Prime reciprocal magic square

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

$$0.\backslash bold\{1\}42857\backslash bold\{1\}42857\backslash bold\{1\}\backslash dots$$

Consequently, multiples of one-seventh exhibit cyclic permutations of these six digits:{{Cite book |last=Wells |first= D. |title=The Penguin Dictionary of Curious and Interesting Numbers |url=https://archive.org/details/penguindictionar0000well_f3y1/mode/2up |url-access=registration |publisher=Penguin Books |location=London |year=1987 |pages=171–174 |isbn=0-14-008029-5 |oclc=39262447 |s2cid=118329153 }}

\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 {{math|1=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 {{val|27}}; however, their mean is {{val|27}}, as one diagonal adds to {{val|23}} while the other adds to {{val|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.

In a full, or otherwise prime reciprocal magic square with $p\; -\; 1$ period, the even number of {{mvar|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 {{mvar|p}} that is divided into {{mvar|n}}−digit strings creates pairs of complementary sequences of digits that yield strings of nines ({{val|9}}) 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.{{Cite book |last1=Rademacher |first1=Hans |author1-link=Hans Rademacher |last2=Toeplitz |first2=Otto |author2-link=Otto Toeplitz |title=The Enjoyment of Mathematics: Selections from Mathematics for the Amateur. |url=https://archive.org/details/enjoymentofmathe0000rade/page/160/mode/2up|url-access=registration |publisher=Princeton University Press |edition=2nd |location= Princeton, NJ |year=1957 |pages=158–160 |isbn=9780486262420 |oclc=20827693 |mr=0081844 |zbl=0078.00114 }}{{Cite journal |last=Leavitt |first=William G. |title=A Theorem on Repeating Decimals |url=http://digitalcommons.unl.edu/mathfacpub/48/ |journal=The American Mathematical Monthly |volume=74 |issue=6 |pages=669–673 |year=1967 |publisher=Mathematical Association of America |location=Washington, D.C. |doi=10.2307/2314251 |jstor=2314251 |mr=0211949 |zbl=0153.06503 }} These complementary sequences are generated between multiples of prime reciprocals that add to 1.

More specifically, a factor {{mvar|n}} in the numerator of the reciprocal of a prime number {{mvar|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 {{val|2}} moves the repeating decimal of {{sfrac|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 squares based on reciprocals of primes {{mvar|p}} in bases {{mvar|b}} with periods $p\; -\; 1$ have magic sums equal to,{{cn|date=January 2024}}

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

The $\backslash bold\{\backslash tfrac\; \{1\}\{19\}\}$ 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:{{Cite book|last=Andrews |first=William Symes |title=Magic Squares and Cubes |url=http://djm.cc/library/Magic_Squares_Cubes_Andrews_edited.pdf |publisher=Open Court Publishing Company |location=Chicago, IL |year=1917 |pages=176, 177 |isbn=9780486206585 |oclc=1136401 |zbl=1003.05500 |mr=0114763 }}{{Cite OEIS |A021023 |Decimal expansion of 1/19. |access-date=2023-11-21 }}

\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{{Cite journal |editor-last=Singleton |editor-first=Colin R.J. |title=Solutions to Problems and Conjectures |url=https://www.tib.eu/en/search/id/olc:OLC1606837575/Solutions-to-Problems-and-Conjectures?cHash=e69a0e2935ea6071c21e685db86a7d91 |journal=Journal of Recreational Mathematics |volume=30 |issue=2 |publisher=Baywood Publishing & Co. |location=Amityville, NY |year=1999 |pages=158–160 }}

:"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, ...} {{OEIS|A072359}}.

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 $\backslash tfrac\; \{1\}\{17\}$ 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:{{Cite journal |last=Subramani |first=K. |title=On two interesting properties of primes, p, with reciprocals in base 10 having maximum period p – 1. |url=https://jmscm.smartsociety.org/volume1_issue2/Paper4.pdf |journal=J. Of Math. Sci. & Comp. Math. |eissn=2644-3368 |volume=1 |issue=2 |year=2020 |pages=198–200 |publisher=S.M.A.R.T. |location=Auburn, WA |doi=10.15864/jmscm.1204 |s2cid=235037714 }}{{Cite OEIS |A007450 |Decimal expansion of 1/17. |access-date=2023-11-24 }}

\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.

• Cyclic number
• Reciprocals of primes

{{Reflist}}

{{Magic polygons|collapse}}

{{DEFAULTSORT:Prime Reciprocal Magic Square}}

Category:Magic squares