In the previous chapter we have passed general properties of Rsystems, especially:
simple Rsystems (r ε P) and composite Rsystems (composite r)I this chapter and some next chapters we will concentrate on special cases.
In this chapter:
Gsystems (r=n^{k}−1) binary Gsystems (r=n^{k}−1, n=2) Msystems (r = (n^{k}−1)/(n−1)) Fsystems (r = n^{h}+1) Ksystems (r = (n^{k}+1)/(n+1))
systems of complex numbers (u ε K) systems of algebraic numbers (u ε A) sign Ssystems (+/− or 0)In the following chapters:
systems of power residues (k=κ(r,v)) systems of primitive roots (k=r−1) Legendre’s systems (k=κ(r,v) where u=v for u in the first row of the system) saturated systems (n=k)  Wieferich ’s systems (r=t²)The area we are studying here is in the combinatorial theory known as Combinatorics of necklaces.
Let us think about arranging n elements into k cells. Let A be a set with n members (elements) with ordinal numbers given by the set E(A) = {0, 1, .., n1}. Let B be a set with k members (cells) and ξ relation from E(B) to E(A).
000 001 010 100 011 110 101 111
ξ (cells) E(B) ─────────────> E(A) (elements)
If numbers E(B) are positions of numbers (digits) from E(A), then written in nnumber system we obtain n^{k} possible numbers.
Let us mark the set of all these numbers I(n,k), members of this set are so called instances. Number n is called base and number k order of the set.
E.g. the sets A={p, q}, B={x, y, z} have their ordinal numbers given by E(A)={0, 1}, E(B)={0, 1, 2}.
Boole’s algebraValues of functions for logical sum (or) and logical product (and) of three variables can be arranged e.g. into the following schema (Y=yes,N=no):
Boole GeorgeBoole George Irish mathematician, creator of the first algebra of mathematical logic. He was engaged also by quaternions and probability theory. His named get the used system of logic and in computer terminology also type of variables that can have just two values (True, False). 
Given values and or ──────────────────────── NNN N N NNA NAN ANN N A NAA AAN ANA N A AAA A A
Let relation G, defined as follows, be called Grelation.
Relation G is equivalence relation, so it will breakup instances u_{i} (j) into classes I(n,k)/G.
Such a set of instances we name Gsystem and mark G(n,k). Because number r = n^{k}−1, total number of instances is r+1 = n^{k},
where n is base and k is order of the Gsystem. Every class has its representative marked by g_{i}., u(i,j) are instance numbers.
Every G(n,k) has m(n,k) classes marked g(i), i=1..m.
Instances of given class are called transpositions. System G(2,3) has 2³=8 instances I(2,3)={000, 001,.., 111} in m=4 classes. In this example (like in the example of Boole’s algebra), the instances are distinguished according to levels only, not according to classes.
Geometrical notionLet M be a set of 3 elements {A,B,C}.
Cube 000 {} 001 010 100 {A} {B} {C} 011 110 101 {A,B} {B,C} {A,C} 111 {A,B,C}
Instances of the system G(2,3) make subsets of the given set of M. Instances G(2,3) represent coordinates of apexes in 3 dimensional cube. Each instance of G(2,3) represents one apex, level instance evaluates distance of the coordinate origin.
In crystallography so called Miller’s indexes (h,k,l) are used.
System G(2,3) represents Miller’s indexes of the octahedron.
In the case of 12tone system, there is n=2, k=12 and
structures make G(2,12). When A={exists, notexists} and B={c, c#,
d, d#, e, f, f#, g, g#, a, a#, b},
then set of ordinal numbers E(A)={0,1}, E(B)={0, 1, .., 11}
and relation ξ makes a set I(2,12).
Diminished seventh chord (c,d#,f#,a) is class of
I(2,12) having 3 transpositions:
0/ 001001001001 [c, d#, f#, a] 1/ 010010010010 [c#, e, g, a#] 2/ 100100100100 [d, f, g#, h]
Particular instances of Gsystem arise as variations of class k from n elements, there is total n^{k } variations.
Among instances all the possible permutation with repetition of selected elements are contained.
E.g. in G(3,5) there is 30 distinct instances with two zeros, two ones and one number two, i.e.. 00112, 01120, 11200, 12001,...
Number of such permutations with repetition is 5!/(2!2!1!) = 30 and because k=5 instance fill just 30/5 = 6 classes.
In binary systems, i.e. systems where n=2, is number of permutation with repetition equal to number of combinations.
E.g. in G(2,5):
1∙
00000
5!/5!0! = 1 =
_{
}
5∙
00001
5!/4!1! = 5 =
_{
}
5∙ 00011 + 5∙ 00101 5!/3!2!
=10 =
_{
}
5∙ 00111 + 5∙ 01011 5!/2!3!
=10 =
_{
}
5∙
01111
5!/1!4! = 5 =
_{
}
1∙
11111
5!/0!5! = 1 =
_{
}
We will distinguish three types of schemas: basic, contrast
and symmetric schema:
Basic schema Contrast schema Symmetric schema ────────────── ──────────────── ───────────────── 0 15 0 1 2 4 8 14 13 11 7 1 2 4 −7 3 6 12 9 12 9 3 6 3 6 −3 −6 5 10 10 5 5 −5 7 14 13 11 8 1 2 4 7 −1 −2 −4 15 0 0
In some cases, some schema is more convenient then the others.
DeformationFor r=15 field Z_{15} is projection (homomorfism) of field Z.
Integer Set of instances and classes ──────────────────────────────────────── 0 0 1 2 4 8 1 Z ────> 3 6 12 9 ────> 3 5 10 5 7 14 13 11 7 15 15
Elements of the field Z_{15} makes e.g. the instances of Gsystem G(2,4).
Numbers of classes in G(2,4) are certain deformations (morfismus) of instance numbers.
Sum of numbers (digits) from instances of a given class g in G(n,k) we call level L(g).
E.g. in systems G(2,k) is level given by number of ones in any of its instances.
In our case classes of G(2,4) have the following levels:
i/ Druh gi Binary schemas Level L(gi) ──────────────────────────────────────────────── 1/ 0 0000 0 2/ 1 0001 0010 0100 1000 1 3/ 3 0011 0110 1100 1001 2 4/ 5 0101 1010 2 5/ 7 0111 1110 1101 1011 3 6/ 15 1111 4
Algorithm for list of numbers of representative classes in Gsystem is similar to Eratosthenes sieve (see Divisibility of numbers).
In our case we need to select numbers of representative classes from numbers of all instances.
Transpositions of instances in Gsystems are analogous to prime multiples in Eratosthenes sieve.
E.g. G(2,4) has base n=2, order k=4 and module r =
n^{k}−1 = 2^{4}−1=15.
We will cross out numbers g∙n^{j} (mod r), j=1,2,...,
always with the smallest unused number g.
We will be interested in crossed numbers, we will write them into the schemas (on the right).
In this way we will step by step cross out all the given numbers (this is a next difference from Eratosthenes' sieve).
