A prime reciprocal magic square is a magic square using the decimal digits of the reciprocal of a prime number.
Introduction Consider a unit fraction, like 1/3 or 1/7. In base ten, the remainder, and so the digits, of 1/3 repeats at once: 0.3333.... However, the remainders of 1/7 repeat over six, or 7−1, digits: 1/7 = 0·142857142857142857... Examining the multiples of 1/7, each is a cyclic permutation of these six digits:
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 ; however, their mean is , as one diagonal adds to while the other adds to .
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.
Decimal expansions In a full, or otherwise prime reciprocal magic square with period, the even number of −th rows in the square are arranged by multiples of — not necessarily consecutively — where a magic constant can be obtained.
For instance, an even repeating cycle from an odd, prime reciprocal of that is divided into −digit strings creates pairs of complementary sequences of digits that yield strings of nines () when added together:
This is a result of Midy's theorem. These complementary sequences are generated between multiples of prime reciprocals that add to 1.
More specifically, a factor 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 moves the repeating decimal of by eight places.
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 constant + 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 10 397 5 792 487 6 593 3 592 631 87 787 13 811 3 810 11 5 11 19 26 2 Magic squares based on reciprocals of primes in bases with periods have magic sums equal to
Full magic squares The 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 −th rows:
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"Fourteen primes less than 1000000 possess this required property [in decimal]". Solution to problem 2420, "Only 19?" by M. J. Zerger.
{19, 383, 32327, 34061, 45341, 61967, 65699, 117541, 158771, 405817, ...} .
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 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:
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 fit in respective −th rows.
See also Cyclic number Reciprocals of primes
References