A NEW PRIME VARIANT BORDER 13x13 SQUARE

A 13x13 prime magic square, discovered by JS Malachy^1,2, consists of internal 3x3, 5x5, 7x7, 9x9, and 11x11 squares which are also magic at their shared borders.

This page depicts another variant having the same 5437 at the center of the square but a different 3x3 square having a difference (Δ) between numbers of 90 instead of 6 as in the original square. In addition prime numbers greater than 10000 are used in generating the new square. The complementary table that is used for generating the new square is shown below.

Construction of a New Prime 13x13 Magic Square

1. Construct the complementary table of prime pairs where each pair of complements, as well as the 85^th number doubled, sums to 10874. Note that the prime numbers start at 7 and move down (follow the arrows) to the center prime 5437 and loop back to the final prime 10867.

Complementary Table

7	13	103	163	211	223	373	421	541	601	907	967	991	1063	1087	1093	1153	1213	1231	1327	⇒
10867	10861	10771	10711	10663	10651	10501	10453	10333	10273	9967	9907	9883	9811	9787	9781	9721	9661	9643	9547	⇐

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1483	1531	1693	1723	1741	1747	1783	1831	1867	1933	1951	1987	2011	2053	2113	2137	2143	2161	2293	2311	2347	2521	⇒
9391	9343	9181	9151	9133	9127	9091	9043	9007	8941	8923	8887	8863	8821	8761	8737	8731	8713	8581	8563	8527	8353	⇐

2557	2683	2713	2857	3001	3121	3187	3253	3271	3301	3313	3457	3463	3541	3631	3637	3697	3847	3877	3907	3967	4003	⇒
8317	8191	8161	8017	7873	7753	7687	7621	7603	7573	7561	7417	7411	7333	7243	7237	7177	7027	6997	6967	6907	6871	⇐

4051	4093	4111	4297	4327	4423	4447	4507	4513	4597	4603	4723	4783	4831	4993	5023	5227	5233	5347	5431	⇓
																				5437
6823	6781	6763	6577	6547	6451	6427	6367	6361	6277	6271	6151	6091	6043	5881	5851	5647	5641	5527	5443	⇐

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.
3. Then generate every odd square starting with 5x5 thru 13x13, by filling each border with the requisite complementary pairs.
4. The final 13x13 square then has an S = 70681.
5. The magic constants of the respective squares are 70681, 59807, 48933, 38059, 27185 and 16311.

Prime Square 7x7   6967 4597 1483 6577 7753 4831 5851 1933 9007 4783 5641 2311 5443 8941 3463 1093 6361 4423 5527 9781 7411 6823 5227 4603 5437 6271 5647 4051 4507 6427 5347 6451 4513 4447 6367 9343 5431 6091 5233 8563 1867 1531 5023 6277 9391 4297 3121 6043 3907

⇒

Prime Square 9x9   4093 7561 3631 3457 7573 9181 2137 3967 7333 7237 6967 4597 1483 6577 7753 4831 5851 3637 2347 1933 9007 4783 5641 2311 5443 8941 8527 6763 3463 1093 6361 4423 5527 9781 7411 4111 6871 6823 5227 4603 5437 6271 5647 4051 4003 3877 4507 6427 5347 6451 4513 4447 6367 6997 7177 9343 5431 6091 5233 8563 1867 1531 3697 7027 5023 6277 9391 4297 3121 6043 3907 3847 3541 3313 7243 7417 3301 1693 8737 6907 6781

⇒

New 13x13 Prime Square

1153	8923	1063	9127	1327	10333	7	9133	9661	1723	991	8887	8353
9967	8161	3253	2857	4723	2143	6547	8821	8713	8317	3001	3271	907
1831	10861	4093	7561	3631	3457	7573	9181	2137	3967	7333	13	9043
9907	4993	7237	6967	4597	1483	6577	7753	4831	5851	3637	5881	967
373	601	2347	1933	9007	4783	5641	2311	5443	8941	8527	10273	10501
10771	2293	6763	3463	1093	6361	4423	5527	9781	7411	4111	8581	103
2011	2683	6871	6823	5227	4603	5437	6271	5647	4051	4003	8191	8863
10711	8761	3877	4507	6427	5347	6451	4513	4447	6367	6997	2113	163
223	211	7177	9343	5431	6091	5233	8563	1867	1531	3697	10663	10651
9643	3187	7027	5023	6277	9391	4297	3121	6043	3907	3847	7687	1231
1783	10453	3541	3313	7243	7417	3301	1693	8737	6907	6781	421	9091
9787	7603	7621	8017	6151	8731	4327	2053	2161	2557	7873	2713	1087
2521	1951	9811	1747	9547	541	10867	1741	1213	9151	9883	1987	9721

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6. For comparisons sake Malachy's prime square is shown below. The square is colored as above but only for comparison. The complementary table for this square is colored differently for some of the primes.
7. For example, the prime 9007 is green in the new square and khaki in the Malachy square.

This concludes the new 13x13 prime square variant.
To see three new variants go to the next web page. Go back to homepage.

References

1. http://www.magic-squares.net/primesqr.htm#Order-13 constant difference
2. I.S.Malachy, Mathematics on Vacation, Thomas Nelson and Sons, 1966, pp92-94.