Prime reciprocal magic square

In mathematics, a reciprocal is a number divided into one, like 1/3 or 1/7. In base ten, the remainder, and so the digits, of 1/3 repeats at once: 0·3333... However, the remainders of 1/7 repeat over six, or 7-1, digits: 1/7 = 0·142857142857142857... If you examine the multiples of 1/7, you can see that each is a cyclic permutation of these six digits:

1/7 = 0.1 4 2 8 5 7... 2/7 = 0.2 8 5 7 1 4... 3/7 = 0.4 2 8 5 7 1... 4/7 = 0.5 7 1 4 2 8... 5/7 = 0.7 1 4 2 8 5... 6/7 = 0.8 5 7 1 4 2...

If the digits are laid out as a square, it is obvious that each row will sum to 1+4+2+8+5+7, or 27, and only slightly less obvious that each column will also do so, and consequently we have a magic square:

1 4 2 8 5 7 2 8 5 7 1 4 4 2 8 5 7 1 5 7 1 4 2 8 7 1 4 2 8 5 8 5 7 1 4 2

However, neither diagonal sums to 27, but all other prime reciprocals in base ten with maximum period of p-1 produce squares in which all rows and columns sum to the same total. In the square from 1/19, with maximum period 18 and row-and-column total of 81, both diagonals also sum to 81, and this square is therefore fully magic:

01/19 = 0·0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1... 02/19 = 0·1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2... 03/19 = 0·1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3... 04/19 = 0·2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8 4... 05/19 = 0·2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5... 06/19 = 0·3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6... 07/19 = 0·3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7... 08/19 = 0·4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6 8... 09/19 = 0·4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9... 10/19 = 0·5 2 6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0... 11/19 = 0·5 7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1... 12/19 = 0·6 3 1 5 7 8 9 4 7 3 6 8 4 2 1 0 5 2... 13/19 = 0·6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3... 14/19 = 0·7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8 9 4... 15/19 = 0·7 8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5... 16/19 = 0·8 4 2 1 0 5 2 6 3 1 5 7 8 9 4 7 3 6... 17/19 = 0·8 9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7... 18/19 = 0·9 4 7 3 6 8 4 2 1 0 5 2 6 3 1 5 7 8...

The same phenomenon occurs with other primes in other bases, and the following table lists some of them, giving the prime, base, and magic total (derived from the formula base-1 x prime-1 / 2):

Prime Base Total 19 10 81 53 12 286 53 34 858 59 2 29 67 2 33 83 2 41 89 19 792 167 68 5,561 199 41 3,960 199 150 14,751 211 2 105 223 3 222 293 147 21,316 307 5 612 383 10 1,719 389 360 69,646 397 5 792 421 338 70,770 487 6 1,215 503 420 105,169 587 368 107,531 593 3 592 631 87 27,090 677 407 137,228 757 759 286,524 787 13 4,716 811 3 810 977 1,222 595,848 1,033 11 5,160 1,187 135 79,462 1,307 5 2,612 1,499 11 7,490 1,877 19 16,884 1,933 146 140,070 2,011 26 25,125 2,027 2 1,013 2,141 63 66,340 2,539 2 1,269 3,187 97 152,928 3,373 11 16,860 3,659 126 228,625 3,947 35 67,082 4,261 2 2,130 4,813 2 2,406 5,647 75 208,902 6,113 3 6,112 6,277 2 3,138 7,283 2 3,641 8,387 2 4,193

Prime	Base	Total
19	10	81
53	12	286
53	34	858
59	2	29
67	2	33
83	2	41
89	19	792
167	68	5,561
199	41	3,960
199	150	14,751
211	2	105
223	3	222
293	147	21,316
307	5	612
383	10	1,719
389	360	69,646
397	5	792
421	338	70,770
487	6	1,215
503	420	105,169
587	368	107,531
593	3	592
631	87	27,090
677	407	137,228
757	759	286,524
787	13	4,716
811	3	810
977	1,222	595,848
1,033	11	5,160
1,187	135	79,462
1,307	5	2,612
1,499	11	7,490
1,877	19	16,884
1,933	146	140,070
2,011	26	25,125
2,027	2	1,013
2,141	63	66,340
2,539	2	1,269
3,187	97	152,928
3,373	11	16,860
3,659	126	228,625
3,947	35	67,082
4,261	2	2,130
4,813	2	2,406
5,647	75	208,902
6,113	3	6,112
6,277	2	3,138
7,283	2	3,641
8,387	2	4,193