Symmetries of the Number Model. ChKSS. Part IV

We continue our acquaintance with the number model and its properties, and specifically, with symmetries at different levels of the model representation: areas of rows, individual rows, elements of one row and elements of different rows. For readers who have read my previous article 1 (On the decomposition of the number model), article 2 (On symmetries…) and others, it is suggested to continue their acquaintance with the problem of modeling and studying numbers. The object of the natural series of numbers (NSN) is so rich in known and completely new properties that their listing would require a lot of space and time.

Consideration of a specific property in detail limits the author on the one hand by the available knowledge, and on the other hand by the limited volume of the publication. Nevertheless, there is a desire to show readers a detailed picture of the manifestations of such a property of the NRCH as symmetry in the behavior of the elements of this remarkable object.

For example, has anyone paid attention to the sequences of quadratic residues (QRR) of elements of the NRF in different modules, when we consider the model as a fragment of the NRF or a finite numerical ring of residues modulo N. These squares follow in pairs R_OR₁ and obtain the form (21 pairs) for N = 1961. Pairs of KVK 484 = 22²; 529 = 23² And
625 = 25²; 676 = 26² are formed by adjacent numbers, for N=1961 they border the average deduction in the 4th layer r_ccc = 0; and for N=2501 in the 5th layer the average deduction r_ccc = 0.

Why in the second case? N=2501 squares follow first with inflections 0then with 1²=1,
4= 2²3²4² ? These squares lie in rows outside the trivial region of the TKVK and there are no multiples among them d_b.

The tables show the order of the KVV = KVK of complete squares, combined into pairs (top\bottom), in total 42 square (for N=1961) And 48 squares (for N=2501). Each square is obtained at some point X_O and implements a decision interval (DI) that ensures the solution of the large number factorization problem (LFP) Ni.e. to calculate divisors N. Based on the law of distribution of divisors, we can write the relationship d_i = X_O ±√КВК and, if necessary, use the Euclidean GCD algorithm.

The purpose of the publication is primarily educational, informative, popularization of science, as well as the desire to attract new young (and not so young) minds to the ranks of researchers, to science, to arouse in such minds the desire to search for answers to emerging questions. The scale of the topic requires the introduction of reasonable restrictions on the presented material after a brief panoramic review of it

Introduction

As the text is presented, abbreviations (abbreviations) are used, which I have collected in one place for the convenience of the reader. I use these abbreviations in all articles on the problem of number factorization and information security.
SMM – list multi-line number model;
ZRD is the law of distribution of divisors of a number;
NRS – natural series of numbers;
Odd numbers sequence;
IB — information security;
KVV is the quadratic residue of a ring element modulo N;
KVK – quadratic residue – perfect square;
ChKSS — a quadruple of multiple adjacent rows;
KCHKV — a finite number ring of residues modulo N; TKVK is a trivial continuous region of SMM rows containing all KVKs;
TCCC is a trivial continuous domain of rows of the CMM containing all average residues that preserve the adjacency of factors;
DC, DIN, D0, are duplicates of the central, first (involutions), last (zero) SMM rows;
Idempotent — idempotent element, element e a ring, semigroup or groupoid equal to its square: e² = e.
Involution (from Latin involutio – folding, curl) – involutive element X rings, the square of which X² =1; a transformation that is its own inverse, and the square of the involution is equal to one in the ring.

It has already been mentioned that in the SM model at least five axes (positions, lines) of symmetry are established (materials for the first three positions have been described, reviewed and published):
– zero (bottom) line of the model;
– the central line of the model;
– a string of non-trivial involutions;
– the dividing line of four adjacent multiple pairs of lines;
– a line separating the lines of idempotents;

Each of the listed positions provides an independent option for composing lines with deductions, which are based on determining the number of the original and duplicated lines.

Dividing line of pairs of quadruple adjacent rows

The task remains basically the same – to localize each line of the HMM using new symmetries. The first two layers of the line border form pairs of multiple lines ChKSS
This dividing line is the common boundary of two pairs (4-fold) of adjacent rows with numbers X_O. Such a 4-ka has common KVV with a triple of multiple adjacent rows (row-double zero) and common values t And t_P with bordered non-trivial involutions.
The symmetry of this line is manifested for the double lines bordering r_ccc average deductions.
With different modules N The line numbers of the border lines are determined by the even involution and the bottom line of the CMM.

