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.
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 successively — 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 (9) when added together:
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 2 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
Magic squares based on reciprocals of primes in bases with periods have
magic sums equal to,[citation needed]
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 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:[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 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 fit in respective −th rows.
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.
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 successively — 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 (9) when added together:
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 2 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
Magic squares based on reciprocals of primes in bases with periods have
magic sums equal to,[citation needed]
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 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:[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 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 fit in respective −th rows.