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Classification of natural numbers based on arithmetic progressions with a difference 6.
G. G. Ryabov, V. A. Serov
Abstract—The article considers a classification of natural numbers based on the submission of the set of all natural numbers as union of six infinite arithmetic progressions. The classes themselves (bijective to the progressions) are considered as members of two finite semigroups with regard to the operations of addition and multiplication. The binary relations between classes and examples of natural numbers properties at such classification are given.
Keywords—Arithmetic relations, classification, numbers, semigroup.
progression, bijection, binary Goldbach prime, set of natural
I. INTRODUCTION
The representation of the set of all natural numbers greater than or equal to 4 N (> 4) as the combination of six infinite arithmetic progressions denoted as S4 = {4 + 6m}, S5 = {5 + 6m}, S6 = {6 + 6m}, S7 = {7 + 6m},S8 = {8 + 6m}, S9 = {9 + 6m}, m e N0was considered in [1]-[3].
In this representation, every natural nt has a unique expression in the form of pairs of numbers (r, mi ), i e N , where r is the index of progression, coinciding with the initial member of progression, and mi is the ordinal number of the number in this progression. It was also proved that all prime numbers P greater than 4, P(> 4) are contained only
in two progressions of these six, namely S5 = {5 + 6 m} and S7 = {7 + 6m}.
Let us agree within the article not to introduce special notations for the proposed classes, which are bijective to six progressions, because substantial part lies in the number-the index of progression, which will be used in the expressions of binary relations between classes in the form of 6 x 6 tables.
In this article, when ensuring [ 1 ] approach, the following six progressions will be considered: S3 = {3 + 6m}; S4 = {4 + 6m}; S5 = {5 + 6m}; S6 = {6 + 6m};
S7 = {7 + 6m}; S8 = {8 + 6m};
The initial fragment of this six progressions is given below (accordingly, and the offered classes natural) Fig.1. Primes in them are marked more in bold, and primes-twins also by underlining.
Properties of progressions go into properties of classes that act as elements of the semigroups under addition and multiplication. So, it should be noted the behavior of primes degrees within this six progressions included in the canonical multiplicative representation of numbers, equal to the index of progressions. So, for S3—3, S4—2, S5—5,
S6 —6, S7 —7, S8 —2 , respectively. It is easy to verify by elementary calculations the following properties inherent to each of these progressions and missing all the rest.
1. S3 contains all ( even and odd) degrees of 3. {3,9,27,81,243,729,2187,...}e S3.
2. S4 contains only even powers of 2. {4,16,64,256,1024,4096,...} e S4.
3. S5 contains only odd powers of 5. {5,125,3125,78125,...} e S5 (1)
4. S6 contains all (even and odd) degrees of 6. {6,36,216,1296,...} e S6.
5. S7 contains all (even and odd) degrees of 7. {7,49,343,2401,...} e S7.
6. S8 contains only odd powers of 2. {8,32,512,2048,...} e S8.
This property can be considered as an invariant of progression and, hence, an invariant of bijective class of the natural. On the other hand, it can be used as a marking of an infinite arithmetic progression by elements of an exponential nature (as automorphic built-in logarithmic scale).
Manuscript received October 28, 2016.
G.G. Ryabov is with the Research Computing Center, M.V. Lomonosov Moscow State University, Moscow, Russia (e-mail: gen-ryabov@ yandex.ru).
V.A. Serov is with the Research Computing Center, M.V. Lomonosov Moscow State University, Moscow, Russia.
International Journal of Open Information Technologies ISSN: 2307-8162 vol. 4, no. 12, 2016 m 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
S3=3 9 15 21 27 33 39 45 51 57 63 69 75 81 87 93 99 105 111 117 123 129 135 141 147
S4=4 10 16 22 28 34 40 46 52 58 64 70 76 82 88 94 100 106 112 118 124 130 136 142 148
S5=_5 11 17 23 29 35 41 47 53 59 65 71 77 83 89 95 101 107 113 119 125 131 137 143 149
S6= 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 126 132 138 144 150
5?=_7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103 109 115 121 127 133 139 145 151
Sg= 8 14 20 26 32 38 44 50 56 62 68 74 80 86 92 98 104 110 116 122 128 134 140 146 152 Fig. 1. The initial fragment of six progressions S3 - S8.
II. Binary relations between classes as elements of
SEMIGROUPS
Classes of natural, bijective to these progressions, are in binary relations to the operations of addition and multiplication, which are shown in a tabular form in Fig.2. It means that for any pair of natural such that nl e and
n2 e S , the sum (nl + n2) e S , where r3 is the index at the intersection of the rl -th column and the r2 -th row (Fig.2a) (or vice versa—the row and the column, since both addition and multiplication are commutative), and the tables are symmetric. Similarly, for multiplication (Fig.2b).
In the semigroup under addition the role of the "0" makes the class S6, and in the semigroup under multiplication the role of "1" performs the class S7.
-)- S3 S4 S5 SS S7 SS S3
54
55 Sfi S7
se
6 7 8 3 4 5
7 S 3 4 5 6
8 3 4 5 6 7
3 4 5 6 7 8
4 5 6 7 8 3
5 6 7 « 3 4
X
53
54
55
56
57
58
S3 S4 S5 S6 S7 S8
3 6 3 6 3 6
6 4 8 6 4 8
3 8 7 6 5 4
6 6 6 6 0 6
3 4 5 6 7 8
(» 8 4 (» 8 4
a)
b)
Fig.2. a). The table of binary relations between the classes S3 - S8 under addition. b). The table of binary relations
between the classes
S3 S8
under multiplication. The cells
corresponding to the class S6 are darkened.
