A prime reciprocal magic square is a magic square using the decimal digits of the reciprocal of a prime number.
In decimal, unit fractions
have no repeating decimal, while
, 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
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 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
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
−th rows in the square are arranged by multiples of
— not necessarily successively — where a magic constant can be obtained.
For instance, an even repeating cycle from an odd, prime reciprocal of
−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.
in the numerator of the reciprocal of a prime number
will shift the decimal places of its decimal expansion accordingly,
In this case, a factor of 2 moves the repeating decimal of
A uniform solution of a prime reciprocal magic square, whether full or not, will hold rows with successive multiples of
Other magic squares can be constructed whose rows do not represent consecutive multiples of
, which nonetheless generate a magic sum.
Magic squares based on reciprocals of primes
have magic sums equal to,[citation needed]
The table below lists some prime numbers that generate prime-reciprocal magic squares in given bases.
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
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] 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.
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