Prime reciprocal magic square

A prime reciprocal magic square is a

.

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,

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

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:

This is a result of Midy's theorem.^[2][3] 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

of its decimal expansion accordingly,

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

]

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

The table below lists some prime numbers that generate prime-reciprocal magic squares in given bases.^[citation needed]

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

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^[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.

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:^[7][8]

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