New invariant of a natural number. Theorem and proof
Earlier on Habré was published the author's work on the invariant number ( here ). Even earlier, in the work [1], the information about the original concept of modeling a natural number of numbers and a single number is given in order to establish properties that are weakly dependent or completely independent of the digit capacity of numbers. Previously, theorems were not cited to prove the truth of the statements used by the author in his works. An analysis of the comments to the works showed how incredulous the readership is to similar works and statements.
The limits of such a distrust from the estimates “Nonsense; Caution: pseudoscientific nonsense; Transfusion from a sieve ... "to" Yes, it certainly seems interesting to me if ... ". These are only a few opinions of readers who have expressed their perception of the work, their opinion about it. Thank them for their attention to the work. By the way, these opinions and assessments were approved by many other readers who did not remain indifferent. Thank them too for your attention. So this prompted me to give a theorem on the φ-invariant and its proof.
I really hope that possible proof of the truth of the assertion about the f-invariant (a new property of a number with a calculated indicator) is possible, will somewhat correct the opinions of readers and raise their doubts about the correctness of their, perhaps, very hasty initial estimates. Each author, who devoted a lot of time and effort to a certain work, perceives his work almost as obvious over time, the presentation of the work, while striving to make it short, contains a lot of details “by default”, which makes it apparently inaccessible for other people to understand.
The topic of this and my other published works directly concerns security and, in particular, information security at all levels where cryptographic protection is used. The fact that publications contain mathematical relationships at the level of elementary mathematics does not make the problem less complex or more accessible. Simply, the author seeks to make the presentation accessible and clear to a wider range of habrovchan readers. My work deals with a completely new approach to solving a private, but important security problem ( here ), which is closely related to many other problems. The author considers the direction in the field of factorization that is developing at the world level to be a dead end, and the practice (criterion of truth) confirms this.
We need a new view on the problem of factorization, on the natural series of numbers (NPS), on the numerical system as a whole, which, as it turned out, is practically not studied. The theory of the NPS and a separate composite odd natural number (snnch) is necessary, models of these objects are necessary, taking into account the requirements and requirements of modern practice. Nothing like this today just copy nowhere. The author, to the best of his abilities and capabilities, creates independently, so far hidden from the attention of others and in a series of publications shares his findings with the public, understanding that everything new, most often, is hosted at all times. The reasons for this phenomenon are very diverse and discuss them here and now is not the time.
')
I really hope to find associates among programmers, since I am not a programmer myself. Material about new results and approaches is distributed in small doses in the articles of the author. For those who find this new and promising direction interesting, there is an opportunity to get acquainted with the content of my works on Habré. The new concepts with their definitions include: classes (left and right) of odd numbers, contour, limit contour, semi-contour, LFD interval, their length, numbering, boundaries, f-invariant number, interval and numbering model of composite odd number [3, 4], formulas for calculating the characteristics of the listed objects. The introduction of these new concepts and designations provides a solution to a new range of problems formulated with respect to natural numbers, the main of which is factorization of the number.
The seemingly simple questions about natural or whole numbers have nowhere to find the answers. Examples of questions are the following.
How to define non-trivial involutions and idempotents of a finite number residue ring modulo snn N without first decomposing N into factors? How to find the comparison roots in absolute value in such a ring, without total sorting of the elements? How to find quadruple quad residues whose values are generated by a single square without factoring N? For now, the author has to look for answers to these and other questions on his own. Waiting for feedback from the interested reader (s) on the substance of the formulated questions and possibly any other. To solve the essence of the questions, a different less elementary mathematics is required, but for the time being, it seems to be premature to touch it here.
Let us introduce the notation for the numbers of the extreme contours of the interval for the snn N: for the smaller number s - start, for the larger number f - finish. Obviously, the left boundary of the multi-contour interval representing the left (N l ≡ 3 (mod4)) odd number coincides with the left border of the smaller s-th contour G l (s) = (2s - 1) 2 , and the right border of this multi-contour interval with the right the boundary of the left semi-contour with a large number f, i.e. Rn (f) = (2f) 2 .
