Office Applications and Entertainment, Magic Squares

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14.0     Special Magic Squares, Prime Numbers

14.13    Consecutive Primes (5)

14.13.41 Simple Magic Squares (13 x 13)

Prime Number Simple Magic Squares of order 13 can be constructed with the Generator Principle, as applied in previous sections.

Suitable Generators (13 Magic Series) can be constructed semi-automatically (ref. CnstrGen13).

A possible Semi Magic Square and resulting Simple Magic Square are shown below, for the smallest consecutive prime numbers {13 ... 1033} for which an Order 13 Simple Magic Square exists (MC13 = 6325):

Semi Magic Square
1033 13 1013 17 1021 19 1019 23 1031 29 1009 31 67
997 37 971 41 953 43 977 47 991 53 983 59 173
929 61 947 71 919 73 967 79 937 83 941 89 229
911 97 907 101 881 103 109 349 107 883 113 887 877
127 863 131 859 137 857 139 853 149 839 151 823 397
157 829 163 827 167 821 811 809 181 179 191 769 421
193 797 197 787 199 773 211 761 223 757 449 227 751
233 479 239 739 241 733 257 727 251 719 263 743 701
269 709 271 557 277 683 281 677 283 673 293 661 691
307 653 311 659 313 643 317 331 641 547 647 619 337
347 631 353 617 359 613 367 607 601 491 379 373 587
383 593 389 541 401 443 409 599 431 569 419 577 571
439 563 433 509 457 521 461 463 499 503 487 467 523
Simple Magic Square
1033 17 1021 19 67 23 1013 29 31 1031 1009 1019 13
911 101 881 103 877 349 907 883 887 107 113 109 97
127 859 137 857 397 853 131 839 823 149 151 139 863
157 827 167 821 421 809 163 179 769 181 191 811 829
307 659 313 643 337 331 311 547 619 641 647 317 653
383 541 401 443 571 599 389 569 577 431 419 409 593
929 71 919 73 229 79 947 83 89 937 941 967 61
347 617 359 613 587 607 353 491 373 601 379 367 631
269 557 277 683 691 677 271 673 661 283 293 281 709
233 739 241 733 701 727 239 719 743 251 263 257 479
193 787 199 773 751 761 197 757 227 223 449 211 797
439 509 457 521 523 463 433 503 467 499 487 461 563
997 41 953 43 173 47 971 53 59 991 983 977 37

The Generator Method has been applied for miscellaneous Magic Sums, for which the first occurring Simple Magic Squares are shown in Attachment 14.13.41.

Each square shown corresponds with numerous Prime Number Magic Squares with the same Magic Sum and variable values.

14.13.42 Simple Magic Squares (13 x 13)
         Single Square Inlays (Lower Order)

Order 13 Simple Magic Squares with single square inlay(s) of lower order can be constructed with the generator method as discussed in detail in Section 14.13.22 through Section 14.13.26.

Examples of order 3 through 7 square inlays, for the consecutive prime numbers {13 ... 1033} with the related Magic Sum s13 = 6325 are shown below:

Order 3 Inlay, s3 = 879
439 17 613 563 601 443 569 449 593 599 431 401 607
257 419 227 233 19 1031 1033 1021 29 31 23 1019 983
997 107 293 479 37 47 1009 331 43 1013 41 991 937
941 353 359 167 53 971 977 967 61 67 397 953 59
191 811 809 193 797 491 179 197 773 173 787 181 743
487 199 769 239 757 229 241 211 739 761 751 223 719
263 251 733 277 709 271 499 653 691 727 701 269 281
631 283 683 677 661 463 313 347 659 673 317 311 307
389 647 641 373 349 619 617 383 409 367 643 379 509
823 857 137 821 829 151 163 157 839 433 827 149 139
103 887 101 853 877 113 131 127 859 881 863 109 421
337 907 947 929 79 919 71 911 89 97 83 883 73
467 587 13 521 557 577 523 571 541 503 461 457 547
Order 4 Inlay, Pan Magic, s4 = 1068
821 607 859 107 853 829 131 103 839 127 113 109 827
23 263 241 383 181 29 19 1019 73 1033 1021 1031 1009
193 257 307 137 367 41 31 983 37 1013 991 997 971
941 151 353 271 293 977 43 347 47 53 953 967 929
911 397 167 277 227 67 907 449 61 919 937 947 59
761 727 773 563 757 229 743 223 751 197 211 199 191
479 487 509 499 521 503 461 541 523 463 443 457 439
547 373 653 643 389 631 619 617 647 409 401 379 17
857 557 887 883 89 877 863 97 881 101 79 83 71
139 769 179 569 811 173 787 823 809 797 163 157 149
13 419 467 593 421 587 571 577 601 599 433 431 613
359 641 691 661 683 673 659 313 337 331 311 317 349
281 677 239 739 733 709 491 233 719 283 269 251 701

Order 5 Inlay, Associated, s5 = 1735
593 443 461 131 107 991 59 47 983 53 977 971 509
191 173 383 431 557 929 73 79 919 911 83 709 887
227 293 347 401 467 71 967 941 947 937 61 599 67
137 263 311 521 503 31 1033 23 1031 29 1021 1013 409
587 563 233 251 101 43 1019 1009 37 41 997 953 491
97 89 857 883 907 877 881 109 113 863 127 103 419
631 647 19 643 499 433 479 397 463 641 577 439 457
821 823 797 211 179 811 197 809 199 787 181 317 193
547 17 487 523 617 541 569 619 607 571 601 13 613
701 739 307 733 653 277 283 313 281 727 271 349 691
257 757 719 769 223 761 241 773 229 239 269 337 751
683 659 661 677 673 421 367 379 353 359 331 373 389
853 859 743 151 839 139 157 827 163 167 829 149 449

Order 5 with Diamond Inlay, s5 = 1735
131 251 557 569 227 61 967 67 947 71 941 599 937
587 101 431 173 443 23 1033 29 1031 31 1021 409 1013
383 401 347 293 311 929 73 919 79 911 83 887 709
167 521 263 593 191 47 991 53 983 59 977 971 509
467 461 137 107 563 37 1019 41 1009 43 997 953 491
907 419 857 97 883 881 109 877 113 863 89 127 103
349 677 647 683 367 673 317 661 373 641 379 17 541
859 359 821 853 179 839 157 829 163 827 149 139 151
479 607 499 601 463 487 503 577 449 523 19 571 547
823 811 757 211 797 809 199 197 193 787 223 181 337
281 353 269 733 739 691 283 701 307 719 659 313 277
653 613 13 643 389 619 433 631 421 617 457 397 439
239 751 727 769 773 229 241 743 257 233 331 761 271

Order 6 Concentric Inlay, s6 = 3312
113 953 941 937 241 127 109 199 107 919 131 619 929
823 881 227 907 193 281 97 83 967 971 103 101 691
821 487 613 461 647 283 997 61 743 67 73 983 89
421 197 911 223 877 683 761 1013 47 53 1009 71 59
157 643 457 617 491 947 773 37 1019 41 1021 43 79
977 151 163 167 863 991 853 23 1031 29 1033 31 13
137 887 149 233 139 859 179 857 173 839 181 883 809
449 587 479 17 463 569 499 557 503 563 541 577 521
191 211 239 349 829 547 257 797 229 787 251 827 811
769 263 733 757 271 269 293 751 277 307 373 739 523
359 353 673 659 367 19 379 653 383 641 571 661 607
389 401 409 601 631 433 419 593 509 431 599 443 467
719 311 331 397 313 317 709 701 337 677 439 347 727