In the list of CM models (in columns with monotonically increasing values X_O X₁) there are values ​​that are multiples of the divisors N without restrictions on the parity of the multiplicity coefficient. In columns (T, T_P) consecutive odd numbers (SON) also have values ​​that are multiples of the divisors N, but only with an odd multiplicity factor k. The generation of the PNC elements occurs by summing a pair of adjacent numbers, and for the column R – by multiplying the terms.

T = {0 + 1 = 1, 1+ 2 = 3, 2 + 3 = 5, 3 + 4 = 7, …, n+(n+1) = 2n + 1},
R = {0 1 = 0, 1 2 = 2, 2 3 = 6, 3 4 = 12, …, n (n+1) = n² + n}.

For such two consecutive sums in T And T_P one of the terms is always common. For example, for t = 5 And t = 7 we have: 2 + 3 = 5, 3 + 4 = 7, common term 3. Variable p = t₁· t_O equal to the product of terms, one or both factors of which may be multiples of the divisor N.

In this case, one factor (this can be one of the divisors N) remains common to two (strings of) consecutive values. This makes a pair of consecutive values R, R_c multiples of a common odd divisor. Such values t are then converted into average deductions r_c and due to the specifics of the transformation of pairs of adjacent lines in positions R And r_c gets values ​​that are multiples of one of the divisors r_cv And r_cн (upper and lower). For the other divisor this is also true.

Such pairs are numerous and there may be different numbers between them in the SMM s non-multiple lines s = 0, 1, 2, …. There is a variant of placing multiples of different divisors of pairs of rows, when two adjacent pairs of multiples of different divisors of rows form a continuous quadruple of multiples of adjacent rows (ChKSS). In such a four, monotonous values t_P repeat (mirror) four column values t in the region of the row of non-trivial involutions. In other words, such a quadruple of rows and the quadruple of rows bordering the row of non-trivial involutions (including it itself) have coinciding counter-directed column segments T, T_P (Table B4).

Symmetry is oriented towards average residues from the TCCC. As before, relative to this line, pairs of rows of numbered model layers symmetrically move away from the line, but duplication of such rows leads to a different result than before. All corresponding pairs of duplicate rows are separated by a single row containing the average residue r_ccc from TSSS with the property of preserving the adjacency of factors.

Table BB4 shows only the beginning of such a list of deductions. r_c = 0, 2, 6, 12… in lines with numbers X_O (r_ccc = 0) = 1128, x_O (r_ccc = 2) = 883, X_O (r_ccc = 6) = 638, x_O (r_ccc = 12) = 393, … with their bordering pairs of duplicate lines. Pairs of SMM lines border the selected line layer by layer, gradually moving away from it.

4. Symmetries of the model. The dividing line of pairs of rows of the quadruple

We have already considered how rows with full squares in the positions of left residues are bordered, and we have considered the mechanisms for the formation and placement of duplicate rows for rows,
bordering lines and rows of symmetry. These were questions concerning the carrier of X. Now let us move on to another carrier of Tn of the elements of interest to us, namely, multiple divisors. The differences are significant, but all the more interesting for study. Let us explain how the duplicate rows with squares rл and average rccc residues are localized. What are the significant differences in localization. Between single multiples of different divisors of rows in CMM, gaps of non-multiple rows arise.

If the interval between multiple lines contains an odd number of lines, then according to the ZRD the middle line of the interval has a quadratic residue КВВ = КВК = r_l a perfect square equal to the square of the distance of the interval boundaries from the center. The boundaries of the interval are rows with elements that are multiples of different values ​​of the divisors (elements X₁ And X_O).
To some extent, a similar phenomenon occurs for average deductions. r_ccc. Also considered are intervals of non-multiple lines, but between doubled (pairs) adjacent lines of multiples of different divisors. If the interval contains an odd number of non-multiple lines, then the central (middle) line has an average residue r_ccc with the property of preserving the adjacency of factors. Let us explain this description with a numerical example.