Given numbers: 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15
Used numbers Selected (crossed out) numbers I/ 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 0 II/ 3,5,6,7,9,10,11,12,13,14,15 1 2 4 8 III/ 5,7,10,11,13,14,15 3 6 12 9 IV/ 7,11,13,14,15 5 10 V/ 15 7 14 13 11 15
Let us mark the set of all crossed numbers X. The
following block diagram describes algorithm for generation of
Gsystems:
empty set X ↓ ++ for g=0 to (n^{(k1)}1)  write the last class g=n^{k}1     ↑ is class g in X ?     + YES+  NO    write g to X  ↓  assign u < g   +> while not u in X +     write u to X      transpose instance   u < n * u  +<+  +<+
This algorithm has the following disadvantage for large
systems. E.g. 2 MB (megabytes) of memory space are needed
for registration of crossed numbers of system G(2,24) (1
cross=1 bit, 1 byte = 2³ bits, 1 megabyte=2^{20}
byte).
Let us define gnumber of a class as minimum of all instance
numbers (all transpositions) of the given class:
g = min(u0,u1,u2,..,uk−1) = min(u, u∙n mod r,
u∙n² mod r,...u∙n^{k}−1 mod r)
For any number u we can determine number g  thereof algorithm:
│ ┌───> for u=0 to (n^{k−1}−1) ──> write the last class g=n^{k}−1 │ │ │ determine g <− min(u,u∙n mod r,...) │ │ │ is g = u ──> write instance of class g │ │ └──────<──────┴──────<──────┼
The second algorithm is timeconsuming. To speed it up, some heuristics are needed  e.g. in case of binary systems g−number of class cannot be even,….
Examples of the output:
G(n,1)G(2,1) G(3,1) G(4,1) 0 0 0 1 1 1 2 2 3
G(2,2) G(3,2) G(4,2) 0 0 0 6 9 1 2 1 3 1 4 7 13 3 2 6 2 8 10 4 3 12 11 14 5 7 5 15 8
G(2,3) G(3,3) G(4,3) 0 0 0 21 1 2 4 1 3 9 1 4 16 22 25 37 3 6 5 2 6 18 2 8 32 23 29 53 7 4 12 10 3 12 48 26 41 38 5 15 19 5 20 17 27 45 54 7 21 11 6 24 33 30 57 39 8 24 20 7 28 49 31 61 55 13 9 36 18 42 14 16 22 10 40 34 43 46 58 17 25 23 11 44 50 47 62 59 26 13 52 19 63 14 56 35 15 60 51
Program for listing of classes
Hint of program for the Gsystem G(n,k) (n=this.Degree, k=this.Order):
var r = (decimal)Math.Pow(this.Degree, this.Order)  1; this.Marked = new BitArray(r + 1, false); var g = 1; while (g ≤ r) { if (g < this.Marked.Count && !this.Marked.Get(g)) { this.AddStructure(g); this.MarkInstancesOfClass(g, r, marked); } g = g + 1; }
void MarkInstancesOfClass(int g, int r) { var u = g; while (u ≥ 0 && u < r && u < this.Marked.Count && !his.Marked.Get(u)) { if (u < this.Marked.Count) { this.Marked.Set(u, true); } u *= this.Degree; while (u > r) { u = r; } } }
Hint of program for the binary Gsystem G(n,k) (n=this.Degree=2, k=this.Order):
var r = (decimal)Math.Pow(this.Degree, this.Order  1)  1; long num; for (num = 1; num ≤ r; num += 2) { decimal classNumber = this.DetermineClassNumber(this.Order, num); if (num == classNumber) { this.AddStructure(num); } } num = (long)Math.Pow(this.GSystem.Degree, this.GSystem.Order)  1; this.AddStructure(num);
long DetermineClassNumber(byte order, long number) { var minNum = number; for (byte n = 1; n < order; n++) { number = this.TransposeLeft(order, number); if (number < minNum) { minNum = number; } } return minNum; } long TransposeLeft(int order, long number) { var shift = (byte)(order  1); return (number & 1) << shift  number >> 1; }
Rules of divisibility
Instance number u_{i} (j) is relative prime with r=(n^{k}−1), just if corresponding class number g_{i} is relative prime to r.
Number of all relative primes with r is φ(r), where φ is Euler’s function.
If g_{i} is relative prime with r, then its class must be self (not nested) class and such class has k instances. Proto k  φ(n^{k}−1), i.e.:
G(2,4) 0 * 1 2 4 8 3 6 12 9 5 10 * 7 14 13 11 15
In G(2,4) we have instance: (1, 2, 4, 8) a (7, 14, 13, 11) relative prime to 2^{4} −1=15,
i.e. rows marked by asterisk. Because φ(2^{4}−1)=φ(15)=8, it holds φ(n^{k}−1) mod k=8 mod 4=0.
Let us call the classes g, (g,r)=1, Euler’s classes, or shortly Eclasses. These have E=φ(r) instances in e=E/k = φ(r)/k classes.
Coefficient g(n,k) = φ(n^{k}−1)/k
n \ k │ 1 2 3 4 5 6 7 8 9 10 ──────┼───────────────────────────────────────────────── 1 │ − − − − − − − − − − 2 │ 1 1 2 2 6 6 18 16 48 60 3 │ 1 2 4 8 22 48 156 320 1008 4 │ 1 4 12 32 120 288 1512 4096 5 │ 2 4 20 48 280 720 5580 6 │ 4 12 56 216 1240 7 │ 2 8 36 160 8 │ 6 18 144 9 │ 4 16 96 10 │ 6 30
n \ k │ 11 12 13 14 15 16 17 18 19 ──────┼────────────────────────────────────────────┼ 2 │ 176 144 630 756 1800 2048 7710 7776 27594
Assumption: for even k (n>2), k  g(n,k).
Let us order instances according to the following rule: any of two adjacent instances have to be different only in one element (digit) [Vrba],
for G(2,4) we get:
Binary numbers Decimal numbers ───────────────────────────────────────────────────────────────── 0000,0001,0011,0010,0110,0111,0101,0100 0, 1, 3, 2, 6, 7,5,4 1100,1101,1111,1110,1010,1011,1001,1000; 12,13,15,14,10,11,9,8
Numbers at odd positions corresponds to numbers relative
prime with module 15:
Odd positions:1,2,7,4,13,14,11,8
Even positions:0,3,6,5,12,15,10,9
Gsystems are based on relation G, made by congruence according to module n^{k}−1., where n is base and k order of the Gsystem.
Relation G is equivalence on set of all instances U. Class of equivalence G is set of all instances of the given class.
Factor set U/G is set of all classes (equivalence classes), i.e. breakup of set U to particular sets of instances according to classes.
Every instance u ε U, belong just to one class g(u). Assignment of an instance to some class p_{G}: G−> U/G is called projection.
Let us assume systems R(n,k,r), where:
We will call these systems Mersenne’s (M) systems and mark
them M(n,k). System M(n,k) has r+1 instances {0,..,r}.
