The model of the natural series of numbers (nrch). Ulam Spiral

The existing approaches to solving the problem of factoring large numbers (ZFSB), which have been used extensively in the world of mathematics over the last 20–30 years, indicate that this task is rather complicated for them, it stubbornly resists the onslaught of specialists and does not give up positions. At the same time, I cannot mention works whose authors would offer a deep analysis of the problem, the state of the question, or would criticize the approach used. The basic principle in the approach - sifting a set of numbers (the sieve principle) dominates in this area, but I think this is not the only way and perhaps not the best. Researchers at ZFBU are pinning great hopes on the computational tools of new types, on new physical principles (quantum, molecular, etc.), but we are not talking about changing the approach. Nevertheless, some conclusions today seem to suggest themselves. In attacks on RSA-like ciphers, SFCI is the main objective.

The RSA test number table published by RSA in 1991 is far from over. The published results of factoring part of RSA numbers from this table allow us to conclude that the applied methods for solving the SFCS are constructed using the properties of numbers that directly depend on the length (length) of numbers. The shorter the number was, the less was the time (in years) spent on factorization.
The second thing that attracts attention is the isolated, autonomous consideration of the factoring number. The situation is now perceived so that the number N exists as if by itself, and not chosen from a coherent, well-organized structure, its relations with the environment are broken, the properties inherent in the number do not inherit the properties of the environment and the location of the number in the environment.
A small example demonstrates the presence of such short-range links. For arbitrary three adjacent numbers, one of them is always a multiple of three, and the product of the extreme numbers is always equal to the square of the average number without one, x = 24963 = 157 159 . The position of the number 158 between 157 and 159 makes it easy to factor the number x.
In order to successfully overcome the crisis in the field of the theory of factorization, it is necessary to consider other approaches, the methods for solving SFChs in which would be based on the properties of numbers that are weakly dependent or not at all dependent on the digit capacity of the number. These approaches are associated with the development of models of numerical systems in general and models of individual numbers within such systems.

Description of the NPS model and its features. Among the famous models of the NPSH helix (Fig. 1), the mathematician of Stanislav Ulam (1909 -1984) occupies a prominent place. She is remarkable for the simplicity of her structure and leaves an unforgettable impression from the first acquaintance with her and her perception. In essence, the model is the cells of the plane, digitized by numbers of the natural series, arranged in turns in the following order. In the middle of a sheet of paper (see Fig. 2), 1 fits into the cell, 2 to the right, 3 above, 4 left, 4 5 down, 6.7 down, 8.9 right and 10, which starts a new turn of the spiral. Each new round, let's call it “ contour ”, increases its length by 8 cells in relation to the previous one. Further, as if continuing the ribbon with a width of a cage, buttwith turn after turn, counterclockwise, geometric squares are wound — the outlines of the model. Each contour (square) will be supplied with a sequence number k, starting from the center of the helix, k = 1,2,3, ... Cells with even and odd numbers are arranged in a spiral like chessboard cells. The diagonals of even and odd numbers alternate with each other.
Surprisingly, such a simple twist of a linear list into a spiral introduces a very rigid order in the organization of the numbers placed in the cells of the model. External signs of such an order attract attention immediately, but what is inside ...?
Clarifying this issue and many others requires the availability of tools for working with the elements and the object of the NPS as a whole. The purpose of the proposed work is the preparation, creation of such tools.

Remarkable features of the model are, firstly, that the model is complete - all the numbers of the natural number are displayed in the cells of the helix, secondly, all cells containing numerical squares are aligned along one direction (along one line passing through the center) in - third, all numerical squares are also divided into even squares on one side of the center, odd squares on the other; fourth, the vertical and horizontal rays, as well as odd diagonals, depart from the central contour k = 1, some of which are formed by cells that cannot contain simple numbers. The cells of these rays and diagonals go to infinity and contain only composite numbers.
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S. Ulam himself noted the dark fill of the cell containing simple numbers, and got a picture (Fig. 1) very similar from space to a multi-million dollar city with its quarters and avenues. To some, the picture may even look like a starry sky, but the abundance of straight line segments and figures with right angles somewhat breaks this similarity. When looking at the spiral, the conclusion is that the number of prime numbers is not so small, and their density per unit area, if it decreases with distance from the center, then the decline is not visually felt.

Figure 1 - Spiral representation of a fragment of the natural row with filled cells for prime numbers (200x200 cells)

At different times, different authors have attempted to modify the appearance of the Ulam spiral. A zero was embedded in the center of the helix, in particular, studies of a hexagonal number helix were carried out. In addition, attempts were made to present a three-dimensional analogue of the Ulam spiral. Thus, for example, a numerical “spiral” in the form of a flat triangle was shown, along the height of which the whole squares were lined. The cone was illustrated as the result of “gluing” the sides of the triangle with the same numerical values. Note that this method of presentation enhances visibility, that is, allows a single view to embrace numbers more than just on the number line. But in such works there is no means of studying, studying the model.
One of the goals of the author was the desire to use, to reveal the pattern of appearance
prime numbers, make it visual and convenient for further work with the model.