Example K. The reduction module is specified N = 989. Fig. 1 shows a fragment of the SMM containing the elements named earlier in the text. Single multiples X₁ And X_O the row dividers in the model have numbers X_O = 253 = 11∙23, X_O = 258 = 6∙43, X_O = 276 = 12∙23 (filled with blue). Between them are intervals of 4-x and 17 non-multiple lines. For a larger interval (17 lines) there is a central point (line, its number X_O = 267). At this point the quadratic residue is equal to r_l = 267 ²(mod N) = 81 – a perfect square. Similarly, in X_O = 277, KVK = 576 = 24², what does the distance of the RI borders from the center mean 24 lines; X_O = 253 = 11∙23, x_O = 301 = 7∙43.

With multiple values ​​by t_P are also two pairs of adjacent lines the first pair with numbers X_O = 264, x_O = 265, where the bottom row of the pair contains the element t_P = 529 = 23∙23 and the second one with numbers X_O = 279, x_O = 280, where the bottom row of the pair contains the element
t_P = 559 = 13∙43. The multiples of the divisors are not the row numbers, but the values ​​of P and r_With = 713 = 31∙23, r_With = 253 = 11∙23 (both are multiples of the divisor d = 23) in the top pair and values r_With = 946 = 22∙43,
r_With = 516 = 12∙43 (both are multiples of the other divisor d = 43) for the bottom pair of lines.

The interval (gap) of non-multiple lines between these adjacent pairs is odd (equal to 13 lines), i.e. has a central point (line with number X_ots = 272). Meaning
r_ccc = 56 = 7·8 – is the product of the number of lines of the interval before the center by the number after the center, including the boundaries of the center itself. On the other hand, this value is equal to the difference of the KVV r_l (272) lines and rlo, i.e. 798 – 742 = 56. In table A4, the values ​​are highlighted in shaded (green) r_ccc intersection lines of trivial regions TKVK ∩ TSSS = {42, 272}
Bordering of duplicate lines with average deductions of the form (R_ccc) in pairs of lines (top\bottom)

Table A4 KVV pairs of lines, layer by layer (11 layers) bordering lines with average deductions (r_sss) is given below. The table is formed starting from the middle (central) line, in the cells of which the values ​​are entered in lexicographic order. r_sss = 0, 2, 6, 12, 20, 30, …, 272, …,812, 870, 930. The values ​​are indicated next to them. t SMM lines containing these r_sss. Then from SMM to the cells of columns (above\below r_sss) are entered layer by layer (i = 1(1)I(N)) KVV ​​border lines.

The layers of the KVV border rows are arranged horizontally in the table. Such a table can be easily transformed into a table with a vertical arrangement of the KVV layers. In the transformed table, each i-th layer of the KVV forms i-th column. The rows (pairs upper\lower) of such a table correspond to the values ​​(names) r_sss. To each r_sss The layer is allocated two rows (cells) in each column. The two central rows are for r_sss = 0 – adjacent. They are adjoined from above and below by the KVV of the bordering rows of the first layer for
r_sss = 2, then for r_sss = 6 and so on until the end of the list r_sss.

Why is such a table transformation considered? The fact is that in the HMM all layered sequences of the KVV (table columns) are already contained in an explicit form. When forming the HMM they arise automatically. It is only necessary to specify the central pair (r_sss = 0) each layer (column.)

Table A4 contains the border KVV (11 layers) of the duplicate rows of the average deductions. Each pair of border rows corresponds to duplicates (table B4, each layer is a vertical). The fifth layer of duplicate rows contains mainly KVV=KVK – complete squares (N=989)

Such pairs are adjacent for the column of the 1st layer and for all other layers. For
r_ccc = 0 this is KVV=443, KVV=54; For r_ccc= 2 this is KVV =834, KVV = 656 in the same 1-m column.

Localization of deductions. Considering the SMM table KVV, let's pay attention to the distribution of duplicate lines containing KVV = KVK = r_l (i.e. duplicates from TKVK) and duplicate lines containing averages (r_ccc) residues (doubles from another trivial region of the TCCC). We will keep in mind that the regularity of the distribution of rows containing multiples of divisors N holds for all components N.

In this case, we will take into account that the elements of the rows are multiples of the divisors in the role X₁ And X_O always meet in single lines (excluding rows with idempotents, in HMM they are adjacent) and can have integer even and odd coefficients, while elements multiples of the same divisors in the role of average residues (r_cr_c-1) always appear double (pairs) of adjacent rows and in the bottom line of such a pair there is always an element t_P multiple of a divisor with an integer odd multiplicity factor.

