Ulam Spiral, prime-ban areas

     Each natural number has very many known and, apparently, even more unknown properties. Even - odd, simple - compound, finite - infinite and other properties contribute to the introduction of the classification of numbers, of some order in their set. The traditional approach assumes that without having the number itself (its value) it is impossible to determine its properties. But it is not so. A number of useful properties for some numbers can be determined without knowing their values, but having data on their position in the natural number series (NRF). Prime numbers, except 2, can only be odd with a flexion of ≠ 5, and their position NRP is determined by the odd position. These positions themselves are not all the same. About some large odd N (x1, x2) numbers (of course, in odd positions in the LDP), you can

     The device of the NRF, the position of numbers in the NRF and their properties

     We will consider the relationship between the properties of natural numbers and their position in the NRF, considering such a relationship existing.
     For example, an odd number equal to an even square (> 2) without a unit is always compound: x² -1 = (x +1) (x -1). The odd square is a composite number. Information about this in a convenient form is precisely provided by the NRF model - the Ulam spiral, where the position of all the squares is uniquely determined. Consequently, the described diagonal rays pressed to the diagonals of squares cannot contain primes. In addition to the lines mentioned below, we point out others that have the same property. This property of the model was not previously noted either by Ulam himself or by other authors who worked with the model or mentioned it in publications. There is no big riddle in this. We identify the point (cell) with the coordinates (x1, x2) of the spiral plane and the number N (x1, x2) in this position.
     The fact of the existence of the forbidden region for primes in the NRF is established from observations of the spiral, and then confirmed (proved) by the author by mathematical means. The usefulness of this property is not obvious. But for odd numbers, the position of the cells (x1, x2) of which in the spiral belongs to the region of the ban of primes, not only their simplicity is established, but they also factorize without problems.
     Based on this fact about some odd numbers N (x1, x2) of even very large bit depths, it can be stated with certainty that they are composite and then easily factorize them.

      Model coordinate system. When using the concept of the contour structure of the NRF in the previous work, “The Model of the Natural Series of Numbers (NRF). Ulam spiral ”the author proposed a coordinate system on the spiral plane. The origin of the coordinates is the center of the spiral. The system is closer to polar than Cartesian. The role of the first coordinate x1 of the number N (x1, x2) is assigned to the contour number (x1 = k) . The second coordinate x2 <L (k), where L (k) is the length of the contour, determines the distance of the cell with the number N (x1, x2) from the beginning of the contour. For the convenience of the reader, the author has saved drawings of a previous work in this.

   Figure 1 - Model of NRF and an enlarged fragment of the central part of the spiral
  In Figure 1, from the fragment of NRP (399x399 cells), the central part with the number of cells 19x19 = 361 is cut out and the enlarged fragment with cells filling (x1, x2) with numbers is shown on the right. In the figure (with numbers), even in a limited volume, vertical and horizontal single and double stripes (highways) that do not contain filled cells are clearly visible, i.e. prime numbers. Below in Figure 2, upon careful examination, these highways can also be seen.

Figure 2 - Representation by a spiral model of a fragment (200x200 cells) of a natural row with filled cells for primes.
     When manipulating primes, it is useful to know and have information about some of their properties and dependencies. For example, the fact that the squares of primes, for example p and q, except 2, when comparing modulo 8, always have one residue 1, р² ≡ 1 (mod8) , and modulo 30 such residues there can only be two 1 and 19: р² ≡ 1 (mod30) or q² ≡ 19 (mod30). We give more detailed information about the number 30, which clarifies something to us regarding such mystery in the behavior of primes, their full squares. This number 30 = 2 ∙ 3 ​​∙ 5 plays a significant role in the study of prime natural numbers.
     Number 30 is the largest number for which all mutually simple with it and its smaller numbers are prime. The number 30 is preceded by the same property, the number 24, which also plays an essential role in number theory.

         The connection of two models of the natural series of numbers of a flat Ulam spiral and linear analytic

     It is more common for us to perceive natural numbers as elements of a linear model. This is convenient, since on the numerical axis all the points have one single coordinate, moreover, the value of this coordinate coincides with the number itself. Manipulating with data you don’t have to think about what is being processed - the coordinate or the number itself. Not so on the flatness. Its point has three characteristics: two coordinates (x1, x2) and the value of the number N (x1, x2) at the point, which in our model is a function of coordinates. Therefore, we establish some relations between the two models of linear and flat spirals. The concept of contour is introduced in both models in the same way - the beginning of the contour is an odd square. The length of the path is a multiple of the number 8. The numbers that make up the path are a segment of the natural number.