Order 6, Concentric Pan Magic Inlay, s6 = 3378
953 317 617 263 659 569 101 97 971 107 103 967 601
179 587 491 881 293 947 73 89 79 977 727 19 983
353 593 797 479 383 773 61 997 991 67 71 647 113
887 419 827 149 857 239 1009 83 1013 53 683 59 47
449 653 137 743 719 677 1019 37 1021 41 769 43 17
557 809 509 863 467 173 1031 1033 23 29 787 31 13
269 227 199 823 811 251 229 839 223 829 241 563 821
233 131 641 127 109 157 919 139 151 937 911 941 929
739 257 751 271 761 277 281 331 283 577 307 757 733
379 541 389 373 367 401 613 607 409 631 397 599 619
503 571 439 433 421 547 461 523 521 499 463 487 457
163 907 181 211 167 883 191 859 197 877 193 853 643
661 313 347 709 311 431 337 691 443 701 673 359 349

Order 7 Concentric Inlay, s7 = 3801
103 937 929 907 677 139 109 941 79 13 89 491 911
859 967 317 367 181 883 227 67 991 19 997 389 61
853 307 863 439 829 277 233 83 71 953 73 373 971
643 577 563 433 619 523 443 53 1013 43 1009 347 59
199 607 313 757 229 809 887 47 1021 41 1019 359 37
167 257 659 719 857 223 919 1031 31 1033 17 383 29
977 149 157 179 409 947 983 101 881 97 107 461 877
271 241 251 733 269 449 263 709 701 739 281 691 727
787 197 193 173 191 509 773 751 211 743 761 239 797
547 569 557 431 503 593 23 467 487 587 499 521 541
331 311 293 641 337 463 683 653 349 647 283 661 673
457 379 617 419 401 397 631 601 353 599 421 571 479
131 827 613 127 823 113 151 821 137 811 769 839 163

Order 7 with Diamond Inlays, s7 = 3213
139 421 941 983 389 227 113 881 79 859 433 787 73
193 953 367 311 109 967 313 919 71 887 67 701 467
107 179 173 523 593 977 661 997 181 59 937 61 877
643 557 863 347 563 137 103 83 1013 43 1009 911 53
823 677 101 223 521 719 149 37 1021 47 1019 947 41
379 163 409 383 907 89 883 17 1033 31 1031 29 971
929 263 359 443 131 97 991 853 127 857 151 751 373
773 241 257 761 797 809 269 233 251 757 239 229 709
293 691 739 277 281 271 733 283 743 353 317 617 727
587 487 499 479 571 461 599 569 547 503 491 509 23
673 397 349 337 659 331 683 647 653 541 401 13 641
619 457 439 431 613 419 631 607 449 577 19 463 601
167 839 829 827 191 821 197 199 157 811 211 307 769

The order 3, 4 and 5 Square Inlays might be moved along the Main Diagonal (top/left to bottom/right) by means of row and column permutations.

As the 169 consecutive prime numbers {13 ... 1033} include the 144 consecutive prime numbers (89 . . . 991), the same inlays as applied in Section 14.13.32 could be used.

The square(s) shown above correspond with numerous Prime Number Magic Squares with the same Magic Sum and variable values, which can be obtained by selecting other aspects of the inlays and variation of the borders.

14.13.43 Simple Magic Squares (13 x 13)
         Order 3 Magic Square Inlays (4 ea)

Comparable with the method discussed in Section 14.13.27 it is possible to construct Order 13 Simple Magic Squares with four order 3 Simple Magic Square Inlays.

