# New invariant of a natural number. Theorem and Proof

Earlier on Habré the author’s work on the invariant of numbers was published ( here ). Even earlier, in [1], information was provided on the original concept of modeling a natural series of numbers and a single number in order to establish properties that weakly depend or do not depend on the bit depth of numbers. Previously, no theorems were presented to prove the truth of the provisions that are used by the author in his works. Analysis of comments on the works showed how incredulously the readership relates to such works and statements.

The limits of such distrust of the estimates are “Bullshit; Caution: pseudoscientific nonsense; Transfusion from empty to empty ... ”to“ Yes, it will certainly seem interesting to me if ... ”. These are just a few of the opinions of readers who expressed their perception of the work, their opinion of it. Thank you for your attention to the work. By the way, these opinions and assessments were approved by many other readers who did not remain indifferent. They also thank you for your attention. So this prompted me to give a theorem on the f-invariant and its proof.

I really hope that the above proof of the truth of the statement about the f-invariant (a new property of a number with a calculated exponent) is possible, it will slightly correct the opinions of readers and raise doubts about the correctness of their possibly hasty initial estimates. Each author, who has devoted a lot of time and effort to a specific work, perceives his work almost as obvious over time, the presentation of the work while striving to make it short, contains a lot of “default” details, which makes it, apparently, inaccessible to other people for understanding.

The theme of this and my other published works is directly related to security and, in particular, information security at all those levels where cryptographic protection is used. The fact that the publications cite mathematical relations at the level of elementary mathematics does not make the problem less complicated or more accessible. It’s just that the author seeks to make the presentation accessible and clear to a wider circle of Habrovsk readers. My work is about a completely new approach to solving a particular but important security problem ( here ), which is closely connected with many other problems. The author considers the direction in the field of factorization, which is currently developing at the world level, to be dead-end, and practice (the criterion of truth) confirms this.

We need a new look at the factorization problem, at the natural series of numbers (NRF), at the number system as a whole, which, as it turned out, has not been studied much. The theory of NRF and a separate composite odd natural number (SNP) is needed, models of these objects are required that take into account the requirements and requirements of modern practice. There is simply nowhere to copy anything like this today. The author, to the best of his abilities and capabilities, creates independently what is so far hidden from the attention of others and in a series of publications he shares his findings with the public, realizing that everything new is most often taken with hostility at all times. The reasons for this phenomenon are very diverse and it is not the time to discuss them here and now.

I really hope to find associates among programmers, since I myself am not a programmer. Material on 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 work on Habré. New concepts with their definitions include: classes of (left and right) odd numbers, a contour, a limit contour, a half-contour, an LDP interval, their lengths, numbering, boundaries, a f-invariant of a number, interval and numbering models of a composite odd number [3, 4], formulas for calculating the characteristics of these objects. The introduction of these new concepts and notation provides a solution to a new range of problems formulated with respect to natural numbers, the main of which is the factorization of numbers.

Today, seemingly simple questions about natural or integer numbers have nowhere to find answers. Examples of questions may include the following.

How to determine non-trivial involutions and idempotents of a finite numerical residue ring modulo SNPS N without first factoring N? How to find the roots of the comparison modulo in such a ring, without total enumeration of elements? How to find quadruples of quadratic residues whose values are generated by a single square without factoring N? While the author has to find the answers to these and other questions independently. I am waiting for feedback from the interested reader (s) on the substance of the questions formulated and possibly some other. Uncovering the essence of the questions requires other less elementary mathematics, but for now, touching it here is apparently premature.

Let us introduce the notation for the numbers of the extreme contours of the interval for snapshot N: for the smaller number s - start, for the larger number f - finish. Obviously, the left border of the multi-loop interval representing the left (N

The left border of the multi -circuit interval representing the right (N

In the previous work ( here ) for the numbering model of the natural odd composite number N, i.e. semicircle in the limit circuit, we obtain the sum of the numbers by switching from the lengths of the contours to their numbers in the form k

Proof we perform separately for the natural odd right N

We start with the right number N = N

s / 2 + (s + 1) + (s + 2) + ... + f => 1/2 (f + s +1) (fs-1 + 1) + s / 2 => 1/2 ( f

We transform the final expression to the form corresponding to the difference of the boundaries of the interval for the studied number N

