Prime reciprocal magic square
Encyclopedia
A prime reciprocal magic square is a magic square
Magic square
In recreational mathematics, a magic square of order n is an arrangement of n2 numbers, usually distinct integers, in a square, such that the n numbers in all rows, all columns, and both diagonals sum to the same constant. A normal magic square contains the integers from 1 to n2...

 using the decimal digits of the reciprocal of a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

.

Consider a number
Number
A number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....

 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
Cyclic permutation
A cyclic permutation or circular permutation is a permutation built from one or more sets of elements in cyclic order.The notion "cyclic permutation" is used in different, but related ways:- Definition 1 :right|mapping of 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
Midy's theorem
In mathematics, Midy's theorem, named after French mathematician E. Midy, is a statement about the decimal expansion of fractions a/p where p is a prime and a/p has a repeating decimal expansion with an even period...



The repeating pattern of an even number of digits [7-1, 11-1, 13-1, 17-1, 19-1, 29-1, ...] 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...

http://upload.wikimedia.org/wikipedia/commons/5/59/MgkSqr_1_over_19_Deva.tif

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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