Prime reciprocal magic square

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

In decimal, unit fractions ⁠1/2⁠ and ⁠1/5⁠ have no repeating decimal, while ⁠1/3⁠ repeats ${\displaystyle 0.3333\dots }$ indefinitely. The remainder of ⁠1/7⁠, on the other hand, repeats over six digits as, $${\displaystyle 0.{\mathbf {1}}42857{\mathbf {1}}42857{\mathbf {1}}\dots }$$

Consequently, multiples of one-seventh exhibit cyclic permutations of these six digits:^[1]

$${\displaystyle {\begin{aligned}1/7&=0.142857\dots \\2/7&=0.285714\dots \\3/7&=0.428571\dots \\4/7&=0.571428\dots \\5/7&=0.714285\dots \\6/7&=0.857142\dots \end{aligned}}}$$

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

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 ${\displaystyle p-1}$ 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.

In a full, or otherwise prime reciprocal magic square with ${\displaystyle p-1}$ period, the even number of k−th rows in the square are arranged by multiples of ${\displaystyle 1/p}$ — 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:

$${\displaystyle {\begin{aligned}1/7=&{\text{ }}0.142\;857\dots \\+&{\text{ }}0.857\;142\ldots =6/7\\&------------\\&{\text{ }}0.999\;999\ldots \\\\1/13=&{\text{ }}0.076\;923\;076\;923\dots \\+&{\text{ }}0.923\;076\;923\;076\ldots =12/13\\&------------\\&{\text{ }}0.999\;999\;999\;999\ldots \\\\1/19=&{\text{ }}0.052631578\;947368421\dots \\+&{\text{ }}0.947368421\;052631578\ldots =18/19\\&------------\\&{\text{ }}0.999999999\;999999999\dots \\\end{aligned}}}$$

This is a result of Midy's theorem.^[2][3] 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,

$${\displaystyle {\begin{aligned}1/23&=0.04347826\;08695652\;173913\ldots \\2/23&=0.08695652\;17391304\;347826\ldots \\4/23&=0.17391304\;34782608\;695652\ldots \\8/23&=0.34782608\;69565217\;391304\ldots \\16/23&=0.69565217\;39130434\;782608\ldots \\\end{aligned}}}$$

In this case, a factor of 2 moves the repeating decimal of ⁠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 ${\displaystyle 1/p}$. Other magic squares can be constructed whose rows do not represent consecutive multiples of ${\displaystyle 1/p}$, which nonetheless generate a magic sum.

Magic squares based on reciprocals of primes p in bases b with periods ${\displaystyle p-1}$ have magic sums equal to,

$${\displaystyle M=(b-1)\times {\frac {p-1}{2}}.}$$

The ${\displaystyle {\mathbf {\tfrac {1}{19}}}}$ 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 ${\displaystyle k}$−th rows:^[4][5]