Introduction to the coordinate system . For practical work with the model and its elements, it is desirable to be able to select any single cell with two coordinates (x1, x2) or their group (contour), and to establish in the cell with given coordinates the numerical value N (x1, x2) assigned to it from the LPS as functions of these coordinates. It is also desirable to be able to solve the inverse problem - for a given value of N (x1, x2) of a natural number in an arbitrary cell, to be able to determine its position in the model, i.e. its coordinates.

We show here how to solve such problems. The model as a whole is a discrete plane. By its well distinguishable elements (points, groups of points) we will consider individual cells and outlines of cells, horizontal (D), vertical (B), diagonal (D) lines and rays .
Note that each contour always contains a pair of cells filled with squares of different parity. We take for the beginning of the contour his cell with an odd square. This square is called the left boundary of the kth contour — the smallest number in the contour. Denote the boundaries of the contour with the symbols Ch (k) = (2k - 1) ² - left (smaller) and Gn (k) = (2k +1) ² - right (larger).
The length of the contour L (k) will be called the number of cells in it. The length of the contour is calculated as L (k) = 8 ∙ k, or as the difference between the values ​​of odd squares — the boundaries of the contours that start a pair of adjacent contours and is always a multiple of 8. This is easy to show, as it follows from the fact that the square of any odd number 2n + 1 has the form 1 + 8 ∙ i , where i is a combinatorial combination of two of n + 1. Therefore, at j> i, the difference of squares of two odd adjacent (and also not adjacent) numbers 1 + 8 ∙ j -1- 8 ∙ i = 8 (j - i) is a multiple of 8. Semi-contours are formed by dividing the contour into two parts: less than m (k) with a length L (m (k)) = Hz (k) -Hl (k) and greater than M (k) with a length L (M (k)) = Hn (k) -Hz (k) , where Hz (k) = (2k) ² is the common boundary of the semi-contour.

The lengths of all successive contours of the model form an infinitely increasing arithmetic progression with the difference d = 8 and the initial term a = 8.

The coordinate system of the model . We take the central cell with the unit as the coordinate origin. In the plane, the position of each point is uniquely described by two coordinates. The assignment of coordinates is the localization of a point. It is convenient to perform such localization of the point at the beginning with the accuracy of the contour, and then within the limits of the fixed contour - with the accuracy of the cell. Coordinates do not have to be Cartesian. With this approach, the role of the first coordinate is assigned to the contour number (x1 = k). The role of the second coordinate x2, which determines the position of the cell in the contour, is assigned to the distance (distance) of the cell from the beginning of the contour.


Figure 2 - Model NPS and an enlarged fragment of the central part of the helix

In Figure 2, viewing a larger fragment of the NPS (399x399 cells), presented in a reduced form, makes it possible to verify that the expansion of the boundaries of the model does not really change the overall picture. From this figure, the central part is cut out with the number of cells 19x19 = 361 and an enlarged fragment is shown on the right with the cells filled with numbers. In the figure (with numbers), even in a limited volume, lanes (highways) that do not contain shaded cells are clearly visible.
This observation of the highways leads to a definite conclusion about the primes of the twins. In each contour there are positions (cells) that cannot be occupied by primes and twins. For example, a pair of simple twins can not be in the cells of the sides of a double highway, and these are four consecutive cells and two of them are odd. Between twins (P> 7) there must always be a cell with an even number. The values ​​of the quadruple of such numbers for any contour are determined uniquely.

Example 1 Let the number N = 39 be given. It is required to determine its position in the model, i.e. (x1, x2) coordinates of the cell with this number.
The decision . We extract the square root of the number N and round it to a smaller odd number. The value of the extracted square root is 6.244. The smallest nearest integer is 6 = 2k, the even number is less than the odd number 6 - 1 = 5. This number is odd and the cell containing its square 25 is the initial cell of the contour containing the number N = 39. This is the smallest number (25) in the contour . The square of the doubled contour number is equal to the only even square in the contour cells; it is obtained from the number per unit of the larger base of the odd square in the contour. Hence the number of the contour and the first coordinate x1 = k = 6/2 = 3 . An analysis of the situation shows that the number N = 39 is inscribed in the cell of the 3rd circuit (with the number k = 3) and this cell lies further than an even square from the initial cell of this circuit. The second coordinate of the cell is found as the difference of the values ​​of a given number N = 39 and the value of an odd square - the beginning of the contour, i.e. x2 = 39 - 25 = 14. The solution is given by (x1, x2) = (3, 14) .
Solve the inverse problem. Let the cell (x1, x2) = (3, 14) be given by its coordinates. Find the number inscribed in the cell. The first coordinate is the contour number x1 = k = 3. The value of the number in the initial cell of the contour is an odd square, determined by the number k from the formula 2k-1 = 2 ∙ 3 ​​-1 = 5 . This square is obviously equal to 25. The value of the number N in a given cell remote from the initial cell of the contour by the value x2 = 14 is determined by the sum N (x1, x2) = 25 + 14 = 39.
Thus, the work proposed a method that provides selection (localization) given by the number N of the NSD point on the NSD planar model, by two coordinates. And the solution of the inverse problem is to find its value by given coordinates of the number N on the plane. A simple solution of these two problems allows you to process data in a variety of problems formulated with respect to the natural series of numbers. In subsequent posts, some of these tasks will be brought to the attention of readers.
Q = [λ] P
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