Examples of subject Inlaid Magic Squares for the consecutive prime numbers {13 ... 1033} with the related Magic Sum s13 = 6325 are shown below:

Simple Magic Type 1, Order 3 Inlays (4 ea)
389 167 233 577 211 313 1033 23 1031 29 1021 307 991
107 263 419 103 367 631 1019 31 1013 37 1009 349 977
293 359 137 421 523 157 997 43 983 41 967 433 971
761 317 449 751 463 409 73 919 79 83 911 887 223
197 509 821 199 541 883 61 941 67 929 71 877 229
569 701 257 673 619 331 947 47 953 937 53 179 59
647 397 439 641 457 643 557 461 467 617 479 503 17
691 311 353 347 659 677 337 661 379 653 431 443 383
773 827 839 853 149 139 151 811 173 829 163 487 131
757 809 787 769 191 797 227 193 239 733 241 401 181
13 19 491 613 563 521 571 593 547 601 607 587 599
857 907 823 109 863 97 101 859 113 127 89 499 881
271 739 277 269 719 727 251 743 281 709 283 373 683
Simple Magic Type 2, Order 3 Inlays (4 ea)
389 167 233 29 23 991 1033 1021 307 1031 577 211 313
107 263 419 31 1013 1009 977 37 349 1019 103 367 631
293 359 137 41 971 43 997 967 983 433 421 523 157
383 677 353 661 379 311 337 443 431 691 653 347 659
829 487 839 811 173 773 131 139 827 151 163 853 149
757 809 787 193 241 733 227 191 401 239 797 769 181
461 397 439 457 647 479 557 617 503 17 643 641 467
13 19 491 593 547 599 601 607 587 571 521 613 563
857 881 823 859 89 907 127 113 499 101 97 109 863
709 739 277 743 283 271 251 373 281 683 727 269 719
761 317 449 47 947 59 953 53 179 937 751 463 409
197 509 821 941 929 71 61 877 67 229 199 541 883
569 701 257 919 83 79 73 887 911 223 673 619 331
s3
789 1101
1527 1623

Miscellaneous (main) diagonal sets are possible for both square types.

Type 1 (left), each resulting square corresponds with 84 = 4096 solutions, which can be obtained by selecting other aspects of the four inlays.

Type 2 (right), each resulting square corresponds with 84 * (3!)4 = 5.308.416 solutions, which can be obtained by selecting other aspects of the four inlays and variation of the border (window).

The order 3 Square Inlays of Type 2 might be moved along the Main Diagonals by means of row and column permutations (1, 2 or 3 positions).

Potential order 3 Simple Magic Square Inlays (unique) for the consecutive prime numbers (13 ... 1033), resulting in numerous valid sets of four, are shown in Attachment 14.13.43.

14.13.44 Simple Magic Squares (13 x 13)
         Order 4 Pan Magic Square Inlays (4 ea)

Comparable with the method discussed in Section 14.13.27 it is possible to construct Order 13 Simple Magic Squares with four order 4 Pan Magic Square Inlays.

Examples of subject Inlaid Magic Squares for the consecutive prime numbers {13 ... 1033} with the related Magic Sum s13 = 6135 are shown below:

Simple Magic Type 1, Order 4 Inlays (4 ea)
227 647 307 787 263 523 439 839 43 1019 53 997 181
211 883 131 743 349 929 173 613 31 29 1033 179 1021
677 197 757 337 593 193 769 509 47 59 1013 983 191
853 241 773 101 859 419 683 103 1031 37 41 277 907
457 443 499 641 167 751 283 887 1009 61 67 83 977
401 739 359 541 223 947 107 811 71 991 73 109 953
521 379 563 577 761 157 877 293 79 971 139 919 89
661 479 619 281 937 233 821 97 967 113 827 127 163
823 271 313 251 269 311 829 257 643 733 809 797 19
601 461 599 463 587 571 467 449 503 491 569 547 17
367 331 719 691 23 709 317 631 373 383 701 353 727
389 397 487 673 431 433 409 607 617 557 653 13 659
137 857 199 239 863 149 151 229 911 881 347 941 421
Simple Magic Type 2, Order 4 Inlays (4 ea)
227 647 307 787 29 31 1033 179 1021 263 523 439 839
211 883 131 743 1031 37 41 277 907 349 929 173 613
677 197 757 337 53 1019 181 997 43 593 193 769 509
853 241 773 101 47 983 59 1013 191 859 419 683 103
823 251 271 313 643 733 19 809 797 829 311 269 257
601 463 461 599 569 491 571 17 547 467 503 587 449
367 691 331 719 383 373 701 709 727 317 353 23 631
389 673 397 487 433 557 653 13 617 409 659 431 607
137 239 857 199 149 881 911 941 421 151 347 863 229
457 443 499 641 977 61 83 1009 67 167 751 283 887
401 739 359 541 73 953 991 109 71 223 947 107 811
521 379 563 577 971 79 919 139 89 761 157 877 293
661 479 619 281 967 127 163 113 827 937 233 821 97
s4
1968 2064
2040 2088

