Office Applications and Entertainment, Magic Squares

17.0   Special Magic Squares, Big Primes

17.1   Introduction

Certain Prime Number Magic Squares can only be constructed based on very large prime numbers, further referred to as Big Primes.

Following sections will discuss how Prime Number Magic Squares can be constructed based on Big Primes selected from the first 2 billion prime numbers as previously available on www.primos.mat.br/2T_en.html.

It will be illustrated that for smaller orders the La Hire method as discussed in Section 14.12 (Magic Squares, Sum of Latin Squares) is very suitable for this purpose.

17.2   Magic Squares (3 x 3)

17.2.1 Consecutive Prime Numbers

A well known example is the 3 x 3 Prime Number Magic Square of consecutive prime numbers, with magic sum s1 = 4440084513, as published by Harry Nelson (ref. Journal of Recreational Mathematics, 1988, pages 214-216):

1480028201	1480028129	1480028183
1480028153	1480028171	1480028189
1480028159	1480028213	1480028141

A very impressive result for those days, although these days the result can be easily found by scrolling through databases as available from the internet.

The 3 x 3 Magic Square shown above can be rewritten as:

1480028 * 103 *

1 1 1 1 1 1 1 1 1

+

201 129 183 153 171 189 159 213 141

Or shorter:

        1480028 * 10³ * A1 + B1

with A1 the Unit Square [1] and B1 a Simple Magic Square with magic sum s2 = 513.

The Magic Square B1 can be found by means of routine Priem3a, which checked the last 3 digits of sets of 9 consecutive prime numbers selected from the range {2 ... 1611623773} within 48 minutes.

17.2.2 Non Consecutive Prime Numbers

As an illustration of the method described in Section 17.2.1 above, Order 3 Simple Magic Squares of non consecutive prime numbers have been generated for the range {1480020013 ... 1480029919}.

Subject squares can be defined as C = 148002 * 10⁴ * A1 + B1 with A1 the Unit Square [1] and B1 a Simple Magic Square with magic sum s2. The resulting magic sum s1 = 3 * 148002 * 10⁴ + s2.

Attachment 17.2.1 shows a few (2 unique) Prime Number Simple Magic Squares of order 3 for the occurring magic sums s2 as generated with routine Priem3b.

17.2.3 Latin Squares

The elements of two suitable selected Order 3 Latin Squares A1 and B1, result in an Order 3 Prime Number Magic Square C with elements c_i = a_i + b_i, i = 1 ... 9.

Following self explanatory example illustrates how this principle can be applied for Big Primes:

The Magic Series {a_i, i = 1 ... 3} and {b_j, j = 1 ... 3} are selected from the first resp. the second part of the Broken Primes such that c_ij = a_i * 10⁴ + b_j (i,j = 1 ... 3) are distinct prime numbers (9 ea).

Attachment 17.2.2 shows a few examples of such series, selected from the range {1792002847 ... 1796888657} with an automatic filter.

Attachment 17.2.3 shows the resulting unique Prime Number Magic Squares with the related Magic Sums. Each square shown corresponds with 8 Order 3 Prime Number Magic Squares (ref. CnstrSqrs3b).

17.3   Magic Squares (4 x 4)

17.3.1 Consecutive Prime Numbers

Even more impressive are the 4 x 4 Prime Number Pan Magic Squares of consecutive prime numbers as published by:

J. Wroblewski (2013) with magic sum s1 = 1282288088665523520

320572022166380 * 103 *

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

+

833 921 849 917 909 857 893 861 911 843 927 839 867 899 851 903

and M. Alekseyev (2014) with magic sum s1 = 682775764735680

17069394118 * 104 *

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

+

3817 3933 3949 3981 3979 3951 3847 3903 3891 3859 4023 3907 3993 3937 3861 3889

The square shown above is, according to M. Alekseyev, the minimum solution e.g. has the minimum magic sum for 4 x 4 Prime Number Pan Magic Squares of consecutive prime numbers.

17.3.2 Non Consecutive Prime Numbers

As an illustration of the method described in Section 17.2.1 above, Order 4 Pan Magic Squares of non consecutive prime numbers have been generated for the range {1480020013 ... 1480029919}.

Subject squares can be defined as C = 148002 * 10⁴ * A1 + B1 with A1 the Unit Square [1] and B1 a Pan Magic Square with magic sum s2. The resulting magic sum s1 = 4 * 148002 * 10⁴ + s2.

Attachment 17.3.1 shows a few (unique) Prime Number Pan Magic Squares of order 4 for the occurring magic sums s2 as generated with routine Priem4a.