For n=2 Msystems melt with Gsystems, because divisor (n−1)=1. So systems M(2,k) have r+1 = (2^{k}−1)/(2−1)+1 = 2^{k} instances.
Algorithm for generation of Msystems is the same as algorithm for Gsystems (see Algorithm for Gsystems)
it uses different value r only.
Examples of outputM(2,1) M(3,1) M(4,1) 0 0 0 1 1 1 M(2,2) M(3,2) M(4,2) 0 0 0 1 2 1 3 1 4 3 2 2 3 4 5 M(2,3) M(3,3) M(4,3) 0 0 0 1 2 4 1 3 9 1 4 16 3 6 5 2 6 5 2 8 11 7 4 12 10 3 12 6 7 8 11 5 20 17 13 7 9 15 18 10 19 13 14 21
If kεP then Msystem has (n^{k}−n)/(n−1) self classes and therefore k  (n^{k}−n)/(n−1) − because is in this case (n−1,k)=1,
we realize no more, than Fermat’s little theorem says.
Numbers relative prime with r always belong to subset of self instances, corresponding classes have always k transpositions.
It holds: k  φ(r) i.e. k  φ((n^{k}−1)/(n−1))
It follows from the property of Gsystems
k\φ(n^{k}−1) and from the Fermat’s little theorem
k\(n^{k}−n) for kεP, (k,n)= 1:
(n^{k}−n)−φ(n^{k}−1)≡0 (mod
k) i.e.
(n^{k}−1+1−n)−φ(n^{k}−1)≡0 (mod k)
Let use mark the expression n^{k}−1−φ(n^{k}−1) by ψ(n^{k}−1) (see Euler’s function φ). So, it holds:
Count of numbers divisible with n^{k}−1 belongs to the same class (mod k) as number n−1.
E.g.:
ψ(5³−1) ≡ 124 ≡ 1 (mod 3) (5−1)≡ 1
(mod 3)
ψ(2^{7}−1) ≡ 127 ≡ 1 (mod 7) (2−1)≡ 1
(mod 7)
Lucas, Edouard A., 18421891, French mathematician. He studied number theory, especially Fibonacci sequences. He showed that the 2^{127} 1 is prime. 
Numbers L_{k}, defined by recurrent rule: L_{2}=4, L_{k+1}=L_{k}² − 2 are called LucasLehmer’s numbers.
Lehmer, Derrick HenryLehmer, Derrick Henry , 19051991, American mathematician. He was interested in testing of primality, in Lucas's functions, Bernoulli numbers, numerical analysis theory, in computers ... He introduced a new method of decomposition of numbers to a multiple of primes using the quadratic residues. 
k Lk Mk Lk/Mk ──────────────────────── 1 − 1 − 2 4 3 − 3 14 7 2 4 194 15 − 5 37634 31 1214 6 1416317954 63 −
To find if Mersenne’s number r=M_{k}=2^{k}−1
is prime or not, so called LucasLehmer’s test(for odd k) is
used:
number M_{k} is prime if it divides L_{k}.
Let d be the smallest divisor d\M(n,k), that d ≡ 1 (mod k). Then R(n,k,r) for r=M(n,k)/d makes trivial subsystem M(n,k).
E.g. for n=2:
R(2,2,2): 0 R(2,3,7): 0 R(2,4,5): 0 1 2 1 2 4 1 2 4 3 3 3 6 5 5 7
Numbers r=M(n,k)/d takes for given n,k the following values:
n\k│ 2 3 4 5 6 7 8 ───┼───────────────────────── 2 │ 3 7 5 31 7 127 17 3 │ 1 13 5 121 ... 4 │ 5 7 17 ... 5 │ 3 31 .. 6 │ 7 43 .. 7 │ 3 19 .. 8 │ 9 ...
R(7,3,19): 0 1 7 11 2 14 3 4 9 6 5 16 17 8 18 12 10 13 15 19
R(4,4,17): 0 1 4 16 13 2 8 15 9 3 12 14 5 6 7 11 10 17
Let us assume systems R(n,k,r), where:
for hεN.
These systems we will call Fermat’s (F) systems a mark F(n,h). System F(n,h) has r+1 instances {0,..,r}.
Algorithm for generation of Fsystems is again the same as algorithms for Gsystems or Msystems)
it uses different value r only.
Examples of the outputF(2,1) F(3,1) F(4,1) 0 0 0 1 2 1 3 1 4 3 2 2 3 4 5 F(2,2) F(3,2) F(4,2) 0 0 0 1 2 4 3 1 3 9 7 1 4 16 13 5 2 6 8 4 2 8 15 9 5 3 12 14 5 10 6 7 11 10 17 F(2,3) F(3,3) F(4,3) 0 0 .... 1 2 4 8 7 5 1 3 9 27 25 19 3 6 2 6 18 26 22 10 9 4 12 8 24 16 20 5 15 17 23 13 11 7 21 14 28
0 1 4 3 12 9 10 2 8 6 11 5 7 13
Systems with odd and even number h are distinct. It holds (n+1)(n^{h}+1) for odd h, so odd systems can be divided by (n+1).
In a similar sense we have used division by (n−1) in Msystems.
E.g. v F'(4,3) is r=(4³+1)/(4+1) = 65/5 = 13, see Kummer’s Ksystems.
Rules of divisibility
Because in Fsystems is number of transpositions of each
self class equal to 2k, in systems F(2, 2^{t}) is equal to
2∙2^{t} = 2^{t+1}.
Euler has proved formula that was supposedly known to
P.Fermat:
With its help (for t=5) Euler has found decomposition of
number 4294967297.
For t=0,1,2 have systems F(2, 2^{t}) this structure:
F(2,1) F(2,2) F(2,4) 0 0 0 1 2 1 2 4 3 1 2 4 8 16 15 13 9 3 5 3 6 12 7 14 11 5 10 17
Euler’s formula say that each divisor of numbers r=2^{(2t)}+1 must be of the form s∙2^{t+1} + 1.
So every divisor must be made as number of instances in s rows plus one.
Starting with t=2 classes can make pairs.
E.Lucas has notes, that number s in the form s∙2^{t+1} + 1 must be (for t≥2) even. Let dεP, d>2 is some divisor of number F(2,2^{t}).
So called Lucas' rule holds:
Every integer divisor of number F(2,2^{t}) have form s∙2^{t+2} + 1 (sεN).
Goldbach, Christian Russian mathematician. He focused primarily on the theory of numbers. Furthermore he studied endless series, theory of curves and theory of equations. 
Each two distinct numbers of the form F(2,2^{t}) are
relative primes, i.e. they does not have a common divisor:
In addition to this, F(2,2^{t}) are also
relative prime with t, i.e.:
Numbers F(n,2^{t−2}) have always residue 2 according
to module 2^{t}, i.e. it holds F(n,2^{t−2}) ≡ 2 mod
2^{t}.
E.g. F(3,4)=82 ≡ 2 mod 16, F(3,8)=6562 ≡ 2 mod 32, ...
If F(2,2^{t})εP then it is a divisor of number F(3,2^{2t−1}).