     Natural numbers can be represented by such an analytical model:
30k, 30k ± 1, 30k ± 2, 30k ± 3, ..., 30k ± 15, k = 1 (1) ∞, of which primes can only be represented as 30k ± 1, 30k ± 7, 30k ± 11 , 30k ± 13 . Of these, 8 primes with k = 0.1, the numbers 7,11,13,17,19,23,29 and 31 ≡ 1 (mod30) form a multiplicative group of residues modulo 30. The order of such a group is equal to the value of the Euler function φ (30 ) = φ (2) φ (3) φ (5) = 8. Such groups are called Euler groups.
      Now about 24. Let us show that for primes p, q> 3, the square difference of two primes p² - q² is always divided by 24. Consider two triples of adjacent numbers (p-1) p (p + 1) and a triple (q-1 ) q (q + 1), where p and q are simple. In each triple, one of the numbers is a multiple of three and this is not p and not q. Therefore, both products (p -1) (p + 1) and (q -1) (q + 1) are multiples of threewhich are formed by pairs of consecutive even numbers. But for such numbers it is known that one of them is a multiple of two, and the other four. It follows that both works are divided by 8. But they are also divided into three. Then each of the products is divided by 24. The difference between the products of the brackets (p -1) (p + 1) - (q -1) (q + 1) should also be divided by 24, and this is the square difference p² - q². Knowing this fact makes it possible to test another with one known square. This does not exclude the multiplicity of the difference of squares of ordinary odd numbers to 24, for example, 225 - 81 = 144 = 6 ∙ 24, but does not guarantee it 1225 - 441 = 784 ≠ 24k , for the difference of squares of ordinary odd and prime numbers , 225 - 49 = 176 ≠ 24k . For squares of primes, this is guaranteed.

      And also about the values ​​of consecutive nth and (n + 1) -th primes. So, Euclid proved the validity of the relation for primes p (n + 1) <p (1) ∙ p (2) ∙ p (3) ∙ .. ∙ p (n), where the values ​​in brackets denote the serial numbers of primes. This is a rather rough estimate. It was later shown that for n> 4, the relation takes the form p (n + 1) ² <p (1) p (2) p (3) ... p (n), i.e. already for
p (5) ² = 11² = 121 <p (1) ∙ p (2) ∙ p (3) ∙ p (4) = 2 ∙ 3 ​​∙ 5 ∙ 7 = 210;
p (5) = 11 <√ (p (1) ∙ p (2) ∙ p (3) ∙ p (4)) = √ (2 ∙ 3 ​​∙ 5 ∙ 7) = √210 . This is a significantly improved rating.
P.L. Chebyshev proved an even stronger result p (n + 1) <2p (n).

             Areas of a flat model that exclude the appearance of primes in them

     Now we’ll take a closer look at the rays emerging from the central part of the spiral and others that were mentioned earlier and also do not contain cells with primes
      We call the vertical and horizontal lines of the spiral “highways”, and the prime numbers in their cells are “traffic lights” that limit the speed along the lines. Then on a spiral you can indicate specific “high-speed” highways running in the direction from the center in four directions: “North - South” and “West - East”, which do not contain traffic lights at all. The cells of these highways are filled only with composite numbers, that is, they do not contain prime numbers. The fact itself is quite remarkable and even surprising, perhaps it contains a “hint” of the nature and distribution of primes in the NDF. Even more surprisingly, the highways to the South and East contain two adjacent lanes, and to the North and West, one each. Within the k-th circuit, we denote the cells belonging to the highways with the symbols of the cardinal points: C (k) - northern highway; Z (k) - western highway; U1 (k) is the southern first highway and U2 (k) is the southern second highway; B1 (k) is the eastern first highway and B2 (k) is the eastern second highway.

     Theoretically, trunks of three bands are unlikely, since a strip of directions perpendicular to the trunks (like all others) in three adjacent cells contains adjacent natural numbers. Among these numbers, one of the three is always a multiple of three, i.e., compound, and two of them are either even or odd. Let a multiple of three in a cell on the side of a two-lane highway. Then the fourth number in the circuit through two cells will also be a multiple of three. Therefore, after two cells of the number, the cell on the other side of the road also contains a composite number, which in some position may turn out to be simple. The even and odd numbers in the cells of the two-lane highway are staggered and, apparently, there is a law governing the filling of the odd cells of the highways, eliminating the appearance of primes in them.
     Pairs of odd numbers in adjacent cells of different highways turn out to be multiples of successively increasing 3,5,7,9, ... odd numbers:
(27,51): 3; (85.125): 5; (175,231): 7; (297, 369): 9 ...... (east direction); 51 = 27 + 24; 125 = 85 +40;
231 = 175 + 56; 369 = 297 +72 increment of values ​​is a multiple of 8 with multiplicity divisor of numbers.
(21.45): 3; (75.115): 5; (161.217): 7; (279.351): 9; ... (southbound).
     The description of the whole infinite set of cells of the highways is simpler than for cells outside the highways. One coordinate - the number of the contour k of the spiral is enough to determine the number in the cell of the highway that belongs to the given number k of the contour. The following dependencies are used (see table).

