Prime Reciprocal Magic Square

A prime reciprocal magic square is a magic square using the decimal digits of the reciprocal of a prime number.

Consider 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.

Other properties of Prime Reciprocals: Midy's theorem

The repeating pattern of an even number of digits in the quotients when broken in half are the nines-complement of each half:

1/7 = 0.142,857,142,857 ... +0.857,142 --------- 0.999,999 1/11 = 0.09090,90909 ... +0.90909,09090 ----- 0.99999,99999 1/13 = 0.076,923 076,923 ... +0.923,076 --------- 0.999,999 1/17 = 0.05882352,94117647 +0.94117647,05882352 ------------------- 0.99999999,99999999 1/19 = 0.052631578,947368421 ... +0.947368421,052631578 ---------------------- 0.999999999,999999999

Ekidhikena Purvena From: Bharati Krishna Tirtha's Vedic mathematics#By one more than the one before

Concerning the number of decimal places shifted in the quotient per multiple of 1/19:

01/19 = 0.052631578,947368421 02/19 = 0.1052631578,94736842 04/19 = 0.21052631578,9473684 08/19 = 0.421052631578,947368 16/19 = 0.8421052631578,94736

A factor of 2 in the numerator produces a shift of one decimal place to the right in the quotient.

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

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