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):

PrimeBaseTotal
191081
5312286
5334858
59229
67233
83241
8919792
167685,561
199413,960
19915014,751
2112105
2233222
29314721,316
3075612
383101,719
38936069,646
3975792
42133870,770
48761,215
503420105,169
587368107,531
5933592
6318727,090
677407137,228
757759286,524
787134,716
8113810
9771,222595,848
1,033115,160
1,18713579,462
1,30752,612
1,499117,490
1,8771916,884
1,933146140,070
2,0112625,125
2,02721,013
2,1416366,340
2,53921,269
3,18797152,928
3,3731116,860
3,659126228,625
3,9473567,082
4,26122,130
4,81322,406
5,64775208,902
6,11336,112
6,27723,138
7,28323,641
8,38724,193

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