The middle and right sides of the equality are converted to the form of the difference of the boundaries representing the number of intervals

N

This completes the proof for the case N = N

Now we perform the proof for the case of N = N

s + (s + 1) + (s + 2) + ... + (f-1) + f / 2 => 1/2 (f-1 + s) (fs) + f / 2 => 1/2 (f

We perform the transformation of the final expression, leading it to the form of the difference of the boundaries of the studied number N = N

The middle and right sides of the equality are converted to the form of the difference of the boundaries representing the number of the interval

N

This proof for the case N = N

The basis of the proof is the calculation of the boundaries for a multi-circuit interval and it is shown that the difference of such boundaries at an arbitrary interval leads to the same result for an interval with a length equal to N. Another way of proving the theorem is possible using the principle of forming partitions of the number k

k

Proof is carried out using Theorem 1 by directly constructing such sums.

In the theorem, statements are made about several facts:

- firstly, for SNCH N there are (i> 1) more than one representative SNCH N interval in different parts of the NRF;

- secondly, each of these intervals corresponds to a continuous sequence of circuits (with numbers k

- thirdly, the boundaries of all intervals are different pairs of squares of natural numbers of different parity;

- fourthly, the length of all representing the intervals is the same, equal to N, and the corresponding sums of the circuit numbers are also constant and are determined only by the value of the f-invariant of the number N, and not by the location of the interval in the LFD;

- fifth, all terms in different sums (except the extreme half-loop number) represent the NRF segment, and monotonically increase by one from one smaller term to the next larger term (k

In other words, the value of the sum of the numbers of the contours and the number of the additional half-circuit for the SNCH N is invariant with respect to the number of terms, to the location of the interval in the LFD, i.e. to the values of the boundaries of the interval, to the values of the terms in the sum.

First, we write out the representation of the sn-function sn-invariant N by the sums of contour numbers. There are three such sums:

29 = 3 + 4 + 5 + 6 + 7 + 8/2;

29 = 7 + 8 + 9 + 10/2;

29 = 19 + 20/2.

Since the numbers of the contours following one after another and the half of the number of the extreme contour corresponding to the left semicircle of the contour with a large number are summed up, the represented number N is easily determined as the sum of the lengths of the contours adjacent to each other and the semicircle:

- for the first partition 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 partition, N = 8 • 19 + 8 •

20/2 - 1 = 152 + 80 - 1 = 231. Such calculations show that the number 231 is representable in the LFD by two multi-contour intervals and the third interval consists of one full contour and one semi-contour limit contour adjacent to the right. The length of each interval equal to the difference of its boundaries r

Further, recalling that the boundaries of the contours and half-circuits are full squares and using the formulas for the boundaries of the contours and half-circuits, we determine the values for the boundary elements of each representing interval.

For the 1st interval we have 29 = 3 + 4 + 5 + 6 + 7 + 8/2; G

For the 2nd interval, we have 29 = 7 + 8 + 9 + 10/2; G

For the 3rd interval, we have 29 = 19 + 20/2; G

With boundaries (pairs squares) for each of the three intervals for one snnch N, decomposition easily obtain different number N into factors.

There is one more interval with borders — squares, between which lies a composite odd number N = 231. Its borders — the boundaries of the limiting contour for the number 231 — are quite simple: left Г

N = Г

The considered material is a novelty different from the existing traditional approach for modeling NRF and SNCH N. A numerical example illustrates an action diagram that can be easily converted to an algorithm with the possibility of eliminating enumeration of options. It is only necessary to close the question (creating a program) to obtain special partitions (or) even the only special partition of the f-invariant. A way to solve this problem is proposed ( here ).

Literature

[1] Vaulin A.E., Pilkevich S.V. “Fundamental structures of the natural series of numbers” - Intelligent systems. Proceedings of the Seventh International Symposium. Ed. K.A. Pupkova. - M.: RUSAKI, 2006. - p. 384-387

[2] Bronstein I. N., Semendyaev K.A. Handbook of mathematics for engineers and students of technical schools. Moscow: GITTL, 1954. -608s

[3] Hall M. Combinatorics. -M .: Mir, 1970 .-- 424 p.

[4] Andrews G. Partition Theory. -M .: Science GRFML, 1982. - 256 s.

