Prime reciprocal magic square explained

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

\tfrac{1}{2}

and

\tfrac{1}{5}

have no repeating decimal, while

\tfrac{1}{3}

repeats

0.3333...

indefinitely. The remainder of

\tfrac{1}{7}

, on the other hand, repeats over six digits as,

0.\bold42857\bold42857\bold\dots

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

$\begin1/7 & = 0.1 4 2 8 5 7\dots \\2/7 & = 0.2 8 5 7 1 4\dots \\3/7 & = 0.4 2 8 5 7 1\dots \\4/7 & = 0.5 7 1 4 2 8\dots \\5/7 & = 0.7 1 4 2 8 5\dots \\6/7 & = 0.8 5 7 1 4 2\dots\end$

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

class=wikitable style="text-align: center;width:12em;height:12em;table-layout:fixed"
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

−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

that is divided into

−digit strings creates pairs of complementary sequences of digits that yield strings of nines (9) when added together:

$\begin1/7 = & \text 0.142\;857\dots \\ + & \text 0.857\;142\ldots = 6/7\\ & ------------ \\ & \text 0.999\;999\ldots \\\\1/13 = & \text 0.076\;923\;076\;923\dots \\ + & \text 0.923\;076\;923\;076\ldots = 12/13\\ & ------------ \\ & \text 0.999\;999\;999\;999\ldots \\\\1/19 = & \text 0.052631578\;947368421\dots \\ + & \text 0.947368421\;052631578\ldots = 18/19\\ & ------------ \\ & \text 0.999999999\;999999999\dots \\ \end$

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

in the numerator of the reciprocal of a prime number

will shift the decimal places of its decimal expansion accordingly,

$\begin1/23 & = 0.04347826\;08695652\;173913\ldots \\2/23 & = 0.08695652\;17391304\;347826\ldots \\4/23 & = 0.17391304\;34782608\;695652\ldots \\8/23 & = 0.34782608\;69565217\;391304\ldots \\16/23 & = 0.69565217\;39130434\;782608\ldots \\ \end$

In this case, a factor of 2 moves the repeating decimal of

\tfrac{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

in bases

with periods

p-1

have magic sums equal to,

$M = (b-1) \times \frac .$

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

\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

−th rows:^[4] ^[5]

$\begin1/19 & = 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 \dots \\2/19 & = 0.1 \text \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 \text 2 \dots \\3/19 & = 0.1 \text 5 \text \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 \text 6 \text 3 \dots \\4/19 & = 0.2 \text 1 \text 0 \text \text 2 \text 6 \text 3 \text 1 \text 5 \text 7 \text 8 \text 9 \text 4 \text 7 \text \text 6 \text 8 \text 4 \dots \\5/19 & = 0.2 \text 6 \text 3 \text 1 \text \text 7 \text 8 \text 9 \text 4 \text 7 \text 3 \text 6 \text 8 \text \text 2 \text 1 \text 0 \text 5 \dots \\6/19 & = 0.3 \text 1 \text 5 \text 7 \text 8 \text \text 4 \text 7 \text 3 \text 6 \text 8 \text 4 \text \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 \text 5 \text 2 \text 6 \text 3 \text \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 \text 1 \text 5 \text \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 \text \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 \text \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 \text 8 \text 4 \text \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 \text 4 \text 7 \text 3 \text 6 \text \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 \text 5 \text 2 \text 6 \text 3 \text 1 \text 5 \text \text 8 \text 9 \text 4 \text 7 \text 3 \dots \\14/19 & = 0.7 \text 3 \text 6 \text 8 \text \text 2 \text 1 \text 0 \text 5 \text 2 \text 6 \text 3 \text 1 \text \text 7 \text 8 \text 9 \text 4 \dots \\15/19 & = 0.7 \text 8 \text 9 \text \text 7 \text 3 \text 6 \text 8 \text 4 \text 2 \text 1 \text 0 \text 5 \text 2 \text \text 3 \text 1 \text 5 \dots \\16/19 & = 0.8 \text 4 \text \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 \text 3 \text 6 \dots \\17/19 & = 0.8 \text \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 \text 7 \dots \\18/19 & = 0. \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 \dots \\\end$

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]

