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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

In decimal, unit fractions ${\tfrac {1}{2}}$ and ${\tfrac {1}{5}}$ have no repeating decimal, while ${\tfrac {1}{3}}$ repeats $0.3333\dots$ indefinitely. The remainder of ${\tfrac {1}{7}}$ , on the other hand, repeats over six digits as $0.{\mathbf {1}}42857{\mathbf {1}}42857{\mathbf {1}}\dots$

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

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

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

$1$	$4$	$2$	$8$	$5$	$7$
$2$	$8$	$5$	$7$	$1$	$4$
$4$	$2$	$8$	$5$	$7$	$1$
$5$	$7$	$1$	$4$	$2$	$8$
$7$	$1$	$4$	$2$	$8$	$5$
$8$	$5$	$7$	$1$	$4$	$2$

In contrast with its rows and columns, the diagonals of this square do not sum to 27.

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.

Midy's theorem

An even repeating cycle from an odd, prime reciprocal that is divided into $k$ −digit strings creates pairs of complementary sequences of digits that yield strings of 9s when added:

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

This is a result of Midy's theorem.^[2]^[3] These complementary sequences are generated between multiples of prime reciprocals that add to 1.

A factor $n$ in the numerator of the reciprocal of a prime number $p$ will shift the decimal places of its decimal expansion accordingly,

${\begin{aligned}1/23&=0.04347826\;08695652\;173913\ldots \\2/23&=0.08695652\;17391304\;347826\ldots \\4/23&=0.17391304\;34782608\;695652\ldots \\8/23&=0.34782608\;69565217\;391304\ldots \\16/23&=0.69565217\;39130434\;782608\ldots \\\end{aligned}}$

In this case, a factor of 2 moves the repeating decimal of ${\tfrac {1}{23}}$ by eight places.

Magic constant

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

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

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 ${\mathbf {\tfrac {1}{19}}}$ 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:^[4]^[5]

${\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 are^[6]

{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, respectively.

References

^ 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.
^ 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.
^ 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.
^ 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.
^ 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.
^ 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 [in decimal]".

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

[Wells-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 [in decimal]".

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

[1]

[2]

[3]

[4]

[5]

[6]

@@ Line 64: / Line 64: @@
 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 prime reciprocals that add to 1.
-A factor <math>m</math> in the numerator of the reciprocal of a prime number <math>n</math> will shift the [[decimal place]]s of its decimal expansion accordingly,
+A factor <math>n</math> in the numerator of the reciprocal of a prime number <math>p</math> will shift the [[decimal place]]s of its decimal expansion accordingly,
 <math display=block>

v t e Magic polygons
Types	Magic circle Magic hexagon Magic hexagram Magic square Magic star Magic triangle
Related shapes	Alphamagic square Antimagic square Geomagic square Heterosquare Pandiagonal magic square Prime reciprocal magic square Most-perfect magic square
Higher dimensional shapes	Magic cube classes Magic hypercube Magic hyperbeam
Classification	Associative magic square Pandiagonal magic square Multimagic square
Related concepts	Latin square Word square Number Scrabble Eight queens puzzle Magic constant Magic graph Magic series