KsystemsLet us assume systems R(n,k,r), where:
these systems we will call Kummer’s (K) systems and mark K(n,h).
Usually (for h<r) Ksystems have order k=2h and correspond to systems R(n,2h,r).
Exception makes system K(2,3) with order k=2(h=r).
Into systems of prime r are just two classes nested, into systems with composite r more classes.
E.g. into K(4,3)= R(4,6,65) the system R(4,2,5) is nested:
n= 4 k= 6 r= 65 n= 4 k= 2 r= 5 0 0 1 4 16 64 61 49 1 4 2 8 32 63 57 33 2 3 3 12 48 62 53 17 5 5 20 15 60 45 50 6 24 31 59 41 34 7 28 47 58 37 18 9 36 14 56 29 51 10 40 30 55 25 35 11 44 46 54 21 19 13 52 22 23 27 43 42 38 26 39 65
Let us consider systems R(n,k,r), where both numbers n or r can be complex. Generally we will speak about systems of complex numbers, Csystems.
Systems with prime module R=a+bi have the same structure as systems with module r, which is norm of the numbers R, i.e. r=a²+b².
In case r=1+i it holds i≡2 mod (1+i):R(1,1,1+i) R(1,1,2) │ R(i,2,1+i) R(2,2,3) 0 0 │ 0 0 1 1 │ 1 i 1 2 i 2 │ 1+i 3 1+i 3 │
In case r=2+i is similarly i≡3 mod (2+i):
R(1,1,2+i) R(1,1,5) │ R(2,4,2+i) R(2,4,5) 0 0 │ 0 0 1 1 │ 1 2 1+i i 1 2 4 3 2 2 │ 2+i 5 i 3 │ 1+i 4 │ R(i,4,2+i) R(3,4,5) 2+i 5 │ 0 0 │ 1 i 1+i 2 1 3 4 2 R(1+i,2,2+i) R(4,2,5) │ 2+i 5 0 0 1 1+i 1 4 2 i 2 3 2+i 5
System R(i,2,1+i) resp. R(2,2,3)=G(2,2) depicts structure of complex numbers.
There is zero element in the first row, real and imaginary
unit in the second row and smallest complex prime 1+I in the third
row:
00 0i+0 0 01 10 0i+1 1i+0 1 i 11 1i+1 1+ i
In two rows systems there must be 1 and also −1 always
in the first row, because both 1 and −1 are complex
powers i.e. quadratic residua.
R(1+i,2,2+i) R(4,2,5) 0 0 0 1 1+i 1 4 1 −1 2 i 2 3 −i i 2+i 5 0
Two systems R(n1,k,r), R(n2,k,r) are complexconjugated, if
their bases n1 and n2 are complexconjugated.
For example systems R(1+i,3,3+2i) and R(1−i,3,3+2i):
R(1+i,3,3+2i) R(9,3,13) R(1−i,3,3+2i) 0 0 0 1 1+i 2i 1 9 3 1 1−i −2i 2 −i 1−i 2 5 6 2 i 1+i −1−i −2i −1 4 10 12 −1+i 2i −1 −1+i −2 +i 7 11 8 −1−i −2 −i 3+2i 13 3+2i R(9,3,13): 0 1 2 3 4 5 6 7 8 9 10 11 12 13 R(1+i,3,3+2i): 0 1 2 2i −1−i −i 1−i −1+i +i 1+i −2i −2 −1 3+2i R(1−i,3,3+2i): 0 1 2 −2i −1+i +i 1+i −1−i −i 1−i 2i −2 −1 3+2i
Each selfrow of the system R(1+i,3,3+2i) has one of complex
units {1,−i,−1,+i}.
R(1+i,3,3+2i) 0 1 1+i 2i +1 ≡ 1 2 −i 1−i −i ≡ 5 −1−i −2i −1 −1 ≡ 12 −1+i −2 +i +i ≡ 8 3+2i
These units correspond to numbers 1,5,12,8 in the system
R(9,3,13), i.e. to numbers of the first row of the system
R(5,4,13).
Because 5 ≡ −i (3+2i), we find complex units in the first row
of the system R(−i,4,3+2i):
R(5,4,13) R(−i,4,3+2i) 0 0 1 5 12 8 1 −i −1 +i 2 10 11 3 2 −2i −2 2i 4 7 9 6 −1−i −1+i 1+i 1−i 13 13
In the second row there are smallest complex even numbers and on the third row smallest complex halfeven numbers.
In case nεN is module of Gsystems the number r=n^{k}−1. For complex number N=a+bi a similar computation of the module does not hold.
(At least not in the sense of analogy to system with base n=a²+b²).
E.g. let n=5, k=3, r= 5³−1 = 124. Then according to it R= (2+i)³−1 = (2+11i)−1 = 1+11i. But norm of the expression 1+11i is number 122!?
Number of classes in S(k) is equal to number of classes in G(2,k) minus number of distinct contrast classes (i.e. excluding inner contrast classes).
It holds:
Let level of instance is absolute value of the sum of positive and negative units, e.g. −−−++ has level −1−1−1+1+1 = 1.
Neutral class is class with zero level.
Outline of instances of systems S(k)
Classes Z(k): Instances Classes Level g−numbers classes transp. total v w m neutr. ────────────────────────────────────────────────────────── k=1: − 1 0,1 2 1 2 2 1 0 1 0 ────────────────────────────────────────────────────────── k=2: −− 2 0,3 2 1 2 −+ 0 1,2 1 2 2 2² 1 1 2 1 ────────────────────────────────────────────────────────── k=3: −−− 3 0,7 2 1 2 −−+ 1 1,3 2 3 6 2³ 1 1 2 0 ────────────────────────────────────────────────────────── k=4: −−−− 4 0,15 2 1 2 −−−+ 2 1,7 2 4 8 −−++ 0 3 1 4 4 −+−+ 0 5 1 2 2 2^{4} 2 2 4 2
Outline of instances of systems S0(k)
System Level Instances Total ───────────────────────────────────────────────────────── k=1: ( 2 classes ) 0 0 1 − + 0 2 31 ───────────────────────────────────────────────────────── k=2: ( 4 classes ) 00 0 1 * −+ +− 0 2 0− 0+ −0 +0 1 4 −− ++ 2 2 * 3² ───────────────────────────────────────────────────────── k=3: ( 6 classes ) 000 0 1 * 0−+ −0+ +0− 0+− +−0 −+0 0 6 00− 00+ 0−0 0+0 −00 +00 1 6 −++ +−− +−+ −+− ++− −−+ 1 6 0−− 0++ −0− +0+ −−0 ++0 2 6 +++ −−− 3 2 * 3³
Let systems of power residues are the systems R(n,k,r), where
for v>1.
Specially in case v=2 we will speak about quadratic systems
(systems of quadratic residues),
resp. in case v=3 about cubic systems, and so on.
All numbers a² mod r are so called quadratic residues (residues with exponent m=2) according to module r. Others numbers (mod r) are so called quadratic nonresidues.