Miscellaneous (main) diagonal sets are possible for both square types.

Type 1 (left), each resulting square corresponds with 2 * 3843 = 113.246.208 solutions, which can be obtained by selecting other aspects of the four inlays.

Type 2 (right), each resulting square corresponds with 3844 * (4!)4 = 7,214 1015 solutions, which can be obtained by selecting other aspects of the four inlays and variation of the border (window).

The order 4 Square Inlays of Type 2 might be moved along the Main Diagonals by means of row and column permutations (1 or 2 positions).

Potential order 4 Pan Magic Square Inlays (unique) for the consecutive prime numbers (13 ... 1033), resulting in numerous valid sets of four, are shown in Attachment 14.13.44.

14.13.45 Inlaid Magic Squares (13 x 13)
         Order 3 (Semi) Magic Square Inlays (9 ea)

Comparable with the method mentioned in Section 14.13.43 above it is possible to construct Order 13 Simple Magic Squares with nine order 3 (Semi) Magic Square Inlays.

An example of subject Inlaid Magic Squares for the consecutive prime numbers {13 ... 1033} with the related Magic Sum s13 = 6325 is shown below:

Simple Magic, Order 3 Inlays (9 ea)
647 593 461 173 727 659 433 487 509 523 1033 37 43
953 281 467 617 131 811 449 373 607 53 653 883 47
101 827 773 769 701 89 547 569 313 1031 23 41 541
347 673 271 971 113 479 293 881 661 19 31 619 967
193 269 829 29 521 1013 631 263 941 59 73 683 821
751 349 191 563 929 71 911 691 233 1009 61 67 499
421 677 599 383 787 397 439 277 709 491 983 79 83
557 379 761 331 317 919 877 409 139 103 613 823 97
719 641 337 853 463 251 109 739 577 947 179 199 311
197 359 307 283 223 239 571 839 127 1019 367 907 887
389 419 211 457 743 401 757 137 149 107 587 977 991
241 601 937 733 227 643 151 229 503 167 859 13 1021
809 257 181 163 443 353 157 431 857 797 863 997 17
s3
1701 1559 1429
1291 1563 1835
1697 1567 1425

The nine Square Inlays are selected as follows:

  • One Magic Center Square selected from the Potential Square Inlays shown in Attachment 14.13.43
  • Eight Semi Magic Border Squares (7 Magic Lines) selected from the collection Semi Magic Squares discussed in Section 14.13.35

and result in a 9th order Magic Square Inlay with Magic Sum s9 = 4689.

The resulting square shown above corresponds with 8 * 125 * 23 * {(4! * 5!)}2 = 1,321 1014 solutions, which can be obtained by selecting other aspects of the nine inlays and variation of the (eccentric) border.

14.13.46 Composed Magic Squares (13 x 13)
         Order 6 and 7 Magic Sub Squares

When the Magic Sum s13 is a multiple of 13 (e.g. 11947), order 13 (Semi) Magic Squares might be composed of:

  • One 7th order Simple Magic Corner Square with Magic Sum s7 = 7 * s13 / 13 = 6433 (top/left)
  • One 6th order Simple Magic Corner Square with Magic Sum s6 = 6 * s13 / 13 = 5514 (bottom/right)
  • Two Magic Rectangles order 6 x 7 with s6 = 5514 and s7 = 6433

An example is shown below for the consecutive prime numbers {359 ... 1511} with the related Magic Sums mentioned above.