$${\displaystyle {\begin{aligned}1/19&=0.{\color {red}0}{\text{ }}5{\text{ }}2{\text{ }}6{\text{ }}3{\text{ }}1{\text{ }}5{\text{ }}7{\text{ }}8{\text{ }}9{\text{ }}4{\text{ }}7{\text{ }}3{\text{ }}6{\text{ }}8{\text{ }}4{\text{ }}2{\text{ }}{\color {red}1}\dots \\2/19&=0.1{\text{ }}{\color {red}0}{\text{ }}5{\text{ }}2{\text{ }}6{\text{ }}3{\text{ }}1{\text{ }}5{\text{ }}7{\text{ }}8{\text{ }}9{\text{ }}4{\text{ }}7{\text{ }}3{\text{ }}6{\text{ }}8{\text{ }}{\color {red}4}{\text{ }}2\dots \\3/19&=0.1{\text{ }}5{\text{ }}{\color {red}7}{\text{ }}8{\text{ }}9{\text{ }}4{\text{ }}7{\text{ }}3{\text{ }}6{\text{ }}8{\text{ }}4{\text{ }}2{\text{ }}1{\text{ }}0{\text{ }}5{\text{ }}{\color {red}2}{\text{ }}6{\text{ }}3\dots \\4/19&=0.2{\text{ }}1{\text{ }}0{\text{ }}{\color {red}5}{\text{ }}2{\text{ }}6{\text{ }}3{\text{ }}1{\text{ }}5{\text{ }}7{\text{ }}8{\text{ }}9{\text{ }}4{\text{ }}7{\text{ }}{\color {red}3}{\text{ }}6{\text{ }}8{\text{ }}4\dots \\5/19&=0.2{\text{ }}6{\text{ }}3{\text{ }}1{\text{ }}{\color {red}5}{\text{ }}7{\text{ }}8{\text{ }}9{\text{ }}4{\text{ }}7{\text{ }}3{\text{ }}6{\text{ }}8{\text{ }}{\color {red}4}{\text{ }}2{\text{ }}1{\text{ }}0{\text{ }}5\dots \\6/19&=0.3{\text{ }}1{\text{ }}5{\text{ }}7{\text{ }}8{\text{ }}{\color {red}9}{\text{ }}4{\text{ }}7{\text{ }}3{\text{ }}6{\text{ }}8{\text{ }}4{\text{ }}{\color {red}2}{\text{ }}1{\text{ }}0{\text{ }}5{\text{ }}2{\text{ }}6\dots \\7/19&=0.3{\text{ }}6{\text{ }}8{\text{ }}4{\text{ }}2{\text{ }}1{\text{ }}{\color {red}0}{\text{ }}5{\text{ }}2{\text{ }}6{\text{ }}3{\text{ }}{\color {red}1}{\text{ }}5{\text{ }}7{\text{ }}8{\text{ }}9{\text{ }}4{\text{ }}7\dots \\8/19&=0.4{\text{ }}2{\text{ }}1{\text{ }}0{\text{ }}5{\text{ }}2{\text{ }}6{\text{ }}{\color {red}3}{\text{ }}1{\text{ }}5{\text{ }}{\color {red}7}{\text{ }}8{\text{ }}9{\text{ }}4{\text{ }}7{\text{ }}3{\text{ }}6{\text{ }}8\dots \\9/19&=0.4{\text{ }}7{\text{ }}3{\text{ }}6{\text{ }}8{\text{ }}4{\text{ }}2{\text{ }}1{\text{ }}{\color {red}0}{\text{ }}{\color {red}5}{\text{ }}2{\text{ }}6{\text{ }}3{\text{ }}1{\text{ }}5{\text{ }}7{\text{ }}8{\text{ }}9\dots \\10/19&=0.5{\text{ }}2{\text{ }}6{\text{ }}3{\text{ }}1{\text{ }}5{\text{ }}7{\text{ }}8{\text{ }}{\color {red}9}{\text{ }}{\color {red}4}{\text{ }}7{\text{ }}3{\text{ }}6{\text{ }}8{\text{ }}4{\text{ }}2{\text{ }}1{\text{ }}0\dots \\11/19&=0.5{\text{ }}7{\text{ }}8{\text{ }}9{\text{ }}4{\text{ }}7{\text{ }}3{\text{ }}{\color {red}6}{\text{ }}8{\text{ }}4{\text{ }}{\color {red}2}{\text{ }}1{\text{ }}0{\text{ }}5{\text{ }}2{\text{ }}6{\text{ }}3{\text{ }}1\dots \\12/19&=0.6{\text{ }}3{\text{ }}1{\text{ }}5{\text{ }}7{\text{ }}8{\text{ }}{\color {red}9}{\text{ }}4{\text{ }}7{\text{ }}3{\text{ }}6{\text{ }}{\color {red}8}{\text{ }}4{\text{ }}2{\text{ }}1{\text{ }}0{\text{ }}5{\text{ }}2\dots \\13/19&=0.6{\text{ }}8{\text{ }}4{\text{ }}2{\text{ }}1{\text{ }}{\color {red}0}{\text{ }}5{\text{ }}2{\text{ }}6{\text{ }}3{\text{ }}1{\text{ }}5{\text{ }}{\color {red}7}{\text{ }}8{\text{ }}9{\text{ }}4{\text{ }}7{\text{ }}3\dots \\14/19&=0.7{\text{ }}3{\text{ }}6{\text{ }}8{\text{ }}{\color {red}4}{\text{ }}2{\text{ }}1{\text{ }}0{\text{ }}5{\text{ }}2{\text{ }}6{\text{ }}3{\text{ }}1{\text{ }}{\color {red}5}{\text{ }}7{\text{ }}8{\text{ }}9{\text{ }}4\dots \\15/19&=0.7{\text{ }}8{\text{ }}9{\text{ }}{\color {red}4}{\text{ }}7{\text{ }}3{\text{ }}6{\text{ }}8{\text{ }}4{\text{ }}2{\text{ }}1{\text{ }}0{\text{ }}5{\text{ }}2{\text{ }}{\color {red}6}{\text{ }}3{\text{ }}1{\text{ }}5\dots \\16/19&=0.8{\text{ }}4{\text{ }}{\color {red}2}{\text{ }}1{\text{ }}0{\text{ }}5{\text{ }}2{\text{ }}6{\text{ }}3{\text{ }}1{\text{ }}5{\text{ }}7{\text{ }}8{\text{ }}9{\text{ }}4{\text{ }}{\color {red}7}{\text{ }}3{\text{ }}6\dots \\17/19&=0.8{\text{ }}{\color {red}9}{\text{ }}4{\text{ }}7{\text{ }}3{\text{ }}6{\text{ }}8{\text{ }}4{\text{ }}2{\text{ }}1{\text{ }}0{\text{ }}5{\text{ }}2{\text{ }}6{\text{ }}3{\text{ }}1{\text{ }}{\color {red}5}{\text{ }}7\dots \\18/19&=0.{\color {red}9}{\text{ }}4{\text{ }}7{\text{ }}3{\text{ }}6{\text{ }}8{\text{ }}4{\text{ }}2{\text{ }}1{\text{ }}0{\text{ }}5{\text{ }}2{\text{ }}6{\text{ }}3{\text{ }}1{\text{ }}5{\text{ }}7{\text{ }}{\color {red}8}\dots \\\end{aligned}}}$$