One of quadratic residues is always zero. All other quadratic residues and nonresidues for aε(1,r−1), rεP, arise symmetrically, i.e. exists r div 2 residues and r div 2 nonresidues.
Quadratic residues for r=13:
a² 1 4 9 16 25 36 49 64 81 100 121 144 a² mod 13 1 4 9 3 12 10 10 12 3 9 4 1
While looking for quadratic residues, it is not necessary to compute squares (second powers).
0 1 4 3 12 9 10 2 8 6 11 5 7 13
Quadratic residues (mod 13) are concentrated in the first row of the system R(4,6,13), quadratic nonresidues in the second row.
Value r div 2 is used often used in theory of numbers. It appears in various formulas and so it exhort for a name.
We will call it median and mark it κ(r). Generally median of order s means number κ(k,s)=k div s, k,sεN.
For order 2 we will mark it simply (like second square root): κ(k)= κ(k,2)= k div 2.
Quadratic residues a nonresidues according to rεP cover all numbers 1,2,..r−1 and make multiplicative groups Zr of order r.
│ residues │ nonresidues │ 1 2 4 │ 3 5 6 ───┼──────────┼───────── 1 │ 1 2 4 │ 3 5 6 2 │ 2 4 1 │ 6 3 5 4 │ 4 1 2 │ 5 6 3 ───┼──────────┼───────── 3 │ 3 6 5 │ 2 1 4 5 │ 5 3 6 │ 1 4 2 6 │ 6 5 3 │ 4 2 1
Number 1 appears in the table only as product of residue* residue or nonresidue* nonresidue.
If n, n' are dual bases, it holds p\(n∙n'−1), i.e. n∙n' ≡ 1 (mod p).
Therefore, when n is quadratic residue (mod p), is n' also quadratic residue.
Quadratic residues are in the top left quadrant of the table:
Quadratic residues make for rεP subgroups Kr(∙) group Zr(∙).
Order of these subgroups is median κ(r).
K3 K5 K7 K11 K13 1 1 4 1 2 4 1 3 4 5 9 1 3 4 9 10 12 4 1 2 4 1 3 9 1 4 5 3 9 12 1 4 10 4 1 2 4 1 5 9 3 4 12 3 10 1 9 5 4 9 3 1 9 1 10 3 12 4 9 5 3 1 4 10 4 1 12 9 3 12 10 9 4 3 1
Carmichael, Robert Daniel 
R.D.Carmichael (1905) reformulated Wilson’s theorem. Number r is prime, if it holds:
Wilson’s theorem can be rewritten into form:
Sylvester, James Joseph [], 18141897, English mathematician, a leading creator of a new symbolism. He was interested in the theory of elementary divisors, algebra forms, theory of determinants. Along with Cayley he created the first algebraic theory of invariants of quadratic forms. He was also concerned in mechanics. 
Sylvester (1860)
k=1 5∙φ(1) = 5 k=2 2∙φ(2) = 2 k=3 1∙φ(3) = 2 5∙6/2 = 15 k=4 1∙φ(4) = 2 k=5 1∙φ(5) = 4 ─────── 15
Let us write for uεN, rεP:
If ru then it holds:
E.g. Q(14,7) = 14³ mod 7 = 0.
It holds for quadratic residues:Q(u,r)= 1 ──────────────────────────── 1^{6} mod 13 = 1 mod 13 =1 4^{6} mod 13 = 4096 mod 13 =1 3^{6} mod 13 = 729 mod 13 =1 12^{6} mod 13 =2985984 mod 13 =1 9^{6} mod 13 = 531441 mod 13 =1 10^{6} mod 13 =1000000 mod 13 =1 
It holds for quadratic nonresidues:Q(u,r)= −1 ───────────────────────────── 2^{6} mod 13 = 64 mod 13 =−1 8^{6} mod 13 = 262144 mod 13 =−1 6^{6} mod 13 = 46656 mod 13 =−1 11^{6} mod 13 =1771561 mod 13 =−1 5^{6} mod 13 = 531441 mod 13 =−1 7^{6} 
Number u is quadratic residue or nonresidue according to value Q(u,r)=u^{κ(r)} mod r.
It is so called Euler’s criterion.
When Q(u,r)= 1 (resp.−1), is number u quadratic residue (resp. nonresidue) according to module r.
Euler’s criterion is abstract and simple. It is advantageous particularly for smaller numbers r.
Number u is quadratic residue, if it exists such number a, that it holds:
a² ≡ u (mod r).
After powering of this formula to exponent κ(r)=(r−1)/2 we
get:
E.g. for r=17, u=19, a=11 it holds 11² ≡ 19 (mod
17), so 19 is quadratic residue mod 17. After squaring to
κ(17)=8:
11^{16} ≡ 19^{8} ≡ 1 (mod 17).
Euler’s criterion follows from Fermat’s little theorem. Because a^{r−1} ≡ 1 (mod r) then u^{κ(r)} ≡ 1 (mod r) in case when u is quadratic residue.
In power systems R(n,k,r) it holds n^{k} ≡ 1 mod r (see Geometric sequences). Number 1 is always in the first row, therefore for n=1 is n^{κ(r)} ≡ 1 mod r.
Therefore in two rows systems there are numbers n^{κ(r)} ≡ 1 mod r (i.e. quadratic residues) in the first row and numbers n^{κ(r)} ≡ −1 mod r (i.e. quadratic nonresidues) in the second row.
E.g. numbers u=a^{4} mod 13 for all aε(1,12):
a^{4 } 1 16 81 256 625 1296 2401 4096 6561 10000 14641 20736 a^{4} mod 13 1 3 3 9 1 9 9 1 9 3 3 1
Biquadratic residues appear in the first row of systems R(n,κ(r,4),r), e.g. R(5,4,13). Because a^{4} mod r=(a²)² mod r, biquadratic residue is also quadratic residue.
R(5,4,13): R(4,6,13): 0 0 1 3 9 1 4 3 12 9 10 2 6 5 2 8 6 11 5 7 4 12 10 13 7 8 11 13
In analyzes of biquadratic residues, Gauss divide numbers into 4 classes A,B,C,D according to following schema:
A │ 1 g^{4} g^{8} ... │ biquadratic residua B │ g g^{5} g^{10} ... (mod r) │ nonresidua C │ g² g^{6} ... │ quadratic residua D │ g³ g^{7} ... │ nonresidua
where g is such number, that all numbers 1,g,...g^{r−1} are (mod r) are all distinct (see primitive roots of numbers r).
E.g. for r=17, g=3 (on the right with numbers in classes ordered according to values)
A 1 13 16 4 A 1 4 13 16 B 3 5 14 12 i.e. B 3 5 12 14 C 9 15 8 2 C 2 8 9 15 D 10 11 7 6 D 6 7 10 11
Distribution of numbers in classes corresponds to distribution of instances in Rsystem, in our case R(4,4,17):
R(4,4,17) 0 1 4 16 13 │ A (1) 2 8 15 9 │ C (g²=9) 3 12 14 5 │ B (g=3) 6 7 11 10 │ D (g³=27≡10) 17
Result of product of any number of class C (e.g.9 mod 17) with number of class D (e.g.7 mod 17) belongs
to class B (9∙7=63≡12 mod 17). Generally the multiplication is directed by the following table:
│ A B C D Z4+│ 0 1 2 3 ──┼───────────── ──┼──────────── A │ A B C D 0 │ 0 1 2 3 B │ B C D A 1 │ 1 2 3 0 C │ C D A B 2 │ 2 3 0 1 D │ D A B C 3 │ 3 0 1 2
This table corresponds to table of addition in field Z4 (see groups).
