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If
is the order of a most-perfect magic square then
. Assuming the prime decomposition of
is
where
for
and
Let
be the the sum of the exponents:
.
Let
be the canonical base for the representation of the integer numbers from
to
in a positional numeral system based on a mixture of prime numbers. It will contain
-tuples:
is the weight of the canonical base
. Both define together the positional numeral system.
The algorithm to find the representation of an integer
between
and
is as follows:
- initialisation
![{\displaystyle k:=1\,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/c1fe734cfc7110967ef48426c34acfdc4f886c1a)
![{\displaystyle a:=a-1\,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/ab3a4aa4114f9281464afe684eba72458f15b9d9)
![{\displaystyle c:=1\,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/c1d0f7615c38f08ee09857020aa881f8e490a7f6)
computations
, where
is the first value of the
-th 2-tuple in ![{\displaystyle B\,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/d8a72cbbfdbb8b9d0dad053538c330994b308bae)
![{\displaystyle a:={\frac {\left(a-b_{k}\right)}{p_{k}}}\,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/90299d3c9e2a08c63903fe84d63d97649f9d4639)
, this is the computation of ![{\displaystyle C\,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/785a192e3331793e37b1be0c5315d196da1a7049)
![{\displaystyle c:=c\cdot p_{k}\,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/69290fa2336ec15938db076de3bdd60be6996cbb)
![{\displaystyle k:=k+1\,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/d8294e11ce218171cf9ec723f641f857939502f0)
Any number
between
and
can now be written as
where
and
∈
In the following the
-tuple formed by the values
is called coefficients in the positional numeral system defined by
and
.
Constructing auxiliary squares
[edit]
Let
be the set of the integer numbers from
and
. It is possible to choose a random
-tuple named
as follows:
- initialisation
![{\displaystyle k:=1\,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/c1fe734cfc7110967ef48426c34acfdc4f886c1a)
![{\displaystyle S:=S\,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/978b9d19a2f94640056e639e4931ed6df72f189f)
![{\displaystyle c:=1\,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/c1d0f7615c38f08ee09857020aa881f8e490a7f6)
computations
- a random element
of
is assigned to the the
-th element of ![{\displaystyle S_{r}\,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/8ce9ec5259c02fa72b669e816e9bc6dab216d736)
![{\displaystyle S:=S\setminus \{n_{k}\}\,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/e2e38af5f487d6c90ed90d76d033ac425fe65a07)
![{\displaystyle S:=S\setminus \{n+1-n_{k}\}\,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/8898585476b785796d50641978b3aa52af3029dd)
![{\displaystyle k:=k+1\,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/d8294e11ce218171cf9ec723f641f857939502f0)
Now it is possible to decompose each element of the
-tuple
in the positional numeral system defined by
and
calculating the coefficients for each element. Regrouping the appropriate coefficients it is possible to compute
different
-tuples
where
relates to the
-th occurence of the prime number
.
Each
-tuple
is used to construct an auxiliary square
of order
as follows:
- the rows of
with an odd number consist of the
-tuples ![{\displaystyle S_{r,k}\,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/6e2ef0a371120dc59d746094d4f8198610d10b7d)
- the rows of
with an even number consist of the prime number
-complements of
-tuples
- for any value
and
∈
its complement is ![{\displaystyle p_{k^{'}}-1-b_{k^{''}}\,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/db1626bfc9b30465af9ecf44069189a14f54dca4)
All steps from above can be repeated for columns. This is how a random
-tuple named
is choosen which will drive the computation of
different
-tuples
where
relates to the
-th occurence of the prime number
.
Finaly
different auxiliary squares
of order
are generated as follows:
- the columns of
with an odd number consist of the transpose of the
-tuples ![{\displaystyle S_{c,k}\,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/703598d7daa5b337c6bd30ce2e5bac0b29a4394e)
- the columns of
with an even number consist of the prime number
-complements of the transpose of the
-tuples ![{\displaystyle S_{c,k}\,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/703598d7daa5b337c6bd30ce2e5bac0b29a4394e)
The set of auxiliary squares
has many interesting properties:
- the sum of any 2×2 subsquare is constant inside a particular square
- the sum of two cells with a distance of n/2 along a (major) diagonal is constant inside a particular square
- choosing an ordered subset of
auxiliary squares one can generate
(one for each element)
-tuples ; counting all different kind of
-tuples show an equal distribution of these tuples
Parameterizing of the auxiliary squares
[edit]
Magic squares of order
contain all values from
to
.
where
for
and
In order to represent the integer numbers from
to
in a positional numeral system any base similar to the ones already used above needs to have
tuples. Below a noncanocal base is presented::
Let
be a permutation of the
-tuples of
where for two consecutive
-tuples
and ![{\displaystyle \left(p_{i+1},x_{i+1}\right)}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/09807a15c524e5b3465039077549d3072b9a98aa)
implies
Let
be the set of all permutations
of
as described above. The question is what is the number of elements of
. The requested number can be calculated with a function named
which depends only on the number and values of the prime decomposition of
; i.e.
. Examples of its calculation are given here.
Description about the calculation of
will follow. One can see that
if
From the construction one can see that this method allows the construction of
different squares. See matching A051235.
If
where
the function
simplifies to
and the number of most-perfect magic squares of order is
( partialy matching A151932 )
To be continued.