The left border of a multi-contour interval representing the right (N p ≡1 (mod4)) odd number N p coincides with the left border of the right half-contour of the smaller s-th contour G l (s) = (2s) 2 , and the right border of the interval for N coincides with the right boundary of the larger contour number f, i.e. (f) = (2f + 1) 2 . The accepted notation is somewhat cumbersome, but semantically justified and convenient for understanding and memorizing.
In the previous paper ( here ) for the numerical model of the natural odd composite number N, i.e. semi-contour in the limit contour we get the sum of numbers by moving from contour lengths to their numbers in the form k d / 2 + Σ t i k i , where t = m − 1, i = 1 (1) t.
Theorem 1 . (On invariant of composite odd number. Single interval). Arbitrary interval in the NPS with a length of N, composed of successive contours of the numerical axis and with squares of natural numbers as the boundaries of the interval, corresponds to the sum k d / 2 + Σ t i k i = k n (N) / 2, where t = m- 1, i = 1 (1) t, numbers forming an interval of contours, and half a contour number with a complementary k d / 2 semi-contour, equal to half the number of the limit contour (f-invariant) of the number N.
Proof. Let us execute separately for natural odd right N p and left N l composite numbers. The idea of the proof is as follows. On the one hand, it is shown that the number N is represented as the difference of squares (boundary points of the interval) in the most general form. On the other hand, the sum of the contour numbers and the half-contour is calculated, which, by equating it to half the number of the limit contour for a composite number N = N p or N = N l and then converting the boundaries, also reduces the difference of the boundary points of the extreme elements of the sum equal to N.
Let's start with the right number N = N p . Let us substitute in the sum of the numerical model the notation s and f for the numbers of the extreme contours introduced earlier in the model of odd composite numbers. We use the formula for the sum of the elements of the segment NPSH given in [2, p. 160].
s / 2 + (s + 1) + (s + 2) + ... + f => 1/2 (f + s +1) (fs-1 + 1) + s / 2 => 1/2 ( f 2 + sf + f -sf -s 2 -s) + s / 2 = 1/2 (f 2 + fs 2 ).
We transform the final expression to the form corresponding to the difference of the boundaries of the interval for the studied number N p , namely, (2f + 1) 2 - (2s) 2 . This is achieved by equating the expression found to half the number of the limit contour of the number N p , that is, k p (N) / 2 = (N p -1) / 8 = 1/2 (f 2 + fs 2 ).
The middle and right sides of the equality are transformed to the form of the difference of boundaries representing the number of the interval
N p = 4f 2 + 4f + 1 - 4s 2 = (2f +1) 2 - (2s) 2 .
This completes the proof for the case N = Nn.
Now we carry out the proof for the case N = N l . Let us substitute in the sum of the numbering model the notation s and f for the numbers of the extreme contours, as before. Then for this case we have:
s + (s + 1) + (s + 2) + ... + (f-1) + f / 2 => 1/2 (f-1 + s) (fs) + f / 2 => 1/2 (f 2 + sf-f-sf-s 2 + s) + f / 2 = 1/2 (f 2 -s 2 + s).
Perform the transformation of the final expression, leading it to the form of the difference of the boundaries of the investigated number N = N l , namely, to the form (2f) 2 - (2s - 1) 2 . This is achieved by equating the expression found to half the number of the limit contour of the number N l , that is, kn (N) / 2 = (N l + 1) / 8 = 1/2 (f 2 + s - s 2 ).
The middle and right sides of the equality are transformed to the form of the difference of boundaries representing the number of the interval
N l = 4f 2 - 4s - 1 - 4s 2 = (2f) 2 - (2s - 1) 2 .
This completes the proof for the case N = Nl and the theorem is completely proved.
The proof is based on the calculation of boundaries for a multi-contour interval and it is shown that the difference between such boundaries at an arbitrary interval leads to the same result for an interval with a length of N. Another way of proving the theorem is possible using the principle of forming partitions of the number kn (N) / 2 into different the number of parts m i with restrictions of the form (k ij +1 = k i (j + 1) ) on the part of the partition. In this approach, the f-invariant kn (N) / 2 numbers of N plays the role of the split number. Indeed, the sn N itself can be divided into parts (contours) in several ways, while the parts of the partitioning will satisfy the imposed requirements (restrictions on values).
