.10 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:
