Prime reciprocal magic square
Encyclopedia : P : PR : PRI : Prime reciprocal magic square
A prime reciprocal magic square is a magic square using the 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. 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
From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.