Composed Semi Magic Square
449 1307 409 1511 503 1031 1223 557 821 827 1381 439 1489
1447 941 401 887 1163 443 1151 1097 619 563 1249 499 1487
983 967 1367 359 1091 1283 383 947 863 523 991 709 1481
797 461 1499 919 1483 421 853 1289 953 541 487 1433 811
599 839 1319 373 883 1021 1399 617 1259 1327 463 1451 397
1229 509 1019 1153 571 1201 751 467 479 1279 1429 1471 389
929 1409 419 1231 739 1033 673 1459 1439 1373 433 431 379
761 757 1117 769 1069 659 1301 971 1193 593 653 677 1427
1109 907 613 1039 701 787 1277 1013 683 1061 1361 587 809
719 1049 823 647 859 1123 1213 773 881 1103 743 1493 521
997 857 661 977 1129 1181 631 829 691 1237 937 1453 367
607 641 1009 1171 1187 1217 601 1051 643 1063 733 727 1297
1321 1303 1291 911 569 547 491 877 1423 457 1087 577 1093
Composed Simple Magic Square
449 1307 409 1511 503 1031 1223 617 1259 1327 463 1451 397
1447 941 401 887 1163 443 1151 1289 953 541 487 1433 811
983 967 1367 359 1091 1283 383 467 479 1279 1429 1471 389
797 461 1499 919 1483 421 853 947 863 523 991 709 1481
599 839 1319 373 883 1021 1399 1097 619 563 1249 499 1487
1229 509 1019 1153 571 1201 751 557 821 827 1381 439 1489
929 1409 419 1231 739 1033 673 1459 1439 1373 433 431 379
1117 1069 757 1301 761 659 769 971 1193 593 653 677 1427
613 701 907 1277 1109 787 1039 1013 683 1061 1361 587 809
823 859 1049 1213 719 1123 647 773 881 1103 743 1493 521
661 1129 857 631 997 1181 977 829 691 1237 937 1453 367
1009 1187 641 601 607 1217 1171 1051 643 1063 733 727 1297
1291 569 1303 491 1321 547 911 877 1423 457 1087 577 1093

The order 7 Simple Magic Sub Square applied above is composed of:

  • Two each order 4 Simple Magic Squares with Magic Sum s4 = 3676 and
  • Two each order 3 Semi   Magic Squares with Magic Sum s3 = 2757

The order 6 Simple Magic Sub Square applied above is composed of 4 each order 3 Semi Magic Squares with Magic Sum s3 = 2757.

Each square shown corresponds with numerous Prime Number Magic Squares with the same Magic Sum(s) and variable values.

14.13.47 Composed Magic Squares (13 x 13)
         Order 5 and 8 Magic Sub Squares

When the Magic Sum s13 is a multiple of 13 (e.g. 11947), order 13 (Semi) Magic Squares might be composed of:

  • One 8th order Simple Magic Corner Square with Magic Sum s8 = 8 * s13 / 13 = 7352 (top/left)
  • One 5th order Simple Magic Corner Square with Magic Sum s5 = 5 * s13 / 13 = 4595 (bottom/right)
  • Two Magic Rectangles order 5 x 8 with s5 = 4595 and s8 = 7352

An example is shown below for the consecutive prime numbers {359 ... 1511} with the related Magic Sums mentioned above.

