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 |
Famous quotes containing the words prime, reciprocal, magic and/or square:
“Faith in reason as a prime motor is no longer the criterion of the sound mind, any more than faith in the Bible is the criterion of righteous intention.”
—George Bernard Shaw (1856–1950)
“Of course we will continue to work for cheaper electricity in the homes and on the farms of America; for better and cheaper transportation; for low interest rates; for sounder home financing; for better banking; for the regulation of security issues; for reciprocal trade among nations and for the wiping out of slums. And my friends, for all of these we have only begun to fight.”
—Franklin D. Roosevelt (1882–1945)
“The magic of photography is metaphysical. What you see in the photograph isn’t what you saw at the time. The real skill of photography is organised visual lying.”
—Terence Donovan (b. 1936)
“In old times people used to try and square the circle; now they try and devise schemes for satisfying the Irish nation.”
—Samuel Butler (1835–1902)