The limits of such distrust of the estimates are “Bullshit; Caution: pseudoscientific nonsense; Transfusion from empty to empty ... ”to“ Yes, it will certainly seem interesting to me if ... ”. These are just a few of the opinions of readers who expressed their perception of the work, their opinion of it. Thank you for your attention to the work. By the way, these opinions and assessments were approved by many other readers who did not remain indifferent. They also thank you for your attention. So this prompted me to give a theorem on the f-invariant and its proof.

I really hope that the above proof of the truth of the statement about the f-invariant (a new property of a number with a calculated exponent) is possible, it will slightly correct the opinions of readers and raise doubts about the correctness of their possibly hasty initial estimates. Each author, who has devoted a lot of time and effort to a specific work, perceives his work almost as obvious over time, the presentation of the work while striving to make it short, contains a lot of “default” details, which makes it, apparently, inaccessible to other people for understanding.

**About the approach and its novelty**

The theme of this and my other published works is directly related to security and, in particular, information security at all those levels where cryptographic protection is used. The fact that the publications cite mathematical relations at the level of elementary mathematics does not make the problem less complicated or more accessible. It’s just that the author seeks to make the presentation accessible and clear to a wider circle of Habrovsk readers. My work is about a completely new approach to solving a particular but important security problem ( here ), which is closely connected with many other problems. The author considers the direction in the field of factorization, which is currently developing at the world level, to be dead-end, and practice (the criterion of truth) confirms this.

We need a new look at the factorization problem, at the natural series of numbers (NRF), at the number system as a whole, which, as it turned out, has not been studied much. The theory of NRF and a separate composite odd natural number (SNP) is needed, models of these objects are required that take into account the requirements and requirements of modern practice. There is simply nowhere to copy anything like this today. The author, to the best of his abilities and capabilities, creates independently what is so far hidden from the attention of others and in a series of publications he shares his findings with the public, realizing that everything new is most often taken with hostility at all times. The reasons for this phenomenon are very diverse and it is not the time to discuss them here and now.

I really hope to find associates among programmers, since I myself am not a programmer. Material on 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 work on Habré. New concepts with their definitions include: classes of (left and right) odd numbers, a contour, a limit contour, a half-contour, an LDP interval, their lengths, numbering, boundaries, a f-invariant of a number, interval and numbering models of a composite odd number [3, 4], formulas for calculating the characteristics of these objects. The introduction of these new concepts and notation provides a solution to a new range of problems formulated with respect to natural numbers, the main of which is the factorization of numbers.

Today, seemingly simple questions about natural or integer numbers have nowhere to find answers. Examples of questions may include the following.

How to determine non-trivial involutions and idempotents of a finite numerical residue ring modulo SNPS N without first factoring N? How to find the roots of the comparison modulo in such a ring, without total enumeration of elements? How to find quadruples of quadratic residues whose values are generated by a single square without factoring N? While the author has to find the answers to these and other questions independently. I am waiting for feedback from the interested reader (s) on the substance of the questions formulated and possibly some other. Uncovering the essence of the questions requires other less elementary mathematics, but for now, touching it here is apparently premature.

**Rationale for the approach**

Let us introduce the notation for the numbers of the extreme contours of the interval for snapshot N: for the smaller number s - start, for the larger number f - finish. Obviously, the left border of the multi-loop interval representing the left (N

_{l}≡3 (mod4)) odd number coincides with the left border of the smaller s-th loop Г_{л}(s) = (2s - 1)^{2}, and the right border of this multi-loop interval with the right the boundary of the left semicircle of the contour with a large number f, i.e. R_{n}(f) = (2f)^{2}.The left border of the multi -circuit interval representing the right (N

_{п}≡1 (mod4)) odd number N_{п}coincides with the left border of the right semicircle of the smaller s-th contour Г_{л}(s) = (2s)^{2}, and the right boundary of the interval for N coincides with the right boundary of the larger contour with number f, i.e. R_{n}(f) = (2f + 1)^{2}. The accepted notation is somewhat cumbersome, but semantically justified and convenient for understanding and remembering.In the previous work ( here ) for the numbering model of the natural odd composite number N, i.e. semicircle in the limit circuit, we obtain the sum of the numbers by switching from the lengths of the contours 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 the invariant of a composite odd number. Single interval). An arbitrary interval in the LFD with a length N composed of consecutive 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_{p}(N) / 2, where t = m- 1, i = 1 (1) t, of the numbers forming the interval of the contours, and half of the number of the contour with a complementary k_{d}/ 2 semicircle equal to half the number of the limiting contour (f-invariant) of N.Proof we perform separately for the natural odd right N