Composed Semi Magic Square
479 1307 379 1511 727 883 577 1489 683 643 1223 587 1459
419 1429 389 1439 433 1399 463 1381 757 739 1187 593 1319
1451 449 1409 367 1453 661 1153 409 701 769 1237 599 1289
1327 491 1499 359 1063 733 1483 397 761 647 907 1129 1151
1091 653 439 1493 1117 617 461 1481 953 1181 619 1279 563
853 929 467 1427 1019 797 487 1373 937 1213 607 1291 547
1013 997 1283 383 863 1031 1361 421 1259 911 601 1303 521
719 1097 1487 373 677 1231 1367 401 1301 1249 971 571 503
1039 773 691 751 1009 839 1049 1201 1471 829 541 1297 457
743 821 967 823 859 977 991 1171 509 1423 1123 557 983
887 1021 947 877 857 1033 811 919 569 523 827 1229 1447
709 787 881 1051 1061 1087 1103 673 613 1321 941 443 1277
1217 1193 1109 1093 809 659 641 631 1433 499 1163 1069 431
Composed Simple Magic Square
479 1307 379 1511 727 883 577 1489 953 1181 619 1279 563
419 1429 389 1439 433 1399 463 1381 701 769 1237 599 1289
1451 449 1409 367 1453 661 1153 409 761 647 907 1129 1151
1327 491 1499 359 1063 733 1483 397 757 739 1187 593 1319
1091 653 439 1493 1117 617 461 1481 683 643 1223 587 1459
853 929 467 1427 1019 797 487 1373 937 1213 607 1291 547
1013 997 1283 383 863 1031 1361 421 1259 911 601 1303 521
719 1097 1487 373 677 1231 1367 401 1301 1249 971 571 503
1009 773 839 751 1039 691 1049 1201 1471 829 541 1297 457
859 821 977 823 743 967 991 1171 509 1423 1123 557 983
857 1021 1033 877 887 947 811 919 569 523 827 1229 1447
1061 787 1087 1051 709 881 1103 673 613 1321 941 443 1277
809 1193 659 1093 1217 1109 641 631 1433 499 1163 1069 431

The order 8 Simple Magic Sub Square applied above is composed of four each order 4 Simple Magic Squares with Magic Sum s4 = 3676.

Each square shown corresponds with numerous Prime Number Magic Squares with the same Magic Sum(s) and variable values.

14.13.48 Composed Magic Squares (13 x 13)
         Order 4 and 9 Magic Sub Squares

When the Magic Sum s13 is a multiple of 13 (e.g. 11947), order 13 (Semi) Magic Squares might be composed of:

  • One 9th order Simple Magic Corner Square with Magic Sum s9 = 9 * s13 / 13 = 8271 (top/left)
  • One 4th order Simple Magic Corner Square with Magic Sum s4 = 4 * s13 / 13 = 3676 (bottom/right)
  • Two Magic Rectangles order 4 x 9 with s4 = 3676 and s9 = 8271

An example is shown below for the consecutive prime numbers {359 ... 1511} with the related Magic Sums mentioned above.

Composed Semi Magic Square
1433 457 1447 431 499 631 1459 443 1471 761 1021 383 1511
1423 509 1319 523 1321 929 503 521 1223 571 1217 389 1499
547 1303 541 1301 1297 877 569 1279 557 491 1277 419 1489
743 1109 739 971 773 991 751 1091 1103 433 907 853 1483
659 1171 1151 1009 1129 1069 719 691 673 947 479 1439 811
769 1063 821 839 1061 881 977 809 1051 1193 653 1451 379
619 1201 643 1093 647 1039 1181 1187 661 1123 727 1453 373
829 859 1033 941 937 857 883 1013 919 1259 563 1487 367
1249 599 577 1163 607 997 1229 1237 613 1493 1427 397 359
953 683 587 701 709 1283 887 1087 1381 1117 1019 863 677
827 983 1097 911 787 641 1049 823 1153 617 797 1031 1231
467 601 593 757 1213 1289 1291 1327 733 461 487 1361 1367
1429 1409 1399 1307 967 463 449 439 409 1481 1373 421 401
Composed Simple Magic Square
1433 457 1447 431 499 631 1459 443 1471 433 907 853 1483
1423 509 1319 523 1321 929 503 521 1223 491 1277 419 1489
547 1303 541 1301 1297 877 569 1279 557 571 1217 389 1499
743 1109 739 971 773 991 751 1091 1103 761 1021 383 1511
659 1171 1151 1009 1129 1069 719 691 673 947 479 1439 811
769 1063 821 839 1061 881 977 809 1051 1193 653 1451 379
619 1201 643 1093 647 1039 1181 1187 661 1123 727 1453 373
829 859 1033 941 937 857 883 1013 919 1259 563 1487 367
1249 599 577 1163 607 997 1229 1237 613 1493 1427 397 359
709 1283 683 701 953 587 887 1087 1381 1117 1019 863 677
787 641 983 911 827 1097 1049 823 1153 617 797 1031 1231
1213 1289 601 757 467 593 1291 1327 733 461 487 1361 1367
967 463 1409 1307 1429 1399 449 439 409 1481 1373 421 401