The described multiple lines, single and paired (highlighted by filling), containing elements that are multiples of the divisors, serve as a kind of background, a grid, on which complete squares appear r_l = KVK and average residues with the property of preserving the adjacency of factors after reduction r_c = r_ccc. The squares form the relationships of the law of distribution of divisors (ZRD) in the NRC, and the average residues ensure the speed of searching for squares and the uniqueness of the solution ZFBC. The task remains basically the same – to localize each line of the HMM using new symmetries. The first two layers of the line border form pairs of multiple lines ChKSS.

Regions of left residues, relative to the average, preserving the adjacency of factors

Previously, we considered the regions of attraction for the KVK with centers at points that are multiples of a larger divisor. A_i = id_b. The interrelationship of the regions, manifested by symmetrical inclusion in the numbered layers of the KVV, was described in detail. The KVV elements of the rows in the columns often change places. The dependencies look completely different when considering the borders of the average residues of the type r_sss. Although everything r_sss are distributed according to the SMM list for different N in a unique way, but for tables everything r_sss are placed in the central line in lexicographically ordered order.

The main difference is that each column-layer of table B4 of vertically arranged KVV layers starts not in the rows where the average deductions are placed, but from some dividing lines between pairs of SMM rows, specified by the current X_O numbers. The first layer of border lines starts from the dividing line of four adjacent lines that are multiples of different divisors. The line runs between the lines with numbers X_O =194 And X_O=195. The complement to the involution of this pair of rows is the pair X_O = 106 And X_O = 105. The line between this pair of lines generates a second layer of borders for r_sss.

Next, we increase the numbers of this pair by an even involution (In = 300) and we get another pair of lines with a dividing line X_O = 406 And X_O = 405, generating line of the third layer. The next layer with an odd number is layer five, generated by the line between the lines with numbers X_O = 16 And X_O = 17. Even layers fourth and sixth are formed by lines between complementary lines before involution (X_O = 284 And X_O = 283) and increased by involution (In = 300), i.e. X_O = 316 And X_O = 317.

Further, by analogy, we can continue, alternating even and odd layer numbers. The 8th layer is generated by the line between the rows with numbers X_O = 72 And X_O = 73. Layers 7 and 9 are defined by the lines between the rows with X_O= 372 And X_O= 373, x_O= 227 And X_O= 228, i.e. for the 7th
X_O = 300 +72 = 372 And X_O = 300 = 73 = 373, and for the 9th X_O = 300 – 72 = 228 And X_O = 300 –73 = 227.

The next three layers 10, 11,12 we set it first for eleven-th layer X_O = 161 And X_O = 162, for the 10th we add involution, and for 12 let's define the complement to the involution X_O = 461 And X_O = 462,
X_O = 300 – 161 = 139 And X_O = 300 – 162 = 138.

Representation of symmetrical regions of the KVV associated with average residues r_sss SM-models differ significantly from those considered earlier, both in their structure and in the dependencies of their elements. It is convenient to perform a description using the table of attraction areas of the KVV lines, bordering layer-by-layer (in 6 layers) ordered average residues r_sss

We find intervals between pairs of multiple double lines of even length. The center of the interval is determined by the line dividing it in half, passing between the lines of the upper one with the number X_donkey and lower X_donkey.

The first such line is the dividing line of the ChKSS. Lines with such KVV are bordered r_ccc = 0 in the 1st layer. The numbers of these lines X_o1 = X_1o –In = 495 –300 = 195 And X_o1 = X_oo –In = 494 –300 = 194. The KVV column from this line (up\down) borders everything r_ccc in lex order by lines of layer 1. The second line (for the 2nd border layer) lies between the complementary lines up to module N, i.e. from above X_o2 = 300 –195 = 105 and from below X_o2 = 300 – 194 = 106. The third line increases the line numbers by In, i.e. X_o3=300+105 = 405 And X_o3= 300+106 = 406. In general, it would be more correct to write the conditions like this X_o2 = 300 ± 105 and x_o2 = 300 ± 106, Where
X_o2 = 105 And X_o2 = 106. Then for layers 4 and 6 we find numbers X_o5 = 16 and x_o5 = 17, x_o4.6 = 300 ± 16 and x_o4.6 = 300 ± 17, Where X_o5 = 272 = 16∙17 = r_ccc the highest average deduction.