Given in the form of tables binary relations between classes can be used to calculate tables for higher arities. Here we will limit ourselves to the example tables of ternary relationships between classes. such tables are no longer flat but three-dimensional 6 x 6 x 6 (Fig.3). For a flat rendering will use the so-called "slice-type".
We are interested in the question: To which class belongs the operation result of "random" pairs or triples of natural. It was natural therefore to consider the table of binary relations as the set of possible outcomes at a probability of 1/6 of
each class of the operand. Then the ratio of the number of indexes of each class to 36 is equal to the probability to get the result in this class of natural.
For binary relations under addition the probability of the result from each class is equal to 1/6, since #(r, Tb) = 6 . For binary relations under multiplication we have different picture, because:
#(3, Tb) = 5; #(4, Tb) = #(8, Tb) = 6; #(5,T) = #(7, Tb) = 2; #(6, Tb) = 15;
Hence the probability of the product of two randomly selected natural to be in the class px (rt, b):
px (3, b) = 5/36; px (4, b) = px (8, b) = 1/6;
px (5, b) = px (7, b) = 1/18; px (6, b) = 5/12;
similar calculations for ternary relations under multiplication give the following results:
#(3,T) = 19 ; #(4, Tt) = #(8,T) = 28 ; #(5,Tt) = #(7, Tt) = 4 ;
#(6,T) = 133 ;
Probability to the product of three randomly selected natural to be in the class px (rt, t): px(3,t) = 19/216;px(4,t) = 7/54 ;px(5,t) = 1/54; px (6,t) = 133/216; px (7,t) = 1/54 ; px (8,t) = 7/54;
At the construction of natural of some random class in the degree the response to a similar question about the probability of the class for the result is contained in the table on the main diagonal. In particular for the construction of the natural square (a binary relation under multiplication) the diagonal is a set D2b = {S3, S4, S7, S6, S7, S4}. which corresponds to (1).
s5, S8 i A.
Thusp2(3,b) = 1/6; p2(4,b) = 1/3;p2(5,b) = 0 ; p2(6, b) = 1/6; p2 (7, b) = 1/3; p2(8, b) = 0; Similarly: = {S3, S4, S5, S6, S7, S8}
P3(3, t ) = p3(4, t ) =... = p3(8, t) = 1/6;
and
Fig. 3. The tables of ternary relationships between classes under multiplication (the top row) and under addition (the lower row). In the top row the cells corresponding to the class S6 are darkened, to emphasize the property of idemponentiality.
III. New classification and the Goldbach primes Prime pairs g1 and g2, the total of which is equal to a given even natural number, play a significant role in researches related to the binary Goldbach problem. These pairs are called the Goldbach primes.
In the proposed classification, as shown in [1], all primes more 4 belong to the classes S5 and S7. As well as compound of these classes they satisfy to the binary relations under addition (Fig.2). All even greater than or equal to 4 belong only to the classes S4, S6, S8. It is easy to see (Fig. 4) that even from the class S4 are Goldbach prime numbers conjecture only from S5 (G(S4) e S5), and the even-numbered from S8 only from S7 (G(S8) e S7). Even from S6 always have one Goldbach prime from S5, and the other from S7, including this applies to primes-twins.
S3 S4 S5 SiG^SS
S3 S4@ SfiQra S8
53 54 is?) 6 7 s 3 i 5 53 54 0 S6 6 7 3 5
7 8 1 1 4 6 7 8 5 4 \ i 6
1 5 c 7 7
S6 © S8
3 4 5 6 8 3 4 6 1 8
ii 3 \ S 7 Q O 3
(ST) S8
5 6 7 8 3 4 5 6 7 8 3 4
a)
b)
Fig 4. a). Dedicated binary relations under addition: (S5 + S5) e S4 and (S7 + S7) e S8; b). Selected binary
relations (S5 + S7 ) e S6 and (S7 + S5 ) e S6 ;
Fig 5. The General scheme of the proposed classification of natural numbers.
The above natural classification allows to estimate the number of joint properties of primes and compound when operating on very large natural numbers. And at the same time to raise the question about the architecture of superprocessor focused on work with very large integers.
References
[1] G. G. Ryabov, V. A. Serov, "On natural numbers structure on the basis of six arithmetical progressions," International Journal of Open Information Technologies, 2016, vol. 4, no. 4, pp. 49-53. Available (in russian): http://inioit.org/index.php/i1/article/view/277
[2] G. G. Ryabov, V. A. Serov, "Composition of Infinitary Structures," Numerical methods and programming, 2015, vol. 16, pp. 557-565. Available (in russian):
http://num-meth.srcc.msu.ru/zhurnal/tom_2015/pdf/v16r452.pdf
[3] G. G. Ryabov, V. A. Serov, "On composition of infinitary structures and symmetries between primes," International Journal of Open Information Technologies, 2015, vol. 3, no. 12, pp. 4-6. Available: http://inioit.org/index.php/i1/article/view/248/197
IV. Conclusion
The General structure of the proposed classification is shown in Fig.5, in which:
P(S5 ) —primes of the class S5 ; P(S7 ) —primes of the class
S7 ; P(S5 u S7 ) — twin primes;
