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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 ${\displaystyle {\tfrac {1}{2}}}$ and ${\displaystyle {\tfrac {1}{5}}}$ have no repeating decimal, while ${\displaystyle {\tfrac {1}{3}}}$ repeats ${\displaystyle 0.3333\dots }$ indefinitely. The remainder of ${\displaystyle {\tfrac {1}{7}}}$, 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:

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

If the digits are laid out as a square, each row and column sums to ${\displaystyle 1+4+2+8+5+7=27.}$ This yields the smallest base-10 non-normal, prime reciprocal magic square

${\displaystyle 1}$	${\displaystyle 4}$	${\displaystyle 2}$	${\displaystyle 8}$	${\displaystyle 5}$	${\displaystyle 7}$
${\displaystyle 2}$	${\displaystyle 8}$	${\displaystyle 5}$	${\displaystyle 7}$	${\displaystyle 1}$	${\displaystyle 4}$
${\displaystyle 4}$	${\displaystyle 2}$	${\displaystyle 8}$	${\displaystyle 5}$	${\displaystyle 7}$	${\displaystyle 1}$
${\displaystyle 5}$	${\displaystyle 7}$	${\displaystyle 1}$	${\displaystyle 4}$	${\displaystyle 2}$	${\displaystyle 8}$
${\displaystyle 7}$	${\displaystyle 1}$	${\displaystyle 4}$	${\displaystyle 2}$	${\displaystyle 8}$	${\displaystyle 5}$
${\displaystyle 8}$	${\displaystyle 5}$	${\displaystyle 7}$	${\displaystyle 1}$	${\displaystyle 4}$	${\displaystyle 2}$

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 ${\displaystyle p-1}$ 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.

Decimal expansions

In a full, or otherwise prime reciprocal magic square with ${\displaystyle p-1}$ period, the even number of ${\displaystyle k}$−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 ${\displaystyle p}$ that is divided into ${\displaystyle n}$−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. These complementary sequences are generated between multiples of prime reciprocals that add to 1.

More specifically, a factor ${\displaystyle n}$ in the numerator of the reciprocal of a prime number ${\displaystyle p}$ 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 ${\displaystyle {\tfrac {1}{23}}}$ by eight places.

A uniform solution of a prime reciprocal magic square, whether full or not, will hold rows with successive multiples of ${\displaystyle 1/p}$. Other magic squares can be constructed whose rows do not represent consecutive multiples of ${\displaystyle 1/p}$, which nonetheless generate a magic sum.

Magic constant

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

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

The table below lists some prime numbers that generate prime-reciprocal magic squares in given bases.

Full magic squares

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:

$${\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

{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 ${\displaystyle {\tfrac {1}{17}}}$ 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:

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

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.

See also

• Cyclic number
• Reciprocals of primes

References

1. Wells, D. (1987). The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin Books. pp. 171–174. ISBN 0-14-008029-5. OCLC 39262447. S2CID 118329153.
2. Rademacher, Hans; Toeplitz, Otto (1957). The Enjoyment of Mathematics: Selections from Mathematics for the Amateur (2nd ed.). Princeton, NJ: Princeton University Press. pp. 158–160. ISBN 9780486262420. MR 0081844. OCLC 20827693. Zbl 0078.00114.
3. Leavitt, William G. (1967). "A Theorem on Repeating Decimals". The American Mathematical Monthly. 74 (6). Washington, D.C.: Mathematical Association of America: 669–673. doi:10.2307/2314251. JSTOR 2314251. MR 0211949. Zbl 0153.06503.
4. Andrews, William Symes (1917). Magic Squares and Cubes (PDF). Chicago, IL: Open Court Publishing Company. pp. 176, 177. ISBN 9780486206585. MR 0114763. OCLC 1136401. Zbl 1003.05500.
5. Sloane, N. J. A. (ed.). "Sequence A021023 (Decimal expansion of 1/19.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-21.
6. Singleton, Colin R.J., ed. (1999). "Solutions to Problems and Conjectures". Journal of Recreational Mathematics. 30 (2). Amityville, NY: Baywood Publishing & Co.: 158–160.
   

   "Fourteen primes less than 1000000 possess this required property ".
   

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

7. Subramani, K. (2020). "On two interesting properties of primes, p, with reciprocals in base 10 having maximum period p – 1" (PDF). J. Of Math. Sci. & Comp. Math. 1 (2). Auburn, WA: S.M.A.R.T.: 198–200. doi:10.15864/jmscm.1204. eISSN 2644-3368. S2CID 235037714.
8. Sloane, N. J. A. (ed.). "Sequence A007450 (Decimal expansion of 1/17.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-11-24.

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Related shapes	Alphamagic square Antimagic square Geomagic square Heterosquare Pandiagonal magic square Prime reciprocal magic square Most-perfect magic square
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Classification	Associative magic square Pandiagonal magic square Multimagic square
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