# 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·__1__42857__1__42857__1__42857... 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 [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 = 0·05 2 6 3 1 5 7 8 9 4 7 3 6 8 4 21... 02/19 = 0·105 2 6 3 1 5 7 8 9 4 7 3 6 842... 03/19 = 0·1 578 9 4 7 3 6 8 4 2 1 0 526 3... 04/19 = 0·2 1 052 6 3 1 5 7 8 9 4 736 8 4... 05/19 = 0·2 6 3 157 8 9 4 7 3 6 842 1 0 5... 06/19 = 0·3 1 5 7 894 7 3 6 8 421 0 5 2 6... 07/19 = 0·3 6 8 4 2 105 2 6 315 7 8 9 4 7... 08/19 = 0·4 2 1 0 5 2 631 578 9 4 7 3 6 8... 09/19 = 0·4 7 3 6 8 4 2 1052 6 3 1 5 7 8 9... 10/19 = 0·5 2 6 3 1 5 7 8947 3 6 8 4 2 1 0... 11/19 = 0·5 7 8 9 4 7 368 421 0 5 2 6 3 1... 12/19 = 0·6 3 1 5 7 894 7 3 684 2 1 0 5 2... 13/19 = 0·6 8 4 2 105 2 6 3 1 578 9 4 7 3... 14/19 = 0·7 3 6 842 1 0 5 2 6 3 157 8 9 4... 15/19 = 0·7 8 947 3 6 8 4 2 1 0 5 263 1 5... 16/19 = 0·8 421 0 5 2 6 3 1 5 7 8 9 473 6... 17/19 = 0·894 7 3 6 8 4 2 1 0 5 2 6 3 157... 18/19 = 0·94 7 3 6 8 4 2 1 0 5 2 6 3 1 5 78...

[1]

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 |

## See also

## References

Rademacher, H. and Toeplitz, O. The Enjoyment of Mathematics: Selections from Mathematics for the Amateur. Princeton, NJ: Princeton University Press, pp. 158–160, 1957.

Weisstein, Eric W. "Midy's Theorem." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/MidysTheorem.html