_{p}and left N_{l}compound 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 numbers of the contours and the half-circuit 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 the subsequent transformation of the boundaries also reduces the difference of the boundary points of the extreme elements of the sum equal to the number N.We start with the right number N = N

_{p}. We substitute the notation s and f for the numbers of the extreme contours introduced earlier in the model of odd composite numbers into the sum of the numbering model. We use the formula for the sum of the elements of the NRF segment 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 found expression with half the number of the limit contour of the number N_{p}, i.e., k_{p}(N) / 2 = (N_{p}-1) / 8 = 1/2 (f^{2}+ fs^{2}).The middle and right sides of the equality are converted to the form of the difference of the boundaries representing the number of intervals

N

_{p}= 4f^{2}+ 4f + 1 - 4s^{2}= (2f +1)^{2}- (2s)^{2}.This completes the proof for the case N = N

_{n}ends.Now we perform the proof for the case of N = N

_{l}. We substitute the notation s and f for the numbers of the extreme contours in the sum of the numbering model, 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).We perform the transformation of the final expression, leading it to the form of the difference of the boundaries of the studied number N = N

_{l}, namely, to the form (2f)^{2}- (2s - 1)^{2}. This is achieved by equating the found expression with half the number of the limiting contour of the number N_{l}, i.e., k_{p}(N) / 2 = (N_{l}+ 1) / 8 = 1/2 (f^{2}+ s - s^{2}).The middle and right sides of the equality are converted to the form of the difference of the boundaries representing the number of the interval

N

_{l}= 4f^{2}- 4s - 1 - 4s^{2}= (2f)^{2}- (2s - 1)^{2}.This proof for the case N = N

_{l}is completed, and the theorem is completely proved.The basis of the proof is the calculation of the boundaries for a multi-circuit interval and it is shown that the difference of such boundaries at an arbitrary interval leads to the same result for an interval with a length equal to N. Another way of proving the theorem is possible using the principle of forming partitions of the number k

_{p}(N) / 2 into different the number of parts m_{i}with restrictions of the form (k_{ij}+1 = k_{i (j + 1)}) into parts of the partition. In this approach, the role of the split number is played by the f-invariant k_{p}(N) / 2 of the number N. Indeed, the snnch N itself can be divided into parts (contours) in several ways, while the parts of the split will satisfy the imposed requirements (restrictions on values).**Theorem 2**. (On the invariant of a composite odd number. Many 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 areas of the numerical axis, having as squares the squares of positive integers of different parities, correspond to the sum of different numbers m_{i of}terms (numbers k_{ij}) of contours following one after another continuously (k_{ij}+1 = k_{i (j + 1)}) and one extreme half-loop 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_{p}(N) / 2, where t = m_{i}-1, j = 1 (1) t, i = 1 (1) ...Proof is carried out using Theorem 1 by directly constructing such sums.

In the theorem, statements are made about several facts:

- firstly, for SNCH N there are (i> 1) more than one representative SNCH N interval in different parts of the NRF;

- secondly, each of these intervals corresponds to a continuous sequence of circuits (with numbers k

_{ij}), each sum of the numbers of which and the half-_{circuit}numbers k_{di}/ 2, contains a different number m_{i}and the composition of the terms from the sums of other intervals ;- thirdly, the boundaries of all intervals are different pairs of squares of natural numbers of different parity;

- fourthly, the length of all representing the intervals is the same, equal to N, and the corresponding sums of the circuit numbers are also constant and are determined only by the value of the f-invariant of the number N, and not by the location of the interval in the LFD;

- fifth, all terms in different sums (except the extreme half-loop number) represent the NRF segment, and monotonically increase by one from one smaller term to the next larger term (k

_{ij}+1 = k_{i (j + 1)}).In other words, the value of the sum of the numbers of the contours and the number of the additional half-circuit for the SNCH N is invariant with respect to the number of terms, to the location of the interval in the LFD, i.e. to the values of the boundaries of the interval, to the values of the terms in the sum.