Each square shown corresponds with 384 Order 4 Prime Number Pan Magic Squares.

17.3.3 Latin Squares

The elements of two suitable selected Latin Squares A1 and B1, with latin main diagonals, result in a Prime Number Magic Square C with elements c_i = a_i + b_i, i = 1 ... 16 (ref. Attachment 14.8.1a).

Following self explanatory example illustrates how this principle can be applied for Big Primes:

The Magic Series {a_i, i = 1 ... 4} and {b_j, j = 1 ... 4} are selected from the first resp. the second part of the Broken Primes such that c_ij = a_i * 10³ + b_j (i,j = 1 ... 4) are distinct prime numbers (16 ea).

Attachment 17.3.2 shows a few examples of such series, selected from the range {179300047 ... 179399903} with an automatic filter.

Attachment 17.3.3 shows the resulting unique Prime Number Magic Squares with the related Magic Sums. Each square shown corresponds with 1152 Prime Number Magic Squares (ref. CnstrSqrs4b).

17.3.4 Balanced Series
       Pan Magic Squares

The construction of Prime Number Pan Magic Squares based on Latin Squares, requires Balanced Magic Series as defined in Attachment 14.8.1b.

Following self explanatory example illustrates how this principle can be applied for Big Primes, selected from the range {190745257 ... 196130903}:

Attachment 17.3.4 shows a few examples of balanced series, selected from a wider range with an automatic filter.

Attachment 17.3.5 shows the resulting unique Prime Number Pan Magic Squares with the related Magic Sums. Each square shown corresponds with 384 Prime Number Pan Magic Squares (ref. CnstrSqrs4b).

17.3.5 Balanced Series
       Associated Magic Squares

Prime Number Associated Magic Squares can be constructed based on Semi-Latin Squares and Balanced Magic Series.

Following self explanatory example illustrates how this principle can be applied for Big Primes, selected from the range {190745257 ... 196130903}:

Attachment 17.3.4 shows a few examples of balanced series, selected from a wider range with an automatic filter.

Attachment 17.3.7 shows the resulting unique Prime Number Associated Magic Squares with the related Magic Sums. Each square shown corresponds with 384 Prime Number Associated Magic Squares (ref. CnstrSqrs4b).

17.4   Magic Squares (5 x 5)

17.4.1 Non Consecutive Prime Numbers

As an illustration of the method described in Section 17.2.1 above, Order 5 Associated Magic Squares of non consecutive prime numbers have been generated for the range {1480020013 ... 1480029919}.

Subject squares can be defined as C = 148002 * 10⁴ * A1 + B1 with A1 the Unit Square [1] and B1 a Pan Magic Square with magic sum s2. The resulting magic sum s1 = 5 * 148002 * 10⁴ + s2.

Attachment 17.4.3 shows a few (unique) Prime Number Associated Magic Squares of order 5 for the occurring magic sums s2 as generated with routine Priem5a.

17.4.2 Latin Squares

The elements of two suitable selected Latin Squares A1 and B1, with latin (pan) diagonals, result in a Prime Number Pan Magic Square C with elements c_i = a_i + b_i, i = 1 ... 25 (ref. Attachment 14.8.1a).

Following self explanatory example illustrates how this principle can be applied for Big Primes:

The Magic Series {a_i, i = 1 ... 5} and {b_j, j = 1 ... 5} are selected from the first resp. the second part of the Broken Primes such that c_ij = a_i * 10³ + b_j (i,j = 1 ... 5) are distinct prime numbers (25 ea).

Attachment 17.4.1 shows a few examples of such series, selected from the range {179201149 ... 179685817} with an automatic filter.

Attachment 17.4.2 shows the resulting unique Prime Number Pan Magic Squares with the related Magic Sums. Each square shown corresponds with 28800 Prime Number Pan Magic Squares (ref. CnstrSqrs5b).

17.5   Magic Squares (6 x 6)

17.5.1 Non Consecutive Prime Numbers

As an illustration of the method described in Section 17.2.1 above, Order 6 Simple Magic Squares of non consecutive prime numbers have been generated for the range {1480020013 ... 1480029919}.

Subject squares can be defined as C = 148002 * 10⁴ * A1 + B1 with A1 the Unit Square [1] and B1 a Simple Magic Square with magic sum s2. The resulting magic sum s1 = 6 * 148002 * 10⁴ + s2.

Routine Priem6a generated 1866 order 6 Prime Number Simple Magic Squares (Symmetrical Diagonals) within 886 sec. of which a few are shown in Attachment 17.5.1.