Theorem 2 . (On the invariant of a composite odd number. The set of intervals). All multi-contour intervals with a constant length N and current numbers i = 1 (1) ..., representing a composite odd positive integer N, in different regions of the number axis, having as their borders squares of natural numbers of different parity, correspond to the sum of different numbers m i (numbers k ij ) of successively following one contour (k ij +1 = k i (j + 1) ) and one extreme semi-contour such that the values of the sums are constant and depend only on the number N.
k di / 2 + Σ t j k ij = k n (N) / 2, where t = m i -1, j = 1 (1) t, i = 1 (1) ...
The proof is carried out using Theorem 1 by directly constructing such sums.
The theorem states statements about several facts:
- first, for snn N there is (i> 1) more than one NnCh representing NnCh interval in different parts of the NPS;
- secondly, each of such intervals corresponds to a continuous sequence of contours (with numbers k ij ), each sum of which numbers and semicontour numbers k di / 2, contains the number m i and the composition of the terms different from the other intervals;
- thirdly, the boundaries of all intervals are different pairs of squares of natural numbers of different parity;
- fourth, the length of all representing intervals is the same, equal to N, and the corresponding sums of contour numbers are also constant and are determined only by the value of the Φ-invariant of N, but not by the location of the interval in the NPS;
- fifth, all the terms in different amounts (except for the number of the extreme semi-contour) represent the NPS segment, and monotonously increase by one from one smaller term to the next higher term (k ij +1 = k i (j + 1) ).
In other words, the value of the sum of contour numbers and the number of additional half-contour for snn N is invariant with respect to the number of terms, to the location of the interval in the NPS, ie to the values of the boundaries of the interval, to the values of the terms in the amount.
Example 1 Set to snnch N = N l = 231, the extreme half of the right. The number 231 corresponds to the number of the limit contour kn (N = 231) = 58. The F-invariant snnch is equal to kn (N) / 2 = 29, i.e. half the number of limit contour. It is required to form partitions of the number 29, subject to the restrictions into parts, for the Φ-invariant.
Beforehand, we write out the representation of the f-invariant snn N by the sums of the contour numbers. There are three such amounts:
29 = 3 + 4 + 5 + 6 + 7 + 8/2;
29 = 7 + 8 + 9 + 10/2;
29 = 19 + 20/2.
Since the contour numbers that follow one after another, and the half of the number of the extreme contour corresponding to the left contour of the contour with a large number, are summed up, the represented number N is easily determined as the sum of the lengths of adjacent contours and the semi-contour:
- for the first partitioning N = 8 • 3 + 8 • 4 + 8 • 5 + 8 • 6 + 8 • 7 + 8 • 8/2 -1 = 24 +32 +40 +48 +56 +32 -1 = 231;
- for the second partition, N = 8 • 7 + 8 • 8 + 8 • 9 + 8 • 10/2 - 1 = 56 + 64 + 72 + 40 - 1 = 231;
- for the third partitioning N = 8 • 19 + 8 • 20/2 - 1 = 152 + 80 - 1 = 231.
Such calculations show that the number 231 is representable in the NPS by two multi-contour intervals and the third interval consists of one complete contour and one half-contour of the limit contour adjacent to the right. The length of each interval is equal to the difference of its boundaries (i) - (i), i.e. the number N is represented by the difference of three (i = 1 (1) 3) different pairs of squares (interval boundaries).
Further, recalling that the boundaries of the contours and half-contours are full squares and using formulas for the boundaries of the contours and half-contours, we define the values for the boundary elements of each representative interval.
For the 1st interval, we have 29 = 3 + 4 + 5 + 6 + 7 + 8/2; G P1 (8/2) = (2 • 8) 2 = 256; G l1 (3) = (2 • 3 - 1) 2 = 25; and (i = 1) - (i = 1) = (16) 2 - (5) 2 = 256 - 25 = 231 = (16 + 5) (16 - 5) = 21 • 11 = 231.