**Example 1**. Given snc N = N_{l}= 231, extreme semicircuit on the right. The number 231 corresponds to the number of the limiting contour k_{p}(N = 231) = 58. The f-invariant of the snc is k_{p}(N) / 2 = 29, i.e. half of the limit loop number. It is required to form partitions of the number 29 subject to restrictions on parts for the f-invariant.First, we write out the representation of the sn-function sn-invariant N by the sums of contour numbers. There are three such sums:

29 = 3 + 4 + 5 + 6 + 7 + 8/2;

29 = 7 + 8 + 9 + 10/2;

29 = 19 + 20/2.

Since the numbers of the contours following one after another and the half of the number of the extreme contour corresponding to the left semicircle of the contour with a large number are summed up, the represented number N is easily determined as the sum of the lengths of the contours adjacent to each other and the semicircle:

- for the first partition 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 partition, N = 8 • 19 + 8 •

20/2 - 1 = 152 + 80 - 1 = 231. Such calculations show that the number 231 is representable in the LFD by two multi-contour intervals and the third interval consists of one full contour and one semi-contour limit contour adjacent to the right. The length of each interval equal to the difference of its boundaries r

_{n}(i) - T_{l}.. (I), ie the number N represented by the difference three (i = 1 (1) 3) of different pairs of squares (slot boundaries).Further, recalling that the boundaries of the contours and half-circuits are full squares and using the formulas for the boundaries of the contours and half-circuits, we determine the values for the boundary elements of each representing interval.

For the 1st interval we have 29 = 3 + 4 + 5 + 6 + 7 + 8/2; G

_{p1}(8/2) = (2 • 8)^{2}= 256; Г_{л1}(3) = (2 • 3 - 1)^{2}= 25; and Г_{п}(i = 1) - Г_{л}(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; Г_{л2}(7) = (2 • 7 - 1)^{2}= 169; and Г_{п}(i = 2) - Г_{л}(i = 2) = (20)^{2}- (13)^{2}= 400 - 169 = 231 = (20 + 13) (20 - 13) = 33 • 7 = 231.For the 3rd interval, we have 29 = 19 + 20/2; G

_{p3}(20/2) = (2 • 20)^{2}= 1600; Г_{л3}(19) = (2 • 19 - 1)^{2}= 1369; and then Г_{п}(i = 3) - Г_{л}(i = 3) = (40)^{2}- (37)^{2}= 1600 - 1369 = 231 = (40 + 37) (40 - 37) = 77 • 3 = 231 .With boundaries (pairs squares) for each of the three intervals for one snnch N, decomposition easily obtain different number N into factors.

There is one more interval with borders — squares, between which lies a composite odd number N = 231. Its borders — the boundaries of the limiting contour for the number 231 — are quite simple: left Г

_{л4}(k = 58) = ((231 - 1) / 2)^{2}= (115)^{2}= 13225, right Г_{п4}(k = 58) = ((231 + 1) / 2)^{2}= (116)^{2}= 13456, The length of the half-circuit interval in the limit circuit isN = Г

_{п4}(k = 58) - Г_{л4}(k = 58) = ((231 + 1) / 2)^{2}- ((231-1) / 2)^{2}= (116)^{2}- (115)^{2}= 13456 - 13225 = 231.**Conclusion**

The considered material is a novelty different from the existing traditional approach for modeling NRF and SNCH N. A numerical example illustrates an action diagram that can be easily converted to an algorithm with the possibility of eliminating enumeration of options. It is only necessary to close the question (creating a program) to obtain special partitions (or) even the only special partition of the f-invariant. A way to solve this problem is proposed ( here ).

Literature

[1] Vaulin A.E., Pilkevich S.V. “Fundamental structures of the natural series of numbers” - Intelligent systems. Proceedings of the Seventh International Symposium. Ed. K.A. Pupkova. - M.: RUSAKI, 2006. - p. 384-387

[2] Bronstein I. N., Semendyaev K.A. Handbook of mathematics for engineers and students of technical schools. Moscow: GITTL, 1954. -608s

[3] Hall M. Combinatorics. -M .: Mir, 1970 .-- 424 p.

[4] Andrews G. Partition Theory. -M .: Science GRFML, 1982. - 256 s.