In solution of problem of biquadratic residues Gauss have
used complex numbers. He assigned complex units 1,i,−1,−i to
classes A,B,C,D.
Let us mark A,B,C,D by numbers (characters) c=0,1,2,3 and numbers
(instances) in classes by letter u. It holds:
where i is imaginary number, i=√−1.
k=2: k= 3: k = 4: k = 5: k = 6: 1 1 2 1 2 3 1 2 3 4 1 2 3 4 5 2 1 2 0 2 2 4 1 3 2 4 0 2 4 3 2 1 3 1 4 2 3 0 3 0 3 4 3 2 1 4 2 0 4 2 5 4 3 2 1
In the centre of multiplicative tables (mod k) there is just one number (if k is even) or there are four numbers (if k is odd).
These four numbers contains numbers κ(k)² mod k and κ(k)∙(κ(k)+1) mod k.
For even k:
For k=4s (i.e. k is divisible by 4) is in the centre of the table zero. For k=4s−2 is in the centre number 2s−1.For odd k:
For k=4s−1 are the four numbers build from s and 3s−1. For k=4s+1 similarly from s and 3s+1.Numbers in the centers of multiplicative tables (mod k) does not depend on number k only, but also on its correspondence to 4 classes: 4s−1, 4s, 4s+1 and 4s+2. 
Odd number n is genuine if its median κ(n) is odd. Genuine odd numbers (p,q,...):
is not a square  second power has always form 4s or 4s+1 genuine odd primes (so called Gauss' primes) cannot be in complex field decomposed(e.g. 3,7,11,19,23,31,43,47,59,67,71,79,83,...)
congruence x²+y²≡0 (mod p) is not solvable (pεP) number aεN can be written as sum of two (relative prime) squares, if there is no genuine odd prime in its partitionNongenuine odd numbers (p,q,...) have the following properties:
make square, i.e.(2k+1)² has always form 4s+1 Euler has proved, that primes of form 4s+1 are always unique sum of two squares
Euler has proved some more sophisticated theorems about
classes (mod 4):
Let (a1,b1)=1, (a2,b2)=1 and numbers p1,p2 ε P. Let us mark f(a,b)
= a²+n∙b².
E.g.
for n=1: 5(1+4),
13(4+9) a 5 ≡ 13 (mod 4);
or 13(1+25),
17(9+25) a 13 ≡ 17 (mod 4).
for n=2: 11(4+2∙9), 59(9+2∙25) a 11 ≡ 59 (mod 8).
Let us consider Rsystems (rεP), that have exactly v selfrows. Selfrow is row, that have just k instances (rows 0 and r are not self rows, they are nested).
Such systems have order k=r div v = κ(k,v).
In the first row are residues of exponent v, i.e. so called. vtic residues (for v=2 quadratic residues, v=3, cubic residues,...).
In other rows there are nonresidues (of exponent v).
E.g. cubic residues, i.e. numbers u=a³ mod r are in the first row of schemas R(n,κ(r,3),r):
For r=7: For r=13: a3 1 8 27 64 125 216 a3 1 8 27 64 125 216 343 512 729 a3 mod 7 1 1 6 1 6 6 a3 mod 13 1 8 1 12 8 8 5 5 1 R(6,2,7): 0 R(5,4,13): 0 1 6 1 5 12 8 2 5 2 10 11 3 3 4 4 7 9 6 7 13
Systems with v selfrows, have order k=r div v =
κ(r,v), i.e. these are systems R(n,κ(r,v),r).
E.g. R(5,4,13) has 3 selfrows, i.e. v=3 and order k=13 div 3 =
4.
R(5,4,13): 0 1 5 12 8 2 10 11 3 4 7 9 6 13
Number u is vtic residue, if exists such a, that it holds a^{v} ≡ u (mod r).
After powering to κ(r,v) we get:
a^{v(r−1)/v} ≡ a^{r−1} ≡
u^{κ(r)} (mod r).
This is an analogy of formula in the previous paragraph.
Number u is vtic residue, when Qv(u,r)= 1 where:
Number u is vtic residue (mod r), if it is in the first row
of the system R(n,k,r), where k=κ(r,v) (see Euler’s generalized
criterion).
In case u=v, is number u is by itself utic residue. We call
Legendre’s systems the systems R(n,k,r), that have v
selfrows
and in the first row number v.
0 1 2 4 │ v = 2 3 6 5 │ 7 ── k=3 ──
E.g. R(2,3,7) is Legendre’s system: it has 2 selfrows
and in the first row number 2 (number 2 is quadratic residue mod 7).
It holds: 2³ ≡ 1 (mod 7).
Generally according to Euler’s criterion must be Qv(v,r) = v^{κ(r,v)} mod r = 1, where exponent v is order of system: k=κ(r,v).
Therefore in Legendre’s systems R(n,κ(r,v),r) it holds:
Legendre, AdrienMarie. French mathematician and physicist. He greatly influenced next development of number theory: he discovered quadratic reciprocity law and generalized relations concerning large Fermat's theorem. He studied also ellipsoids, elliptic functions and integrals. He worked on meassuring of the Earth, derived new formulas for spherical triangles. He managed compilation of logarithmic and trigonometric tables. 
Because k=κ(r,v), we get k∙v = r−1 (rε P). So (r−1)^{k} = (v∙k)^{k} = k^{k}∙v^{k}, where:
v^{k} ≡ 1 (mod r) (r−1)^{k} ≡ (−1)^{k} (mod r).After substitution we get:
When k is even (k=2t) tak (−1)^{k} = 1 it holds:
Here r= 2tv+1, rε P.
E.g. for t=1,2,3,4 we get values:
t (2t)^{2t} rozklad T=(2t)^{2t}−1 ──────────────────────────────────────────────── 1 2² = 4 3 2 4^{4}= 256 255 = 3∙5∙17 3 6^{6}= 46656 46655 = 5∙7∙31∙43 4 8^{8} = 16777216 16777215 = 3²∙5∙7∙13∙17∙241
Partition of numbers T helps to select numbers r.
We will derive one appropriate Legendre’s system with even order for each t:
t=1 r= 3, v= 1, k=2 0 1 2 3 t=2 r=17, v= 4, k=4 0 1 4 16 13 2 8 15 9 3 12 14 5 6 7 11 10 17 
t=3 r=31, v= 5, k=6 0 1 6 5 30 25 26 2 12 10 29 19 21 3 18 15 28 13 16 4 24 20 27 7 11 8 17 9 23 14 22 31 t=4 r=17, v= 2, k=8 0 1 2 4 8 16 15 13 9 3 6 12 7 14 11 5 10 17 
We will call systems of primitive roots the systems R(n,k,r), where:
Systems of primitive roots supply the case v=1 of systems of power residues.
