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 2However, 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 ---------------------- 0.999999999,999999999Ekidhikena 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 = 0.8421052631578,94736A 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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—Samuel Butler (1835–1902)