For the 2nd interval, we have 29 = 7 + 8 + 9 + 10/2; G n2 (10/2) = (2 • 10) 2 = 400; G l2 (7) = (2 • 7 - 1) 2 = 169; and (i = 2) - (i = 2) = (20) 2 - (13) 2 = 400 - 169 = 231 = (20 + 13) (20 - 13) = 33 · 7 = 231.
For the third interval, we have 29 = 19 + 20/2; 3 (20/2) = (2 • 20) 2 = 1600; G l3 (19) = (2 • 19 - 1) 2 = 1369; and then G p (i = 3) - G l (i = 3) = (40) 2 - (37) 2 = 1600 - 1369 = 231 = (40 + 37) (40 - 37) = 77 • 3 = 231 .
Having bounds (pairs of squares) for each of the three intervals for one night, N, we easily get different decompositions of the number N into factors.
There is another interval with boundaries - squares, between which lies a composite odd number N = 231. Its boundaries - the boundaries of the limit contour for the number 231 are quite simple: the left is H l4 (k = 58) = ((231 - 1) / 2) 2 = (115) 2 = 13225, right 4 (k = 58) = ((231 + 1) / 2) 2 = (116) 2 = 13456, The length of the interval-half-contour in the limit contour is equal to
N = G n4 (k = 58) - G l4 (k = 58) = ((231 + 1) / 2) 2 - ((231-1) / 2) 2 = (116) 2 - (115) 2 = 13456 - 13225 = 231.
The considered material is a different from existing traditional approach of modeling NPS and snn N, which has a novelty. The numerical example illustrates the scheme of actions that is easily transformed into an algorithm with the possibility of eliminating the search of variants. It is only necessary to close the question (creating a program) to obtain special partitions (or) even a single special partition of the Φ-invariant. The way to solve this problem is proposed ( here ).
Literature
[1] Vaulin A.E., Pilkevich S.V. "The fundamental structure of the natural series of numbers" - Intellectual systems. Proceedings of the Seventh International Symposium. Ed. K.A. Pupkova. - Moscow: RUSAKI, 2006. - p. 384-387
[2] Bronshtein I.N., Semendyayev K.A. Handbook of mathematics for engineers and students of high schools. _M .: GITTL, 1954. -608
[3] Hall M. Combinatorics. -M .: Mir, 1970. - 424 p.
[4] Andrews G. Theory of partitions. - M .: Science GRFML, 1982. - 256 s.
The limits of such a distrust from the estimates “Nonsense; Caution: pseudoscientific nonsense; Transfusion from a sieve ... "to" Yes, it certainly seems interesting to me if ... ". These are only a few opinions of readers who have expressed their perception of the work, their opinion about it. Thank them for their attention to the work. By the way, these opinions and assessments were approved by many other readers who did not remain indifferent. Thank them too for your attention. So this prompted me to give a theorem on the φ-invariant and its proof.
I really hope that possible proof of the truth of the assertion about the f-invariant (a new property of a number with a calculated indicator) is possible, will somewhat correct the opinions of readers and raise their doubts about the correctness of their, perhaps, very hasty initial estimates. Each author, who devoted a lot of time and effort to a certain work, perceives his work almost as obvious over time, the presentation of the work, while striving to make it short, contains a lot of details “by default”, which makes it apparently inaccessible for other people to understand.
About the approach and its novelty
The topic of this and my other published works directly concerns security and, in particular, information security at all levels where cryptographic protection is used. The fact that publications contain mathematical relationships at the level of elementary mathematics does not make the problem less complex or more accessible. Simply, the author seeks to make the presentation accessible and clear to a wider range of habrovchan readers. My work deals with a completely new approach to solving a private, but important security problem ( here ), which is closely related to many other problems. The author considers the direction in the field of factorization that is developing at the world level to be a dead end, and the practice (criterion of truth) confirms this.
We need a new view on the problem of factorization, on the natural series of numbers (NPS), on the numerical system as a whole, which, as it turned out, is practically not studied. The theory of the NPS and a separate composite odd natural number (snnch) is necessary, models of these objects are necessary, taking into account the requirements and requirements of modern practice. Nothing like this today just copy nowhere. The author, to the best of his abilities and capabilities, creates independently, so far hidden from the attention of others and in a series of publications shares his findings with the public, understanding that everything new, most often, is hosted at all times. The reasons for this phenomenon are very diverse and discuss them here and now is not the time.