The powers of a given root can pass all other roots only in case, when s=h, i.e. when order (s) of given root is equal to exponent of binomial equation h.
The roots with this property we call primitive roots. Fundamental root α is always one of primitive roots.
Let us select such systems R(n,k,r) (for rεP) that have r−1 transpositions, i.e. that (excluding numbers 0 and r) does not break rows. For r=7 there exist 2 such systems, R(3,6,7) a R(5,6,7):
R(3,6,7): R(5,6,7): 0 0 1 3 2 6 4 5 1 5 4 6 2 3 7 7
Each of these systems has just one selfclass with representative g=1. For α=n, j=0,..,r−1 expressions α^{j} pass all instances
and makes so integer analogy of primitive roots of equation x^{6}−1=0 with exponent h=r−1=6 (see cyclic groups,...)
Number of primitive roots is equal to number of primes relative to exponent h. Binomial equation x^{h}−1 = 0 has therefore φ(h) primitive roots.
For given rεP exists always φ(r−1) systems with primitive root as a base n. E.g. for r=7 there exist φ(6)= 2 primitive roots.
The root x_{j}(h) is primitive root of equation x^{h}−1=0, when numbers j and h are relative prime, (j,h)=1.
E.g. for number h=6, the numbers j=1 and 5 points to primitive roots:
j: 0 1 2 3 4 5 ──────────────────────────────────── R(3,6,7): 1 3 2 6 4 5 R(5,6,7): 1 5 4 6 2 3
For r=11 there exist φ(10)= 4 primitive roots and therefore four distinct systems with full number of transpositions:
R(2,10,11): R(6,10,11): 0 0 1 2 4 8 5 10 9 7 3 6 1 6 3 7 9 10 5 8 4 2 11 11 R(7,10,11): R(8,10,11): 0 0 1 7 5 2 3 10 4 6 9 8 1 8 9 6 4 10 3 2 5 7 11 11
Primitive roots 2,6,7,8 are at positions 1,3,7,9 (i.e. numbers relative prime with r−1=10).
0 1 2 3 4 5 6 7 8 9 ─────────────────────────────────────────────────────── R(2,10,11): 1 2 4 8 5 10 9 7 3 6 R(6,10,11): 1 6 3 7 9 10 5 8 4 2 R(7,10,11): 1 7 5 2 3 10 4 6 9 8 R(8,10,11): 1 8 9 6 4 10 3 2 5 7
For r=13 φ(12)= 4 primitive roots exists: 2,6,7,11. These roots make bases of systems R(2,12,13), R(6,12,13), R(7,12,13) a R(11,12,13).
R(n,k,13) 0 n │ k +−> 1 2 4 8 3 6 12 11 9 5 10 7 R(2,12,13) ────┼──── │ 13 1 │ 1 │ 2 │ 12 ──┼ 0 3 │ 3 +−> 1 6 10 8 9 2 12 7 3 5 4 11 R(6,12,13) 4 │ 6 │ 13 5 │ 4 │ 6 │ 12 −+ 0 7 │ 12 ───> 1 7 10 5 9 11 12 6 3 8 4 2 R(7,12,13) 8 │ 4 13 9 │ 3 10 │ 6 0 11 │ 12 ───> 1 11 4 5 3 7 12 2 9 8 10 6 R(11,12,13) 12 │ 2 13
Let r is odd prime (i.e. rεP, r>2).
According to Fermat’s little theorem is n^{r−1} ≡ 1 mod r, i.e. r (n^{r−1}−1).
When r is odd, number r−1 is even and we can expand
expression n^{r−1}−1 according to formula
a²−b² = (a−b)(a+b):
n^{r−1}−1 = (n^{κ(r)}
−1)(n^{κ(r)} +1)
Difference of coefficients is 2, so the product cannot have divisor greater than 2.
Number r must be therefore a divisor of one of numbers (n^{κ(r)} −1) or (n^{κ(r)} +1).
If (n^{κ(r)} −1) was the divisor then n^{κ(r)} ≡ 1 mod r, but it is for primitive root impossible (system can not break row on the index κ(r)...)
For primitive root n (according to module r) it holds:
So primitive roots keep the same relation as quadratic nonresidues (see Euler’s criterion).
Not all quadratic nonresidues are primitive roots, primitive roots make subset of quadratic nonresidues.
Sequence of numbers B1,B2,....Bn,A1,A2,..An can be reordered to sequence A1,B1,A2,B2,... with help of one permutation.
We are looking for permutation with just one cycle (problem "perfect shuffle" in the theory of sort algorithms).
Let use consider an equivalent problem:
Sequence of 2n numbers 0,1,..,2n−1 have to be reordered by one permutation, that n odd numbers come in front of n even numbers.
We are interested for which n can be such reordering made by permutation with just one cycle (monocycle).
Monocycle exists only when number 2 is primitive root according to module p=2n+1.
Example of a monocycle
For n=5 (p=11) has permutation just one cycle [0 1 3 7 4 9 8 6 2 5]:
j 0 1 2 3 4 5 6 7 8 9 i 1 3 5 7 9 0 2 4 6 8
Origin of cycle:
j 0 1 2 3 4 5 6 7 8 9 2^{j}−1 0 1 3 7 15 31 63 127 255 511 (2^{j}−1) mod 11 0 1 3 7 4 9 8 6 2 5
Cycle has full length because there does not exists j<p−1, that (2^{j}−1) mod p=0.
Example of more cycles
For n=3 (p=7) is permutation composed of two cycles [0 1 3][2 5 4]:
Permutation: Vznik cyklu: j 0 1 2 3 4 5 j 0 1 2 3 4 5 i 1 3 5 0 2 4 2^{j}−1 0 1 3 7 15 31 (2^{j}−1) mod 7 0 1 3 0 1 3
Permutation does not make monocycle: cycle has not full length, because (2³−1) mod 7 = 0.
Briggs RafaelBriggs Rafael, 15261572, English mathematician, published the first table of decimal logarithms. 
Special meaning in math has (for P=a^{p}, Q=a^{q}, a,p,q,P,Q,r ε R)
exponential function, that convert additional operation to multiplication:e.g. 5∙125=5^{1} 5³ = 5^{1+3}= 5^{4} = 625, and
Neper(Napier) JohnNeper(Napier) John Scottish mathematician, inventor of the natural logarithms. Dealt with spherical trigonometry and calculation methods. 
logarithmic function , that convert multiplicative operation to addition
e.g. log_{5}625 = log_{5}5^{4} = 4∙ log_{5}5 = 4.
As bases are used:
In decimal system base a=10 (decimal logarithms…) Music tones in ratio 2:1 (octave) are similar. In this case binary logarithms (base a=2) can be used.(We find these logarithms also e.g. in theory of information…)
In mathematic analysis Euler’s number e=(1+1/n)^{n} is significant (higher n, more precise e).Euler’s number is base of natural logarithms a=e (or a=1/e how J.Napier originally defined).