17.5.2 Semi Latin Squares

The elements of two suitable selected Semi Latin Squares A1 and B1, with latin main diagonals, result in a Prime Number Magic Square C with elements c_i = a_i + b_i, i = 1 ... 36 (ref. Attachment 14.8.1b).

Following self explanatory example illustrates how this principle can - occasionally - be applied for Big Primes:

The Balanced Series {a_i, i = 1 ... 6} and {b_j, j = 1 ... 6} are selected from the first resp. the second part of the Broken Primes such that c_ij = a_i * 10³ + b_j (i,j = 1 ... 6) are distinct prime numbers (36 ea).

17.6   Magic Squares (7 x 7)

17.6.1 Non Consecutive Prime Numbers

As an illustration of the method described in Section 17.2.1 above, Order 7 Bordered Magic Squares of non consecutive prime numbers have been generated for the range {1480020013 ... 1480029919}.

Subject squares can be defined as C = 148002 * 10⁴ * A1 + B1 with A1 the Unit Square [1] and B1 a Bordered Magic Square with magic sum s2. The resulting magic sum s1 = 7 * 148002 * 10⁴ + s2.

With routine Priem7a numerous order 7 Prime Number Bordered Magic Squares could be generated, of which a few are shown in Attachment 17.6.3.

17.6.2 Latin Squares

The elements of two suitable selected Latin Squares A1 and B1, with latin (pan) diagonals, result in a Prime Number Pan Magic Square C with elements c_i = a_i + b_i, i = 1 ... 49 (ref. Attachment 14.8.1a).

Following self explanatory example illustrates how this principle can be applied for Broken Primes:

The Magic Series {a_i, i = 1 ... 7} and {b_j, j = 1 ... 7} are selected from the first resp. the second part of the Broken Primes such that c_ij = a_i * 10³ + b_j (i,j = 1 ... 7) are distinct prime numbers (49 ea).

Attachment 17.6.1 shows a few examples of such series, selected from the range {11 ... 391691} with an automatic filter.

Attachment 17.6.2 shows the resulting unique Prime Number Pan Magic Squares with the related Magic Sums. Each square shown corresponds with 304.819.200 Prime Number Pan Magic Squares (ref. CnstrSqrs7b).

17.7   Magic Squares (8 x 8)

17.7.1 Non Consecutive Prime Numbers

As an illustration of the method described in Section 17.2.1 above, Order 8 Composed Magic Squares of non consecutive prime numbers have been generated for the range {1480020013 ... 1480029919}.

Subject squares can be defined as C = 148002 * 10⁴ * A1 + B1 with A1 the Unit Square [1] and B1 a Composed Magic Square with magic sum s2. The resulting magic sum s1 = 8 * 148002 * 10⁴ + s2.

With routine Priem4c a few order 8 Prime Number Composed Magic Squares could be generated, which are shown in Attachment 17.7.1.

17.7.2 Latin Squares

The elements of two suitable selected Latin Squares A1 and B1, with latin main diagonals, result in a Prime Number Magic Square C with elements c_i = a_i + b_i, i = 1 ... 64 (ref. Attachment 14.8.1a).

Following self explanatory example illustrates how this principle can be applied for Broken Primes:

The Magic Series {a_i, i = 1 ... 8} and {b_j, j = 1 ... 8} are selected from the first resp. the second part of the Broken Primes such that c_ij = a_i * 10³ + b_j (i,j = 1 ... 8) are distinct prime numbers (64 ea).

Attachment 17.7.2 shows a few examples of such series, selected from the range {59 ... 13965773} with an automatic filter.

Attachment 17.7.3 shows the resulting unique Prime Number Simple Magic Squares with the related Magic Sums. Each square shown corresponds with numerous Prime Number Magic Squares (ref. CnstrSqrs8b).

17.8   Magic Squares (10 x 10)

17.8.1 Non Consecutive Prime Numbers

As an illustration of the method described in Section 17.2.1 above, Order 10 Bordered Magic Squares of non consecutive prime numbers have been generated for the range {1480020013 ... 1480029919}.

Subject squares can be defined as C = 148002 * 10⁴ * A1 + B1 with A1 the Unit Square [1] and B1 a Bordered Magic Square with magic sum s2. The resulting magic sum s1 = 10 * 148002 * 10⁴ + s2.

With routine Priem10c a few order 10 Prime Number Bordered Magic Squares could be generated, which are shown in Attachment 17.8.1.

17.9   Summary

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

This is the end of the Chapter 'Prime Number Magic Squares' of this website.