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{{Short description|Type of magic square}}
A '''prime reciprocal magic square''' is a [[magic square]] using the decimal digits of the [[Reciprocal (mathematics)|reciprocal]] of a [[prime number]].
== Formulation ==
=== Basics ===
In [[decimal]], [[unit fraction]]s {{sfrac|1|2}} and {{sfrac|1|5}} have no [[repeating decimal]], while {{sfrac|1|3}} repeats <math>0.3333\dots</math> indefinitely. The remainder of {{sfrac|1|7}}, on the other hand, repeats over six digits as,
<math display=block>0.\bold{1}42857\bold{1}42857\bold{1}\dots</math>
Consequently, multiples of one-seventh exhibit [[cyclic permutation]]s of these six digits:<ref name="Wells">{{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 }}</ref>
<math display=block>
\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}</math>
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]]
{| | class=wikitable style="text-align: center;width:12em;height:12em;table-layout:fixed"
|-
| {{val|1}} || {{val|4}} || {{val|2}} || {{val|8}} || {{val|5}} || {{val|7}}
|-
| {{val|2}} || {{val|8}} || {{val|5}} || {{val|7}} || {{val|1}} || {{val|4}}
|-
| {{val|4}} || {{val|2}} || {{val|8}} || {{val|5}} || {{val|7}} || {{val|1}}
|-
| {{val|5}} || {{val|7}} || {{val|1}} || {{val|4}} || {{val|2}} || {{val|8}}
|-
| {{val|7}} || {{val|1}} || {{val|4}} || {{val|2}} || {{val|8}} || {{val|5}}
|-
| {{val|8}} || {{val|5}} || {{val|7}} || {{val|1}} || {{val|4}} || {{val|2}}
|}
In contrast with its rows and columns, the ''diagonals'' of this square do not sum to {{val|27}}; however, their [[Arithmetic mean|mean]] is {{val|27}}, as one diagonal adds to {{val|23}} while the other adds to {{val|31}}.
All prime reciprocals in any [[Radix|base]] with a <math>p - 1</math> 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 <math>p - 1</math> period, the even number of {{mvar|k}}−th rows in the square are arranged by multiples of <math>1/p</math> — not necessarily successively — where a magic constant can be obtained.
For instance, an [[Parity (mathematics)|even]] repeating [[Cyclic number|cycle]] from an odd, prime reciprocal of {{mvar|p}} that is divided into {{mvar|n}}−digit strings creates pairs of [[Method of complements#Numeric complements|complementary sequences]] of digits that yield strings of nines ({{val|9}}) when added together:
<math display=block>
\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}</math>
This is a result of [[Midy's theorem]].<ref>{{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 }}</ref><ref>{{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 }}</ref> These complementary sequences are generated between multiples of [[Reciprocals of primes|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 place]]s of its decimal expansion accordingly,
<math display=block>
\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}</math>
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 <math>1/p</math>. Other magic squares can be constructed whose rows do not represent consecutive multiples of <math>1/p</math>, which nonetheless generate a magic sum.
== Magic constant ==
{| class="wikitable floatright" style="text-align: right;"
|+ 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 || {{val|1719|fmt=commas}}
|-
| 397 || 5 || 792
|-
| 487 || 6 || {{val|1215|fmt=commas}}
|-
| 593 || 3 || 592
|-
| 631 || 87 || {{val|27090|fmt=commas}}
|-
| 787 || 13 || {{val|4716|fmt=commas}}
|-
| 811 || 3 || 810
|-
| {{val|1033|fmt=commas}} || 11 || {{val|5160|fmt=commas}}
|-
| {{val|1307|fmt=commas}} || 5 || {{val|2612|fmt=commas}}
|-
| {{val|1499|fmt=commas}} || 11 || {{val|7490|fmt=commas}}
|-
| {{val|1877|fmt=commas}} || 19 || {{val|16884|fmt=commas}}
|-
| {{val|2011|fmt=commas}} || 26 || {{val|25125|fmt=commas}}
|-
| {{val|2027|fmt=commas}} || 2 || {{val|1013|fmt=commas}}
|}
Magic squares based on reciprocals of primes {{mvar|p}} in bases {{mvar|b}} with periods <math>p - 1</math> have [[magic sum]]s equal to,{{cn|date=January 2024}}
<math display=block>M = (b-1) \times \frac {p-1}{2}.</math>
== Full magic squares ==
The <math>\bold{\tfrac {1}{19}}</math> 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 <math>k</math>−th rows:<ref>{{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 }}</ref><ref>{{Cite OEIS |A021023 |Decimal expansion of 1/19. |access-date=2023-11-21 }}</ref>
<math display=block>
\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}</math>
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<ref>{{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 }}<br/>
:"Fourteen primes less than 1000000 possess this required property <nowiki>[</nowiki>in decimal<nowiki>]</nowiki>".<br />
:Solution to problem 2420, "Only 19?" by M. J. Zerger.</ref>
:{19, 383, 32327, 34061, 45341, 61967, 65699, 117541, 158771, 405817, ...} {{OEIS|A072359}}.
The smallest prime number to yield such magic square in [[Binary number|binary]] is [[59 (number)|59]] (111011<sub>2</sub>), while in [[Ternary numeral system|ternary]] it is [[223 (number)|223]] (22021<sub>3</sub>); these are listed at [[OEIS:A096339|A096339]], and [[OEIS:A096660|A096660]].
=== Variations ===
A <math>\tfrac {1}{17}</math> 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:<ref>{{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 }}</ref><ref>{{Cite OEIS |A007450 |Decimal expansion of 1/17. |access-date=2023-11-24 }}</ref>
<math display=block>
\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}</math>
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 <math>1/p</math> fit in respective <math>k</math>−th rows.
== See also ==
*[[Cyclic number]]
*[[Reciprocals of primes]]
== References ==
{{Reflist}}
{{Magic polygons|collapse}}
{{DEFAULTSORT:Prime Reciprocal Magic Square}}
[[Category:Magic squares]]
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