Prime reciprocal magic square

A prime reciprocal magic square is a magic square using the decimal digits of the reciprocal of a prime number.

Formulation

Basics

In decimal, unit fractions $\frac{1}{2}$ and $\frac{1}{5}$ have no repeating decimal, while $\frac{1}{3}$ repeats $0.3333\dots$ indefinitely. The remainder of $\frac{1}{7}$, on the other hand, repeats over six digits as, $$0.\mathbf{1}42857\mathbf{1}42857\mathbf{1}\dots$$ Consequently, multiples of one-seventh exhibit cyclic permutations of these six digits: $$\begin{array}{rl}1/7& =0.142857\dots \\ 2/7& =0.285714\dots \\ 3/7& =0.428571\dots \\ 4/7& =0.571428\dots \\ 5/7& =0.714285\dots \\ 6/7& =0.857142\dots \end{array}$$ If the digits are laid out as a square, each row and column sums to $1+4+2+8+5+7=27.$ This yields the smallest base-10 non-normal, prime reciprocal magic square

$1$	$4$	$2$	$8$	$5$	$7$
$2$	$8$	$5$	$7$	$1$	$4$
$4$	$2$	$8$	$5$	$7$	$1$
$5$	$7$	$1$	$4$	$2$	$8$
$7$	$1$	$4$	$2$	$8$	$5$
$8$	$5$	$7$	$1$	$4$	$2$

In contrast with its rows and columns, the diagonals of this square do not sum to 27; however, their mean is 27, as one diagonal adds to 23 while the other adds to 31. All prime reciprocals in any base with a $p-1$ period will generate magic squares where all rows and columns produce a magic constant, and only a select few will be full, such that their diagonals, rows and columns collectively yield equal sums.

Decimal expansions

In a full, or otherwise prime reciprocal magic square with $p-1$ period, the even number of $k$−th rows in the square are arranged by multiples of $1/p$ — not necessarily successively — where a magic constant can be obtained. For instance, an even repeating cycle from an odd, prime reciprocal of $p$ that is divided into $n$−digit strings creates pairs of complementary sequences of digits that yield strings of nines (9) when added together: $$\begin{array}{rl}1/7=& 0.142857\dots \\ +& 0.857142\dots =6/7\\ & ------------\\ & 0.999999\dots \\ \\ 1/13=& 0.076923076923\dots \\ +& 0.923076923076\dots =12/13\\ & ------------\\ & 0.999999999999\dots \\ \\ 1/19=& 0.052631578947368421\dots \\ +& 0.947368421052631578\dots =18/19\\ & ------------\\ & 0.999999999999999999\dots \\ \end{array}$$ This is a result of Midy's theorem. These complementary sequences are generated between multiples of prime reciprocals that add to 1. More specifically, a factor $n$ in the numerator of the reciprocal of a prime number $p$ will shift the decimal places of its decimal expansion accordingly, $$\begin{array}{rl}1/23& =0.0434782608695652173913\dots \\ 2/23& =0.0869565217391304347826\dots \\ 4/23& =0.1739130434782608695652\dots \\ 8/23& =0.3478260869565217391304\dots \\ 16/23& =0.6956521739130434782608\dots \\ \end{array}$$ In this case, a factor of 2 moves the repeating decimal of $\frac{1}{23}$ by eight places. A uniform solution of a prime reciprocal magic square, whether full or not, will hold rows with successive multiples of $1/p$. Other magic squares can be constructed whose rows do not represent consecutive multiples of $1/p$, which nonetheless generate a magic sum.

Magic constant

Magic squares based on reciprocals of primes $p$ in bases $b$ with periods $p-1$ have magic sums equal to, $$M=(b-1)\times \frac{p-1}{2}.$$ The table below lists some prime numbers that generate prime-reciprocal magic squares in given bases.

Prime	Base	Magic sum
19	10	81
53	12	286
59	2	29
67	2	33
83	2	41
89	19	792
211	2	105
223	3	222
307	5	612
383	10	1,719
397	5	792
487	6	1,215
593	3	592
631	87	27,090
787	13	4,716
811	3	810
1,033	11	5,160
1,307	5	2,612
1,499	11	7,490
1,877	19	16,884
2,011	26	25,125
2,027	2	1,013

Full magic squares

The $\frac{1}{19}$ magic square with maximum period 18 contains a row-and-column total of 81, that is also obtained by both diagonals. This makes it the first full, non-normal base-10 prime reciprocal magic square whose multiples fit inside respective $k$−th rows: $$\begin{array}{rl}1/19& =0.052631578947368421\dots \\ 2/19& =0.105263157894736842\dots \\ 3/19& =0.157894736842105263\dots \\ 4/19& =0.210526315789473684\dots \\ 5/19& =0.263157894736842105\dots \\ 6/19& =0.315789473684210526\dots \\ 7/19& =0.368421052631578947\dots \\ 8/19& =0.421052631578947368\dots \\ 9/19& =0.473684210526315789\dots \\ 10/19& =0.526315789473684210\dots \\ 11/19& =0.578947368421052631\dots \\ 12/19& =0.631578947368421052\dots \\ 13/19& =0.684210526315789473\dots \\ 14/19& =0.736842105263157894\dots \\ 15/19& =0.789473684210526315\dots \\ 16/19& =0.842105263157894736\dots \\ 17/19& =0.894736842105263157\dots \\ 18/19& =0.947368421052631578\dots \\ \end{array}$$ The first few prime numbers in decimal whose reciprocals can be used to produce a non-normal, full prime reciprocal magic square of this type are

{19, 383, 32327, 34061, 45341, 61967, 65699, 117541, 158771, 405817, ...} (sequence A072359 in the OEIS).

The smallest prime number to yield such magic square in binary is 59 (111011₂), while in ternary it is 223 (22021₃); these are listed at A096339, and A096660.

Variations

A $\frac{1}{17}$ prime reciprocal magic square with maximum period of 16 and magic constant of 72 can be constructed where its rows represent non-consecutive multiples of one-seventeenth: $$\begin{array}{rl}1/17& =0.0588235294117647\dots \\ 5/17& =0.2941176470588235\dots \\ 8/17& =0.4705882352941176\dots \\ 6/17& =0.3529411764705882\dots \\ 13/17& =0.7647058823529411\dots \\ 14/17& =0.8235294117647058\dots \\ 2/17& =0.1176470588235294\dots \\ 10/17& =0.5882352941176470\dots \\ 16/17& =0.9411764705882352\dots \\ 12/17& =0.7058823529411764\dots \\ 9/17& =0.5294117647058823\dots \\ 11/17& =0.6470588235294117\dots \\ 4/17& =0.2352941176470588\dots \\ 3/17& =0.1764705882352941\dots \\ 15/17& =0.8823529411764705\dots \\ 7/17& =0.4117647058823529\dots \\ \end{array}$$ As such, this full magic square is the first of its kind in decimal that does not admit a uniform solution where consecutive multiples of $1/p$ fit in respective $k$−th rows.

See also

• Cyclic number
• Reciprocals of primes