')
I really hope to find associates among programmers, since I am not a programmer myself. Material about new results and approaches is distributed in small doses in the articles of the author. For those who find this new and promising direction interesting, there is an opportunity to get acquainted with the content of my works on Habré. The new concepts with their definitions include: classes (left and right) of odd numbers, contour, limit contour, semi-contour, LFD interval, their length, numbering, boundaries, f-invariant number, interval and numbering model of composite odd number [3, 4], formulas for calculating the characteristics of the listed objects. The introduction of these new concepts and designations provides a solution to a new range of problems formulated with respect to natural numbers, the main of which is factorization of the number.
The seemingly simple questions about natural or whole numbers have nowhere to find the answers. Examples of questions are the following.
How to define non-trivial involutions and idempotents of a finite number residue ring modulo snn N without first decomposing N into factors? How to find the comparison roots in absolute value in such a ring, without total sorting of the elements? How to find quadruple quad residues whose values are generated by a single square without factoring N? For now, the author has to look for answers to these and other questions on his own. Waiting for feedback from the interested reader (s) on the substance of the formulated questions and possibly any other. To solve the essence of the questions, a different less elementary mathematics is required, but for the time being, it seems to be premature to touch it here.
Substantiation of the approach
Let us introduce the notation for the numbers of the extreme contours of the interval for the snn N: for the smaller number s - start, for the larger number f - finish. Obviously, the left boundary of the multi-contour interval representing the left (N l ≡ 3 (mod4)) odd number coincides with the left border of the smaller s-th contour G l (s) = (2s - 1) 2 , and the right border of this multi-contour interval with the right the boundary of the left semi-contour with a large number f, i.e. Rn (f) = (2f) 2 .
The left border of a multi-contour interval representing the right (N p ≡1 (mod4)) odd number N p coincides with the left border of the right half-contour of the smaller s-th contour G l (s) = (2s) 2 , and the right border of the interval for N coincides with the right boundary of the larger contour number f, i.e. (f) = (2f + 1) 2 . The accepted notation is somewhat cumbersome, but semantically justified and convenient for understanding and memorizing.
In the previous paper ( here ) for the numerical model of the natural odd composite number N, i.e. semi-contour in the limit contour we get the sum of numbers by moving from contour lengths to their numbers in the form k d / 2 + Σ t i k i , where t = m − 1, i = 1 (1) t.
Theorem 1 . (On invariant of composite odd number. Single interval). Arbitrary interval in the NPS with a length of N, composed of successive contours of the numerical axis and with squares of natural numbers as the boundaries of the interval, corresponds to the sum k d / 2 + Σ t i k i = k n (N) / 2, where t = m- 1, i = 1 (1) t, numbers forming an interval of contours, and half a contour number with a complementary k d / 2 semi-contour, equal to half the number of the limit contour (f-invariant) of the number N.
Proof. Let us execute separately for natural odd right N p and left N l composite numbers. The idea of the proof is as follows. On the one hand, it is shown that the number N is represented as the difference of squares (boundary points of the interval) in the most general form. On the other hand, the sum of the contour numbers and the half-contour is calculated, which, by equating it to half the number of the limit contour for a composite number N = N p or N = N l and then converting the boundaries, also reduces the difference of the boundary points of the extreme elements of the sum equal to N.
Let's start with the right number N = N p . Let us substitute in the sum of the numerical model the notation s and f for the numbers of the extreme contours introduced earlier in the model of odd composite numbers. We use the formula for the sum of the elements of the segment NPSH given in [2, p. 160].
s / 2 + (s + 1) + (s + 2) + ... + f => 1/2 (f + s +1) (fs-1 + 1) + s / 2 => 1/2 ( f 2 + sf + f -sf -s 2 -s) + s / 2 = 1/2 (f 2 + fs 2 ).
We transform the final expression to the form corresponding to the difference of the boundaries of the interval for the studied number N p , namely, (2f + 1) 2 - (2s) 2 . This is achieved by equating the expression found to half the number of the limit contour of the number N p , that is, k p (N) / 2 = (N p -1) / 8 = 1/2 (f 2 + fs 2 ).