Notes and References

Book: Wells, D. . The Penguin Dictionary of Curious and Interesting Numbers . registration . . London . 1987 . 171–174 . 0-14-008029-5 . 39262447 . 118329153 .
Book: Rademacher . Hans . Hans Rademacher . Toeplitz . Otto . Otto Toeplitz . The Enjoyment of Mathematics: Selections from Mathematics for the Amateur. . registration . . 2nd . Princeton, NJ . 1957 . 158-160 . 9780486262420 . 20827693 . 0081844 . 0078.00114 .
Leavitt . William G. . A Theorem on Repeating Decimals . . 74 . 6 . 669–673 . 1967 . . Washington, D.C. . 10.2307/2314251 . 2314251 . 0211949 . 0153.06503 .
Book: Andrews, William Symes . Magic Squares and Cubes . . Chicago, IL . 1917 . 176, 177 . 9780486206585 . 1136401 . 1003.05500 . 0114763 .
2023-11-21 .
Subramani . K. . On two interesting properties of primes, p, with reciprocals in base 10 having maximum period p – 1. . J. of Math. Sci. & Comp. Math. . 2644-3368 . 1 . 2 . 2020 . 198-200 . S.M.A.R.T. . Auburn, WA . 10.15864/jmscm.1204 . 235037714 .
2023-11-24 .
Singleton . Colin R.J. . Solutions to Problems and Conjectures . . 30 . 2 . Baywood Publishing & Co. . Amityville, NY . 1999 . 158-160 .
"Fourteen primes less than 1000000 possess this required property [in decimal]".

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

.

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:^[6] ^[7]
$\begin1/17 & = 0. \text 5 \; 8 \; 8 \; 2 \; 3 \; 5 \; 2 \; 9 \; 4 \; 1 \; 1 \; 7 \; 6 \; 4 \; \dots \\5/17 & = 0.2 \; \; 4 \; 1 \; 1 \; 7 \; 6 \; 4 \; 7 \; 0 \; 5 \; 8 \; 8 \; 2 \; \; 5 \dots \\8/17 & = 0.4 \; 7 \; \; 5 \; 8 \; 8 \; 2 \; 3 \; 5 \; 2 \; 9 \; 4 \; 1 \; \; 7 \; 6 \dots \\6/17 & = 0.3 \; 5 \; 2 \; \; 4 \; 1 \; 1 \; 7 \; 6 \; 4 \; 7 \; 0 \; \; 8 \; 8 \; 2 \dots \\13/17 & = 0.7 \; 6 \; 4 \; 7 \; \; 5 \; 8 \; 8 \; 2 \; 3 \; 5 \; \; 9 \; 4 \; 1 \; 1 \dots \\14/17 & = 0.8 \; 2 \; 3 \; 5 \; 2 \; \; 4 \; 1 \; 1 \; 7 \; \; 4 \; 7 \; 0 \; 5 \; 8 \dots \\2/17 & = 0.1 \; 1 \; 7 \; 6 \; 4 \; 7 \; \; 5 \; 8 \; \; 2 \; 3 \; 5 \; 2 \; 9 \; 4 \dots \\10/17 & = 0.5 \; 8 \; 8 \; 2 \; 3 \; 5 \; 2 \; \; \; 1 \; 1 \; 7 \; 6 \; 4 \; 7 \; 0 \dots \\16/17 & = 0.9 \; 4 \; 1 \; 1 \; 7 \; 6 \; 4 \; \; \; 5 \; 8 \; 8 \; 2 \; 3 \; 5 \; 2 \dots \\12/17 & = 0.7 \; 0 \; 5 \; 8 \; 8 \; 2 \; \; 5 \; 2 \; \; 4 \; 1 \; 1 \; 7 \; 6 \; 4 \dots \\9/17 & = 0.5 \; 2 \; 9 \; 4 \; 1 \; \; 7 \; 6 \; 4 \; 7 \; \; 5 \; 8 \; 8 \; 2 \; 3 \dots \\11/17 & = 0.6 \; 4 \; 7 \; 0 \; \; 8 \; 8 \; 2 \; 3 \; 5 \; 2 \; \; 4 \; 1 \; 1 \; 7 \dots \\4/17 & = 0.2 \; 3 \; 5 \; \; 9 \; 4 \; 1 \; 1 \; 7 \; 6 \; 4 \; 7 \; \; 5 \; 8 \; 8 \dots \\3/17 & = 0.1 \; 7 \; \; 4 \; 7 \; 0 \; 5 \; 8 \; 8 \; 2 \; 3 \; 5 \; 2 \; \; 4 \; 1 \dots \\15/17 & = 0.8 \; \; 2 \; 3 \; 5 \; 2 \; 9 \; 4 \; 1 \; 1 \; 7 \; 6 \; 4 \; 7 \; \; 5 \dots \\7/17 & = 0. \; 1 \; 1 \; 7 \; 6 \; 4 \; 7 \; 0 \; 5 \; 8 \; 8 \; 2 \; 3 \; 5 \; 2 \; \dots \\\end$

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
References

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