TWO NEW PRIME VARIANT BORDER 7X7 SQUARES

Picture of a 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.

This page depicts two other variants but unlike the Johnson square which has 1979 at the center of the square, these squares have 1931, the number prior to 1979 im the Johnson complementary table. Both squares have identical 3x3 and 5x5 squares but differ at the 7x7 border where only the 4 corner numbers are identical. Thus it depends what prime pairs one uses, i.e., there can be more than one square coinsisting of positive prime numbers as shown here.

Construction of a 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 3862.
  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 42.
  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, where the S = 13517.
1129 239 281 401449 491 563 653 659 9831229 1319 14391451 1481 15111523 1553 1619 1721 1823 18891913
1931
3851 3833 3623 3581 3461 3413 3371 3299 3209 3203 2999 2633 2543 2423 2411 2381 2351 2339 2309 2243 2141 2039 1973 1949
Prime Square 3x3
 
 
2381 1439 1973
1523 1931 2339
1889 2423 1841
 
 
Prime Square 5x5
 
1619659 172138331823
19132381 1439 19731949
26331523 1931 23391229
14511889 2423 1481 2411
20393203 2141292243
 
Prime Square 7x7
1511 1319 563 9833461 33712309
2391619659 1721383318233623
1119132381 1439 19731949 3851
358126331523 1931 23391229281
341314511889 2423 1481 2411449
320920393203 2141292243 653
155325433299 29994014912351

Construction of a Second Variant of a 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 3862.
  2. Take the 5x5 square constructed above and fill in the 7x7 border with different requisite complementary pairs. The sum again is 13517.
29 101 269 503 641659 743 839 863 1019 1061 11511229 14391451 1481 1511 1523 1553 1619 1721 1823 18891913
1931
3833 3761 3593 3359 3221 3203 3119 3023 2999 2843 2801 27112633 2423 2411 2381 2351 2339 2309 2243 2141 2039 1973 1949
Prime Square 5x5
 
1619659 172138331823
19132381 1439 19731949
26331523 1931 23391229
14511889 2423 1481 2411
20393203 2141292243
 
Prime Square 7x7
1511 503 839 28012843 27112309
6411619659 1721383318233221
10119132381 1439 19731949 3761
311926331523 1931 23391229743
299914511889 2423 1481 2411863
359320393203 2141292243 269
155333593023 1061101911512351

This concludes the new two prime square variants.
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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