The middle and right sides of the equality are transformed to the form of the difference of boundaries representing the number of the interval
N p = 4f 2 + 4f + 1 - 4s 2 = (2f +1) 2 - (2s) 2 .
This completes the proof for the case N = Nn.
Now we carry out the proof for the case N = N l . Let us substitute in the sum of the numbering model the notation s and f for the numbers of the extreme contours, as before. Then for this case we have:
s + (s + 1) + (s + 2) + ... + (f-1) + f / 2 => 1/2 (f-1 + s) (fs) + f / 2 => 1/2 (f 2 + sf-f-sf-s 2 + s) + f / 2 = 1/2 (f 2 -s 2 + s).
Perform the transformation of the final expression, leading it to the form of the difference of the boundaries of the investigated number N = N l , namely, to the form (2f) 2 - (2s - 1) 2 . This is achieved by equating the expression found to half the number of the limit contour of the number N l , that is, kn (N) / 2 = (N l + 1) / 8 = 1/2 (f 2 + s - s 2 ).
The middle and right sides of the equality are transformed to the form of the difference of boundaries representing the number of the interval
N l = 4f 2 - 4s - 1 - 4s 2 = (2f) 2 - (2s - 1) 2 .
This completes the proof for the case N = Nl and the theorem is completely proved.
The proof is based on the calculation of boundaries for a multi-contour interval and it is shown that the difference between such boundaries at an arbitrary interval leads to the same result for an interval with a length of N. Another way of proving the theorem is possible using the principle of forming partitions of the number kn (N) / 2 into different the number of parts m i with restrictions of the form (k ij +1 = k i (j + 1) ) on the part of the partition. In this approach, the f-invariant kn (N) / 2 numbers of N plays the role of the split number. Indeed, the sn N itself can be divided into parts (contours) in several ways, while the parts of the partitioning will satisfy the imposed requirements (restrictions on values).
Theorem 2 . (On the invariant of a composite odd number. The set of intervals). All multi-contour intervals with a constant length N and current numbers i = 1 (1) ..., representing a composite odd positive integer N, in different regions of the number axis, having as their borders squares of natural numbers of different parity, correspond to the sum of different numbers m i (numbers k ij ) of successively following one contour (k ij +1 = k i (j + 1) ) and one extreme semi-contour such that the values of the sums are constant and depend only on the number N.
k di / 2 + Σ t j k ij = k n (N) / 2, where t = m i -1, j = 1 (1) t, i = 1 (1) ...
The proof is carried out using Theorem 1 by directly constructing such sums.
The theorem states statements about several facts:
- first, for snn N there is (i> 1) more than one NnCh representing NnCh interval in different parts of the NPS;
- secondly, each of such intervals corresponds to a continuous sequence of contours (with numbers k ij ), each sum of which numbers and semicontour numbers k di / 2, contains the number m i and the composition of the terms different from the other intervals;
- thirdly, the boundaries of all intervals are different pairs of squares of natural numbers of different parity;
- fourth, the length of all representing intervals is the same, equal to N, and the corresponding sums of contour numbers are also constant and are determined only by the value of the Φ-invariant of N, but not by the location of the interval in the NPS;
- fifth, all the terms in different amounts (except for the number of the extreme semi-contour) represent the NPS segment, and monotonously increase by one from one smaller term to the next higher term (k ij +1 = k i (j + 1) ).
In other words, the value of the sum of contour numbers and the number of additional half-contour for snn N is invariant with respect to the number of terms, to the location of the interval in the NPS, ie to the values of the boundaries of the interval, to the values of the terms in the amount.
Example 1 Set to snnch N = N l = 231, the extreme half of the right. The number 231 corresponds to the number of the limit contour kn (N = 231) = 58. The F-invariant snnch is equal to kn (N) / 2 = 29, i.e. half the number of limit contour. It is required to form partitions of the number 29, subject to the restrictions into parts, for the Φ-invariant.
Beforehand, we write out the representation of the f-invariant snn N by the sums of the contour numbers. There are three such amounts:
29 = 3 + 4 + 5 + 6 + 7 + 8/2;
29 = 7 + 8 + 9 + 10/2;
29 = 19 + 20/2.
