Prime reciprocal magic square

Introduction

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 of these six digits: \begin{align} 1/7 & = 0.1 4 2 8 5 7\dots \ 2/7 & = 0.2 8 5 7 1 4\dots \ 3/7 & = 0.4 2 8 5 7 1\dots \ 4/7 & = 0.5 7 1 4 2 8\dots \ 5/7 & = 0.7 1 4 2 8 5\dots \ 6/7 & = 0.8 5 7 1 4 2\dots \end{align}

If the digits are laid out as a square, each row and column sums to 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 ; however, their mean is , as one diagonal adds to while the other adds to .

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 () when added together:

\begin{align} 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{align}

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{align} 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{align}

In this case, a factor of moves the repeating decimal of 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}.

Full magic squares

The \bold{\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 k−th rows:

\begin{align} 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{align}

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

:"Fourteen primes less than 1000000 possess this required property [in decimal]".

:Solution to problem 2420, "Only 19?" by M. J. Zerger.

:{19, 383, 32327, 34061, 45341, 61967, 65699, 117541, 158771, 405817, ...} .

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 \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:

\begin{align} 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{align}

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.

References

References

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