The main feature of the duplicate lines of such diverging pairs of border lines (see table A4) is that they become adjacent (X_O (r_ccc) = ± 1 upper\lower) with rows that initially include average deductions in duplicate rows in order of monotonous increase of average deduction r_c = 0, 2, 6, 12, …, and after they are exhausted from the TSSS, and for lines with other deductions.

Let us consider in more detail the property of the CM model lines, which manifests itself with duplicate lines, layer by layer bordering the line dividing pairs of lines of a quadruple of adjacent lines, multiples of divisors N lines.

Each line is a duplicate r_ccc ϵ TCCC V 5-m layer of border line pairs contains KVV=KVK, the first degrees of which for r_ccc = 0 adjacent, and their sum is equal to ½ (d_m + d_b) half the sum of the divisors N. For N=2501 in line with r_ccc = 0 this is KVK 25² And 26²and for r_ccc = 2 this is KVK 24² And 27², r_ccc = 6 this is KVK 23² And 28², r_ccc = 12 this is KVK 22² And 29² etc.
25+26 = 24+27 = 23+28 = 22+29 =½(41+61) = 51.

Example 8. (Symmetry of the dividing line of 4 adjacent multiples of divisors N couple of lines). Let N = 2501 = 41 61. This type of symmetry in the SM model is determined by the middle line of 4-fold rows (Table. B4, BB4highlighted by filling). The leading role is played by the middle deductions. The triplets of deductions of the rows of the 1st layer of the quadruple are duplicated in the bordering rows of the “zero” row containing r_sss = 0.

These are the average multiples of the rows 4-ki also contain elements (blocks) X_O = r_With = 246 = 6·41 And
X₁ = r_With = 2257=37·61, the same elements are contained in the bordering lines of the line of non-trivial involutions. Lines 2-th layer (outer rows of the quadruple) contain triplets of row residues that are contained in the border rows of the middle residue r_With = 2.

So middle line of the four (two adjacent pairs of lines, highlighted by filling) multiples of different divisors N rows specifies the symmetry of the position (the value of the distance from it) of other rows of the model, the duplicates of which lie in other areas of the CM model and are separated by only a single row containing the duplicate row of the average residue r_c from the trivial domain of residues preserving the property of adjacency of factors.

In this case, the quadratic residue of the number of the dividing line is bordered by the quadratic residues of the numbers of the original symmetrically (layer-by-layer) distant lines, a multiple of four. In the table B4 zThe two pairs of lines of the CM model are highlighted in bold. The top pair of lines is a multiple of the smaller divisor dm = 41the lower one is a multiple of the larger divisor db = 61. KVV pairs of internal rows of the quadruple border the duplicate row of the last (zero) row (Table BB4). The following pairs of rows bordering the line have duplicate rows with rows placed between them containing average subtractions from the TCCC, observing the order of their following one after another, starting with the values r_c = 0, 2, 6, 12, 20, … until the list of TCSS is exhausted.

In the following table BB4 (top row) the left and right residues from the zero row have changed places; the right residue is KVK= 625 began to play the role of the center of the decisive interval; the average residue took on the value r_c (1123) = 1250 = x_oo (1250). In normal position (double of average deduction 1-th row of the CMM) it is equal to 1 + r_By = 1 + 625 = 626; odd involution (In = 245 = (122 = 2 61) + (123 = 3 41)) is decomposed into a sum of multiples of different divisors N terms. The duplicate of the zero line is bordered by lines with the KVK 25² And 26²placed in 5-th layer.

Between duplicate lines with average deductions r_With with the SSS the distance is equal to an odd involution, 1128 – 883 = 883 – 638 = 638 – 393 = 393 – 148 = … = 245.