Since the contour numbers that follow one after another, and the half of the number of the extreme contour corresponding to the left contour of the contour with a large number, are summed up, the represented number N is easily determined as the sum of the lengths of adjacent contours and the semi-contour:
- for the first partitioning N = 8 • 3 + 8 • 4 + 8 • 5 + 8 • 6 + 8 • 7 + 8 • 8/2 -1 = 24 +32 +40 +48 +56 +32 -1 = 231;
- for the second partition, N = 8 • 7 + 8 • 8 + 8 • 9 + 8 • 10/2 - 1 = 56 + 64 + 72 + 40 - 1 = 231;
- for the third partitioning N = 8 • 19 + 8 • 20/2 - 1 = 152 + 80 - 1 = 231.
Such calculations show that the number 231 is representable in the NPS by two multi-contour intervals and the third interval consists of one complete contour and one half-contour of the limit contour adjacent to the right. The length of each interval is equal to the difference of its boundaries (i) - (i), i.e. the number N is represented by the difference of three (i = 1 (1) 3) different pairs of squares (interval boundaries).
Further, recalling that the boundaries of the contours and half-contours are full squares and using formulas for the boundaries of the contours and half-contours, we define the values for the boundary elements of each representative interval.
For the 1st interval, we have 29 = 3 + 4 + 5 + 6 + 7 + 8/2; G P1 (8/2) = (2 • 8) 2 = 256; G l1 (3) = (2 • 3 - 1) 2 = 25; and (i = 1) - (i = 1) = (16) 2 - (5) 2 = 256 - 25 = 231 = (16 + 5) (16 - 5) = 21 • 11 = 231.
For the 2nd interval, we have 29 = 7 + 8 + 9 + 10/2; G n2 (10/2) = (2 • 10) 2 = 400; G l2 (7) = (2 • 7 - 1) 2 = 169; and (i = 2) - (i = 2) = (20) 2 - (13) 2 = 400 - 169 = 231 = (20 + 13) (20 - 13) = 33 · 7 = 231.
For the third interval, we have 29 = 19 + 20/2; 3 (20/2) = (2 • 20) 2 = 1600; G l3 (19) = (2 • 19 - 1) 2 = 1369; and then G p (i = 3) - G l (i = 3) = (40) 2 - (37) 2 = 1600 - 1369 = 231 = (40 + 37) (40 - 37) = 77 • 3 = 231 .
Having bounds (pairs of squares) for each of the three intervals for one night, N, we easily get different decompositions of the number N into factors.
There is another interval with boundaries - squares, between which lies a composite odd number N = 231. Its boundaries - the boundaries of the limit contour for the number 231 are quite simple: the left is H l4 (k = 58) = ((231 - 1) / 2) 2 = (115) 2 = 13225, right 4 (k = 58) = ((231 + 1) / 2) 2 = (116) 2 = 13456, The length of the interval-half-contour in the limit contour is equal to
N = G n4 (k = 58) - G l4 (k = 58) = ((231 + 1) / 2) 2 - ((231-1) / 2) 2 = (116) 2 - (115) 2 = 13456 - 13225 = 231.
Conclusion
The considered material is a different from existing traditional approach of modeling NPS and snn N, which has a novelty. The numerical example illustrates the scheme of actions that is easily transformed into an algorithm with the possibility of eliminating the search of variants. It is only necessary to close the question (creating a program) to obtain special partitions (or) even a single special partition of the Φ-invariant. The way to solve this problem is proposed ( here ).
Literature
[1] Vaulin A.E., Pilkevich S.V. "The fundamental structure of the natural series of numbers" - Intellectual systems. Proceedings of the Seventh International Symposium. Ed. K.A. Pupkova. - Moscow: RUSAKI, 2006. - p. 384-387
[2] Bronshtein I.N., Semendyayev K.A. Handbook of mathematics for engineers and students of high schools. _M .: GITTL, 1954. -608
[3] Hall M. Combinatorics. -M .: Mir, 1970. - 424 p.
[4] Andrews G. Theory of partitions. - M .: Science GRFML, 1982. - 256 s.
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