In theory of numbers integer logarithms were introduced. Their bases are so called primitive roots.
Let us consider exponential function: y=n^{x} mod r,
that assign numbers {1..r−1} to transpositions xε{0..r−2}.
Inverse function to this is an analogy of logarithmic function: x=
ind(y) mod (r−1).
If n is primitive root, the assignment is definite.
If module of exponential function is r then module of inverse function is (r−1).
E.g. for r=11, n=2:
x 0 1 2 3 4 5 6 7 8 9 y=n^{x} mod r 1 2 4 8 5 10 9 7 3 6 y 1 2 3 4 5 6 7 8 9 10 x=ind(y) mod r−1 0 1 8 2 4 9 7 3 6 5
Some congruences can be solved like logarithmic
equations:
E.g. a³≡9 (mod 11):
a³ ≡ 9 (mod 11) ind(a³) ≡ 3∙ind(a) ≡ ind(9) (mod 10) ind(a) ≡ ind(9)/3 ≡ 6/3 ≡ 2(mod 10) a ≡ 4 (mod 11) Zkouška: 4³ mod 11 = 64 mod 11 = 9
From criterion for primitive roots follows, that index
numbers −1 (mod r) is κ(r).
E.g. for r=11:
y
1 2 3 4 5 6 7 8 9 10
x=ind(y) mod (r−1) 0 1 8 2 4 9 7 3 6 5
ind(−1)= ind(r−1)= ind(10) = 5 κ(r)
= 11 div 2 = 5.
Among the Rsystems with r=5 are φ(4)= 2 systems of primitive roots:
R(2,4,5) 0 R(3,4,5) 0 1 2 4 3 1 3 4 2 5 5
We will call primitive expressions r(x) the polynomials with
coefficients taken from instances of systems of primitive
roots.
r_{1}(x)=
1+2x+4x²+3x³ r_{2}(x)=
1+3x+4x²+2x³
We are interesting in the product of all such expressions for given r when x^{r−1}=x^{k}=1.
For r=5 is: r_{1}(x)∙r_{2}(x) = 5(6+5x+4x²+5x³).
If expression V(x) mod r can be substituted for V(x), then:
[r_{1}(x)∙r_{2}(x)]_{5} =
5(1−x²).
(Expression ∑s² for s=1,..,p−1 in products is always
divisible by p, because ∑s² =
(p−1)p(2p−1)/6,...).
For r=7:
r_{1}(x)=
1+3x+2x²+6x³+4x^{4}+5x^{
5
}
r_{2}(x)=
1+5x+4x²+6x³+2x^{4}+3x^{5}
r_{1}(x)∙r_{2}(x) =
7(13+10x+11x²+8x³+11x^{4}+10x^{5}
and [r_{1}(x)∙r_{2}(x)]_{7} =
7(1−x³)(−1+3x−3x²).
These expressions, after substitution x=α, where α is complex root of equation x^{r}=1, cohere with Kummer’s theory of cyclic numbers.
We call Wieferich's (W) systems the systems R(n,k,r), where:
and tεN. We are interested especially in the case n=2 with t odd prime (tεP).
We will try to observe, what properties the Wsystems have in the case t is so called.Wieferich's prime.
Wsystems  Examples of outputR(2, 20, 5²) 0 1 2 4 8 16 7 14 3 6 12 24 23 21 17 9 18 11 22 19 13 5 10 20 15 25 R(2, 21, 7²) 0 1 2 4 8 16 32 15 30 11 22 44 39 29 9 18 36 23 46 43 37 25 3 6 12 24 48 47 45 41 33 17 34 19 38 27 5 10 20 40 31 13 26 7 14 28 21 42 35 49
Wieferich, Arthur [], 18841954, the German mathematician and teacher. He proved, that validity of the Last Fermat Theorem can  in the first case  distort at most only some of (socalled Wieferich's) primes. 
Wieferich's prime is such p
, pεP,
that
:
2^{
p
}^{
1
}≡ 1 (mod
p^{
2
})
Only known Wieferich's primes are 1093 and 3511.
Orders of Wsystems are in the observed cases (t=3,5,7,..) larger than the value t, so k>t (eg. k=20 for t=5; k=21 for t=7; ...) .
Even, it holds t  k (in our example 5  20; 7  21). Order k ofted have value k = t(t1), so k  t(t1) (e.g. 20  5*4, 21  7*6).
In the system R(n,k,t²) the number W=2^{t1}  1 mod t² must lay in the first row. It determines the order of the system k, k  (t1).
Here is the main distinction of systems R(n,k,t² ) of common prime t from systems with Wieferich's prime t.
In the first case k  t(t1), but in the second case the order k must be necessarily less : k  (t1) .¨
In the system of order k = 364 there are m=3284 classes, from these v=3282 are self classes and w=2 are nested classes (from the order 1).
Thus, in total system contains 3282*364 + 2* 1 = 1194650 instances (=1093²+1).
In the system of order k =1755 there are m=7026 classes, from these v= 7024 are self classes and w=2 are nested classes (from the order 1).
Thus, in total system contains 7024*1755 + 2* 1 = 12327122 instances (=3511²+1).
Order of any system d_{i } nested into R(n,k,t² ) have to divide self order k, so d_{i}  k, while nesting is determined by common divisor 2^{d}1 and t² .
For prime tεP and (2^{d}1, t² )>1 is the only case t  2^{d}1 for some d_{i}  k.
In R(2, 20, 5²) is k= 2²*5 and for d=5 it holds t  2^{5}1. In R(2, 20, 7²) is k=3*7 and for d=3 it holds: t  2³1.
Therefore these systems have nested classes with number k/t of transpositions.
In the case, where for all d_{i}  k holds (2^{d}1, t² )=1, the system has only self classes and trivial classes nested from order d=1.
Because number of trivial classes is always 2 and total number of instances is t²+1, the self classes cover t²1 instances.
Each self instance has k transpositions, so k 
t²1.
These relations hold also in both of the mentioned systems

in R(2, 364, 1093²)
364 
1093²
1
and in R(2, 1755, 3511²)
1755 
3511²1.
In the systems R(n,k, t² ) is 2^{t1} = a* t² + h (a,hεN), where h=1 in case of Wieferich's prime t.
In other cases we get  e.g. for t=5: h=16, for t=7: h=15, ... where it holds: h mod t = 1.
So h = b*t + 1 (bεN) and we get from the conjunction with the previous relation:
2^{t1} = a* t² + b*t + 1
It is clear, that the whole equation is divisible by t only in case t  2^{t1} 1.
Equation 2^{t1} = a* t² + b*t + 1 is satisfied in case b=0 by Wieferich's primes 1093 and 3511.
For the next values b = 1,2,... we get the folowing values t.
b=1: t=3,29,37,3373; b=2: t=7,71,379; b=3: t=13,19,173; b=5: t=11; b=6: t=31, 89; ...