## Wednesday, June 14, 2017

### Magic anti-Fermat Squares

Magic anti-Fermat Squares

Here is the classic magic square:

2       7       6
9       5       1
4       3       8

Reduce these numbers modulo 7:

2       0       6
2       5       1
4       3       1

These have a magic-square constant of 1 mod 7.
In modulo 7, the fifth power is a one-to-one function:
N                0       1       2       3       4       5       6
N^5            0       1       4       5       2       3       6

N^5, mod 7, swaps 2 with 4, and 3 with 5, so it’s period-two:
(N^5)^5  = N  mod 7
Therefore any sum of fifth powers, mod 7, is a fifth power mod 7:
a^5 + b^5   =   some c^5  for any a and b.
So mod 7 defies Fermat’s Last Theorem!
Take fifth powers, mod 7, of the last magic square:

4       0       6
4       3       1
2       5       1

Sums of fifth powers, mod 7, along rows, columns or diagonals always add to 1^5. This is a magic anti-Fermat square.

The classic magic square, mod 5, is:

2       2       1
4       0       1
4       3       3

N^p mod 5 is one-to-one for any odd p:

N                          0       1       2       3       4
N^(4k+1)            0       1       2       3       4
N^(4k+3)            0       1       3       2       4

4k+1 = 1, 5, 9, 13…;  N to these powers is the identity
4k+3 = 3, 7, 11, 15…;  N to these powers has period 2.

Take the 3rd (or 7th, 11th, etc.) power of the above magic square:

3       3       1
4       0       1
4       2       2

3rd (or 7th, etc.) powers summed mod 5 along rows or columns of this square always sums to 0^3 (=0^7, etc.)

Here is the Chautisa Yantra:

7       12     1       14
2       13     8       11
16     3       10     5
9       6       15     4

This remains the same, mod 17; and in mod 17, _all_ odd powers are invertible by other powers:
N = (N^3)^11 = (N^5)^13 = (N^7)^7 = (N^9)^9 = (N^15)^15  mod 17
Therefore all odd powers of this, mod 17, have constant sums of the inverting powers. I.e. the cube of this square, mod 17, has constant 11th powers mod 17; the 11th power has constant cubes, the 5th power has constant 13th powers, the 13th power has constant 5th powers, the 7th power has constant 7th powers, the 9th power has constant 9th powers, and the 15th power has constant 15th powers.
The constant sum is also for subsquares and wraparound diagonals.

Mod 5 it is:
2       2       1       4
2       3       3       1
1       3       0       0
4       1       0       4
4k+1 powers (= identity) have a constant sum of 4 mod 5.

Its third power mod 5 is:
3       3       1       4
3       2       2       1
1       2       0       0
4       1       0       4
4k+3 powers have a constant sum of 4 mod 5.

Here is Durer’s Jupiter:
4       14     15     1
9       7       6       12
5       11     10     8
16     2       3       13

Mod 5 it is:
4       4       0       1
4       2       1       2
0       1       0       3
1       2       3       3
4k+1 powers (= identity) have a constant sum of 4 mod 5.

Its third power mod 5 is:
4       4       0       1
4       3       1       3
0       1       0       2
1       3       2       2
4k+3 powers have a constant sum of 4 mod 5.

Here is Durer’s Melancholia:
16     3       2       13
5       10     11     8
9       6       7       12
4       15     14     1

Mod 5 it is:
1       3       2       3
0       0       1       3
4       1       2       2
4       0       4       1
4k+1 powers (= identity) have a constant sum of 4 mod 5.
Its third power mod 5 is:
1       2       3       2
0       0       1       2
4       1       3       3
4       0       4       1
4k+3 powers have a constant sum of 4 mod 5.

Here is one by Ramanujan:
22     12     18     87
88     17     9       25
10     24     89     16
19     86     23     11

Mod 17 it is
5       12     1       2
3       0       9       8
10     7       4       16
2       1       6       11
This has constant sums for odd powers as noted above.

Mod 5 it is
2       2       3       2
3       2       4       0
0       4       4       1
4       1       3       1
4k+1 powers have a constant sum of 4 mod 5.
This, cubed mod 5, is
3       3       2       3
2       3       4       0
0       4       4       1
4       1       2       1
4k+3 powers have a constant sum of 4 mod 5.
Here is the one and only Magic Hexagram:

3                 17               18
19               7                 1                 11
16               2                 5                 6                 9
12               4                 8                 14
10               13               15

This has a constant sum of 38 on straight lines in three directions.
Here it is mod 5:

3                 2                 3

4                 2                 1                 1

1                 2                 0                 1                 4

2                 4                 3                 4

0                 3                 0

Here is that cubed mod 5:

2                 3                 2

4                 3                 1                 1

1                 3                 0                 1                 4

3                 4                 2                 4

0                 2                 0
Here is a 6 by 6:

35     1       6       26     19     24
3       32     7       21     23     25
31     9       2       22     27     20
8       28     33     17     10     15
30     5       34     12     14     16
4       36     29     13     18     11

Here it is mod 17:
1       1       6       9       2       7
3       15     7       4       6       8
14     9       2       5       10     3
8       11     16     0       10     15
13     5       0       12     14     16
4       2       12     13     1       11

As noted above, the cube of this magic square, mod 17, has constant 11th powers mod 17; its 11th power has constant cubes, its 5th power has constant 13th powers, its 13th power has constant 5th powers, its 7th power has constant 7th powers, its 9th power has constant 9th powers, and its 15th power has constant 15th powers.

For instance here is the square’s 7th power:
1       1       14     2       9       12
11     8       12     13     14     15
6       2       9       10     5       11
15     3       16     0       5       8
4       10     0       7       6       16
13     9       7       4       1       3
Sums of 7th powers along rows, columns and diagonals are constant at 9.