Prime reciprocal magic square
A prime reciprocal magic square is a magic square using the decimal digits of the reciprocal of a prime number.
FormulationEdit
BasicsEdit
In decimal, unit fractions Template:Sfrac and Template:Sfrac have no repeating decimal, while Template:Sfrac repeats <math>0.3333\dots</math> indefinitely. The remainder of Template:Sfrac, 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 permutations of these six digits:<ref name="Wells">Template:Cite book</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 Template:Math 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 Template:Val; however, their mean is Template:Val, as one diagonal adds to Template:Val while the other adds to Template:Val.
All prime reciprocals in any 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 expansionsEdit
In a full, or otherwise prime reciprocal magic square with <math>p - 1</math> period, the even number of Template:Mvar−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 even repeating cycle from an odd, prime reciprocal of Template:Mvar that is divided into Template:Mvar−digit strings creates pairs of complementary sequences of digits that yield strings of nines (Template:Val) 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>Template:Cite book</ref><ref>Template:Cite journal</ref> These complementary sequences are generated between multiples of prime reciprocals that add to 1.
More specifically, a factor Template:Mvar in the numerator of the reciprocal of a prime number Template:Mvar will shift the decimal places 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 Template:Val moves the repeating decimal of Template:Sfrac 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 constantEdit
| 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 | Template:Val |
| 397 | 5 | 792 |
| 487 | 6 | Template:Val |
| 593 | 3 | 592 |
| 631 | 87 | Template:Val |
| 787 | 13 | Template:Val |
| 811 | 3 | 810 |
| Template:Val | 11 | Template:Val |
| Template:Val | 5 | Template:Val |
| Template:Val | 11 | Template:Val |
| Template:Val | 19 | Template:Val |
| Template:Val | 26 | Template:Val |
| Template:Val | 2 | Template:Val |
Magic squares based on reciprocals of primes Template:Mvar in bases Template:Mvar with periods <math>p - 1</math> have magic sums equal to,Template:Cn
<math display=block>M = (b-1) \times \frac {p-1}{2}.</math>
Full magic squaresEdit
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>Template:Cite book</ref><ref>Template:Cite OEIS</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>Template:Cite journal
- "Fourteen primes less than 1000000 possess this required property [in decimal]".
- Solution to problem 2420, "Only 19?" by M. J. Zerger.</ref>
- {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.
VariationsEdit
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>Template:Cite journal</ref><ref>Template:Cite OEIS</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.