All three subtractions of the duplicate row repeat the subtractions of the original row. Thus, for the border of the duplicate row of the zero row, the two middle rows of the 4-ki (table AT 4) repeat the deductions. In the table, these lines are adjacent (not separated as in the table BB4 duplicate of the zero row). The average deductions of a pair of internal duplicate rows of the table are tripled to
0+3 246 = 738, 3·2257= 1769; when edging the next element r_With = 2, increase in 5 once for r_With = 6, that is before 2 + 246 5 =1232 And 2+ 1283 and then grow with odd k coefficient, i.e.
k = 1(2) … until the elements of the TCSS are exhausted. The numbers of rows symmetrical with respect to the middle row in a quadruple of adjacent rows of the model are determined by multiplying half of the even involution by the ordinal coefficient k, corresponding to the average deduction in the TSSS. This coefficient k takes consecutive values ​​of odd numbers from k = 1, 3, 5… and increases until the average deductions in the TSSS, following in order of increasing values, are exhausted.

Example 9. (Symmetry of the dividing line of 4 adjacent multiples of divisors N couple of lines). Let N=989. In the table section A4 symmetries for the middle line of the four multiples of different divisors of the rows (first 17 columns) the initial values ​​are indicated r_With average residues with the property of preserving the adjacency of factors (TCCC). The sets of these residue values ​​can be numbered, they start from r_With = 0, 2, 6…, get numbers n = 1, 2, 3, … and are the same for all numbers N. With growth N only their number in the model list changes. Let the duplicate of the average deduction from the TCSS be selected r_With = 30 = 5·6. His serial number
n = 6. Duplicate lines bordering the middle line of division of multiple adjacent lines 4-ki lying in 6-m layer, have KVV r_lv = 117 And

r_ln= 440 then the coefficient k = 2 6 – 1 = 11These border lines are separated from each other. 10-th intermediate lines. Duplicates of these lines are separated by only one line containing a duplicate of the average deduction r_sss = thirty. Multiplicity factor
k = 2n–1Where n – the current number of the average deduction from the TSSS. This number is for r_sss equal to the greater of 2-x adjacent factors. The value n For r_sss is defined as the larger adjacent factor in the list of average residues. The beginning of this list is as follows:

r_sss = {0·1 = 0, 1·2 = 2, 2·3 = 6, 3 4 = 12, 4 5 = 20, 5 6 = 30, 6 7 = 42, …}.

Thus, the deduction r_sss = thirty gets the current number n(30) = 6. Then, k = 2n – 1 = eleven.
This means that the average deduction r_sss = thirty contained in line number X_O = 661, that is the number X_O = ½k300(mod 989) = 661. Depending on the selected value of the composite module N a duplicate row of the model containing this average deduction r_sss = thirtywill receive a different number.

For N=989 non-trivial even involution is equal to X_O = 300and its half 150. Let's make sure that X_O is an involution: X_O² = 300²(mod 989) = 90000 = 1 + 91 989 = 1. Now let's return to determining the line number X_O (r_With = 30) = 150k = 150·11 = 1650. Let's perform a modulo reduction N, what gives X_O (r_With = 30) = 1650 – N = 1650 – 989 = 661. For a line with this number, you can calculate the average deduction using the formula

r_With (X_O= (N – 661)) = ¼ ((N – 2 x_O)² – 1) = ¼ ((989 – 2 328)² – 1) =27722(mod 989) = thirty.
For this row, the quadratic residue and the residues bordering it are easily determined
r_l (661) = (X_O (r_With = 30))²(mod 989) = 661²(mod 989) = 772.

Conclusion

Here we describe another object of the SM model that generates symmetries. The fact that there is a variety of types of mechanisms that generate row symmetries on a fairly static set of model rows is surprising. Within a given composite number N and within the SM model, the rows do not move and do not exhibit obvious signs of symmetry.

An analysis of the reasons for the manifestation of a number of features in the behavior of individual elements and entire lines led to the need for a careful consideration of particular details. For example, if the divisors of N (prime numbers) are related by a linear dependence, then
p=23, q=1+2p =47module N=1081the idempotent turns out to be equal to q = 47;
p=67, q=1+4p =269module N=18023then the idempotent is equal to q = 269; or
p=13, q=2+3p =41module N=533then the involution turns out to be equal to q – 1= 40; or
p=61, q=2+5p =307module N=18727, then the involution turns out to be equal q – 1= 306.

Proving these facts is a waste of time, distracting from the main task (Fermat and Euler also did not prove everything and made mistakes). But there is benefit from the facts, when studying them, symmetries emerged that I had never heard or read about before, but I managed to understand some of them. Which I submit to the public's judgment.