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^[6]

{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.

A ${\displaystyle {\tfrac {1}{17}}}$ 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:^[7][8]

$${\displaystyle {\begin{aligned}1/17&=0.{\color {blue}0}{\text{ }}5\;8\;8\;2\;3\;5\;2\;9\;4\;1\;1\;7\;6\;4\;{\color {blue}7}\dots \\5/17&=0.2\;{\color {blue}9}\;4\;1\;1\;7\;6\;4\;7\;0\;5\;8\;8\;2\;{\color {blue}3}\;5\dots \\8/17&=0.4\;7\;{\color {blue}0}\;5\;8\;8\;2\;3\;5\;2\;9\;4\;1\;{\color {blue}1}\;7\;6\dots \\6/17&=0.3\;5\;2\;{\color {blue}9}\;4\;1\;1\;7\;6\;4\;7\;0\;{\color {blue}5}\;8\;8\;2\dots \\13/17&=0.7\;6\;4\;7\;{\color {blue}0}\;5\;8\;8\;2\;3\;5\;{\color {blue}2}\;9\;4\;1\;1\dots \\14/17&=0.8\;2\;3\;5\;2\;{\color {blue}9}\;4\;1\;1\;7\;{\color {blue}6}\;4\;7\;0\;5\;8\dots \\2/17&=0.1\;1\;7\;6\;4\;7\;{\color {blue}0}\;5\;8\;{\color {blue}8}\;2\;3\;5\;2\;9\;4\dots \\10/17&=0.5\;8\;8\;2\;3\;5\;2\;{\color {blue}9}\;{\color {blue}4}\;1\;1\;7\;6\;4\;7\;0\dots \\16/17&=0.9\;4\;1\;1\;7\;6\;4\;{\color {blue}7}\;{\color {blue}0}\;5\;8\;8\;2\;3\;5\;2\dots \\12/17&=0.7\;0\;5\;8\;8\;2\;{\color {blue}3}\;5\;2\;{\color {blue}9}\;4\;1\;1\;7\;6\;4\dots \\9/17&=0.5\;2\;9\;4\;1\;{\color {blue}1}\;7\;6\;4\;7\;{\color {blue}0}\;5\;8\;8\;2\;3\dots \\11/17&=0.6\;4\;7\;0\;{\color {blue}5}\;8\;8\;2\;3\;5\;2\;{\color {blue}9}\;4\;1\;1\;7\dots \\4/17&=0.2\;3\;5\;{\color {blue}2}\;9\;4\;1\;1\;7\;6\;4\;7\;{\color {blue}0}\;5\;8\;8\dots \\3/17&=0.1\;7\;{\color {blue}6}\;4\;7\;0\;5\;8\;8\;2\;3\;5\;2\;{\color {blue}9}\;4\;1\dots \\15/17&=0.8\;{\color {blue}8}\;2\;3\;5\;2\;9\;4\;1\;1\;7\;6\;4\;7\;{\color {blue}0}\;5\dots \\7/17&=0.{\color {blue}4}\;1\;1\;7\;6\;4\;7\;0\;5\;8\;8\;2\;3\;5\;2\;{\color {blue}9}\dots \\\end{aligned}}}$$

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 ${\displaystyle 1/p}$ fit in respective ${\displaystyle k}$−th rows.

• Cyclic number
• Reciprocals of primes