Table. The calculated characteristics of the cells of the main lines of the Ulam spiral. The

     formulas given in the table guarantee unlimited continuation of the bands when specifying the loop number k while preserving the marked properties of the bands and the numbers in them. A particularly important property is the divisibility property of odd numbers in the cells forming the stripes. In each circuit, given its number k, the formulas from the table determine the values ​​of the numbers in six cells of the intersection of highways with the circuit. All these numbers, let's call them benchmarks , the constituents and their divisors are known. The formulas in the table already contain descriptions of each divisor.
     Other numbers of the k-th contour (of the type of frame ± t ∙ frame divider) can also be subjected to the factorization procedure if certain conditions are met.

     These conditions include the following. The test number differs from the benchmark by a multiple of a smaller or larger divisor. For such numbers, factorization is performed without problems. An important property is the divisibility of odd numbers in the cells forming the stripes. Therefore, in each circuit there are more than 8 cells containing odd numbers that cannot be prime, since in the expression ( frame ± t ∙ frame divider ) the common factor is taken out of the bracket.

     In addition to the considered highways, there are lines (diagonals) of odd numbers also not containing prime numbers. Firstly, such an odd diagonal is the diagonal of odd squares. Even and diagonals are pressed to it before and after it, which obviously do not contain prime numbers. In this case, three adjacent cells without prime numbers appear in each circuit. Secondly, three more adjacent cells without prime numbers in each circuit appear next to the diagonal of even squares. This diagonal is preceded by an odd diagonal in which, as we now show, all cells are occupied by compound numbers. The values ​​in the cells of this diagonal are equal to the even square (2k) ² - 1 without unity. But this ratio is always decomposed into the product of two brackets - the sum with unity by the difference with unity of the first degree of the even square of the contour, (2k +1) (2k - 1).
     An example . Let the number N = 4294967297 = F5 be given. It is required to determine its position in the model, i.e. (x1, x2) the coordinates of the cell with this number, as well as the values ​​and positions of the six numbers belonging to this circuit on the highways: C = ?; B1 = ?; B2 = ?; S1 =?; U2 =?; З =?;
     Decision. We take the square root of the number N and round it to a smaller odd number. The value of the extracted square root is 65536.0000076 The smallest nearest integer is 65536 = 2k, (hence x1 = k = 32768 ) is an even number, the smaller is an odd number 65536 - 1 = 65535. This number is odd and the cell containing its square H (k) = 4294836225, is the initial cell of the circuit containing the number
N = F5 = 4294967297. The second coordinate x2 = 4294967297 - 4294836225 = 131072. So, for the number F5, a pair of coordinates (x1, x2) = (32768.131072) is obtained. The first coordinate of all the frames is the same - this is the contour number x1 = 32768. The second coordinates of x2 are different for all frames.
      The contour cells belonging to the highways are guaranteed to receive the multiplicative representation
C = 4294934528 = 32768 ∙ 131072; S = 4295000064 = 32768 ∙ 131073; S1 = 4295065599 = 32769 ∙ 131071;
S2 = 4295065600 = 32768 ∙ 131075; B1 = 429 486 8991 = 32767 ∙ 131073; B2 = 4294868992 = 32768 ∙ 131069.
     Highway cells receive the coordinates: the beginning of the NK contour (x1, x2) = (32768, 0);
B1 (x1, x2) = (32768, 32766); B2 (x1, x2) = (32768, 32767); C (x1, x2) = (32768, 98303);
3 (x1, x2) = (32768, 63839); U1 (x1, x2) = (32768, 229374); 10 (x1, x2) = (32768, 229375);
     The number in the circuit before the even square (2k) ² - 1 = (2k +1) (2k - 1) = 65535x65537 = 4294967295; guaranteed compound; the number behind the even square also turned out to be composite (2k) ² + 1 = 641x6700417.

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