The Order 9 Simple Magic Sub Square (s9 = 8271), constructed with the Generator Principle as applied in previous sections, corresponds with 8 * 24/2 * (4!) = 8 * 192 = 1536 Sub Squares.

The resulting square corresponds consequently with 32 * 1536 * (4! * 5!)2 = 4,076 1011 Prime Number Magic Squares with the same Magic Sum(s) and variable values.

14.13.49 Composed Magic Squares (13 x 13)
         Overlapping Sub Squares

Prime Number Composed Magic Squares of order 13 with Overlapping Sub Squares can be constructed when the Magic Sum s13 is a multiple of 13.

An example is shown below, for the consecutive prime numbers {359 ... 1511} with the related Magic Sum s13 = 11947

Composed Magic Square (J)
1423 523 547 643 1459 479 1307 379 1511 727 883 577 1489
599 1319 773 1447 457 419 1429 389 1439 433 1399 463 1381
1249 1181 503 971 691 1451 449 1409 367 1453 661 1153 409
881 743 1471 431 1069 1327 491 1499 359 1063 733 1483 397
443 829 1301 1103 919 1061 1297 809 509 613 761 1109 1193
1091 653 439 1493 647 499 1279 941 1229 1213 1163 659 641
853 929 467 1427 1151 1049 521 1051 823 593 701 1123 1259
1013 997 1283 383 557 1303 911 1223 601 563 619 1217 1277
719 1097 1487 373 1321 683 587 571 1433 1289 1187 631 569
1117 617 461 1481 859 991 1039 1033 937 1291 739 709 673
1019 797 487 1373 787 907 857 821 877 751 1237 1087 947
863 1031 1361 421 1009 811 953 983 1093 757 887 607 1171
677 1231 1367 401 1021 967 827 839 769 1201 977 1129 541

The 13th order Composed Magic Square J shown above, contains following sub squares:

  • Two each other overlapping 5th order Simple Magic Squares G and C (MC5 = 5 * s13 / 13 = 4595),
    with the element a(57) = s13 / 13 = 919 in common;
  • Four 4th order Magic Border Squares (MC4 = 4 * s13 / 13 = 3676): A and B (left), D and E (top);
  • Two each other overlapping 9th order Magic Squares (MC9 = 9 * s13 / 13 = 8271):
    - F composed out of B (left bottom), G (left top), D (right top) and C (right bottom)
    - H with eccentric embedded C (left top).

and corresponds with numerous Prime Number Magic Squares with the same Magic Sum(s) and variable values.

14.13.50 Summary

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

Order

Main Characteristics

Subroutine

Results

-

-

-

-

13

Consecutive Primes, Simple Magic

CnstrSqrs13

Attachment 14.13.41

3

Square Inlays, Consecutive Primes (13 ... 1033)

Prime1343

Attachment 14.13.43

4

Square Inlays, Consecutive Primes (13 ... 1033)

Prime1344

Attachment 14.13.44

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-

-

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Following sections will explain the concept of Prime Number Magic Squares composed of Twin Primes and illustrate how subject squares can be generated with comparable routines as described in previous sections.

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