A prime reciprocal magic square is a

magic square In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number ...

using the decimal digits of the reciprocal of a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...

. Consider a unit fraction, 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 In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set ''X'' which maps the elements of some subset ''S'' of ''X'' to each other in a cyclic fashion, while fixing (that is, ...

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, 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 -1, 11-1, 13-1, 17-1, 19-1, 23-1, 29-1, 47-1, 59-1, 61-1, 73-1, 89-1, 97-1, 101-1, ...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...

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 × prime−1 / 2):

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 {{DEFAULTSORT:Prime Reciprocal Magic Square Recreational mathematics Magic squares

See also

References