A NEW PRIME BORDER SQUARE BASED ON THE JOHNSON SQUARE

Picture of a square

A 7x7 Prime Magic Square

A 7x7 magic square, discovered by A. Johnson, contains only prime numbers and its internal 5x5 and 3x3 squares are also magic at their shared borders. (See square 000 below referenced from Ian Stuart's book1 and A. Johnson's original paper2). The Johnson prime square is also shown in the previous web page.

A second variant employing a different internal 3x3 square as well as other changes is possible if and only if the negative entry is treated as a negative prime. Normally primes are positive integers but since the negative primes give the same divisors upon division, viz., 1 and the number itself in this page they will be treated as prime.

Again because of the structure of the square, this square can be converted into other variants by different degrees of rotation of the individual borders. In this page only rotation by 180° is being employed.

If we depict the original square as 000 (no rotation), then its counterpart is 111 (rotation of all three borders). Similarly 100 (rotation of external border) is equivalent to 011 (rotation of the two internal borders), 010 (rotation of the internal border) equivalent to 101 (rotation of the internal and external border), and 001 (rotation of only the internal border) equivalent to 110 (rotation of the two external borders).

Accordingly, the four equivalencies are summarized as:

000 ≡ 111 100 ≡ 011010 ≡ 101 001 ≡ 110

Construction of the New Prime 7x7 Magic Square

  1. Construct the complementary table of prime pairs where each pair of complements, as well as the 25th number doubled, sums to 3958.
  2. First generate the internal 3x3 square is generated using the nine khaki colored numbers whose difference (Δ) between numbers (as shown in the figure above) is 90. Note that one prime is negative.
  3. Then generate the 5x5 square by filling in the border with the requisite complementary pairs.
  4. Finally finish up by filling in the 7x7 border with the requisite complementary pairs.
-43 11 47 107137 179 257 281 491 569 821 839 1049 1181 12171229 1301 1481 1601 1607 1721 1871 1877 1889
1979
4001 3947 3911 3851 3821 3779 3701 3677 3467 3389 3137 3119 2909 2777 2741 2729 2657 2477 2357 2351 2237 2087 2081 2069
Prime Square 3x3
 
 
3911 -43 2069
137 1979 3821
1889 4001 47
 
 
⇒
Prime Square 5x5
 
5692237 377912292081
23573911 -43 20691601
2351137 1979 38211607
27411889 4001 47 1217
18771721 17927293389
 
⇒
Prime Square 7x7
2777 257 281 14813119 38512087
4915692237 3779122920813467
1123573911 -43 20691601 3947
29092351137 1979 382116071049
265727411889 4001 47 12171301
313718771721 17927293389 821
187137013677 24778391071181

Construction of the Prime 7x7 Rotational Variants

Generate the three other rotational squares.

Prime Square 7x7
2777 257 281 14813119 38512087
4915692237 3779122920813467
1123573911 -43 20691601 3947
29092351137 1979 382116071049
265727411889 4001 47 12171301
313718771721 17927293389 821
187137013677 24778391071181
⇒
100 Square
1181 107 839 24773677 37011871
8215692237 3779122920813137
130123573911 -43 20691601 2657
10492351137 1979 382116072909
394727411889 4001 47 121711
346718771721 17927293389 491
208738513119 14812812572777
+
010 Square
2777 257 281 14813119 38512087
49133892729 179172118773467
1112173911 -43 20692741 3947
29091607137 1979 382123511049
265716011889 4001 47 23571301
313720811229 37792237569 821
187137013677 24778391071181
+
001 Square
2777 257 281 14813119 38512087
4915692237 3779122920813467
11235747 4001 18891601 3947
290923513821 1979 13716071049
2657274147 -43 3911 12171301
313718771721 17927293389 821
187137013677 24778391071181

This concludes the new prime square and rotational variants. To continue to new 7x7 prime square variants.
Go back to homepage.

References

  1. Ian Stuart: Professors Stuart's Hoard of Mathematical Treasures(2009) Page 192
  2. A. W. Johnson, Jr., Journal of Recreational Mathematics 15:2, 1982-83, p. 84

Copyright © 2010 by Eddie N Gutierrez. E-Mail: Fiboguti89@Yahoo.com