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.
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