Factorization dashed lines

Prerequisites:

When constructing a linear dashed line, the sequence of primes is usually used as a list of components, until it stops at one of them, according to a certain criterion. However, this is not a strict rule of dashed line theory, which, in principle, allows us to choose components arbitrarily. Obviously, the way to choose the components is important, because, depending on which and how many are used, the structure of the dashed line changes accordingly, and therefore its properties also change.

One possibility is to assume that, given any positive integer m, it can be decomposed into prime factors. Since prime numbers are used as components in “classic” linear dashed lines, these two aspects can be combined with each other, creating a dashed line which has the prime factors of m as its components. What we get is a particular type of dashed line, which can be formally defined like this:

Factorization dashed line

Given an integer m > 1, we name factorization dashed line of m a linear dashed line which has the prime factors of m as its components, each taken once.

Starting from number m = 10, let’s build for example the related factorization dashed line W = (2, 5), whose components are precisely the prime divisors of 10 (lthe columns highlighted in yellow are its spaces):

0 1 2 3 4 5 6 7 8 9 10
2
5
As the Hypothesis H.1 (Hypothesis of existence of Goldbach pairs based on dashed lines) is formulated, which considers the dashed lines T_k, whose components are the first prime numbers, it can be seen how the same dashed line is used for different starting even numbers. For example, using the dashed line viewer choosing the option “Prime numbers up to smallest p_k where p_{k+1}^2 \gt n“, we can see how the dashed lines of the numbers 10 and 12 are identical, and coincide with the dashed line T_2 = (2, 3) (the only thing that changes is the number of columns displayed, but the dashed line is the same). For this reason, when dealing with dashed lines like these, the order is given more importance than the starting number.
This type of behavior does not apply to factorization dashed lines, in which the number from which they are built takes on more importance than the order; this derives from the fact that, given two natural numbers, their factorization can be very different even if the two numbers are very close.
This does not mean that the factorization dashed lines are always different from each other. For example, in the dashed line viewer, by choosing the “Prime divisors of n” option, we can observe that the dashed lines of the numbers 10 and 20 coincide with each other, because these two numbers have the same prime factors, except for the power at which they are raised (10 = 2 \cdot 5 and 20 = 2^2 \cdot 5). As in the previous case, the only difference between the two dashed lines is that the one related to 20 is more extensive, but their basic structure is still the same.

Factorization dashed line are linear, so they are periodical due to the Property T.4 (Linear dashed lines are periodical). In particular, for factorization dashed lines, the following Property is true:

Length of a period of a factorization dashed line

The length of a period of a factorization dashed line of a number m is the product of the prime factors of m, i.e. the number P := \prod_{p \mid m} p. Furthermore, we haveP \mid m.

The length of a period of the factorization dashed line of m is equal, for the Property T.4 (Linear dashed lines are periodical), to the least common multiple of its components, which in this case are the prime divisors of m. So, in formulas, the length of a period is \mathrm{MCM}_{p \mid m} p. But the prime divisors of m, being precisely prime, are also two by two coprime; therefore their least common multiple coincides with their product \mathrm{MCM}_{p \mid m} p = \prod_{p \mid m} p, which is the number that in the statement is called P. It remains to be shown that P \mid m. To do this, we decompose m into prime factors, obtaining m = q_1^{m_1} \cdot \ldots \cdot q_k^{m_k}, where k \geq 1 and q_1, \ldots, q_k are the distinct prime factors of m. Since we have represented only the prime numbers that appear in the factorization of m, the exponents m_1, \ldots, m_k are all at least 1 (if some m_i were 0, the relative prime factor q_i, being raised to zero and different from all the others, would not be a prime factor of m). Instead P = \prod_{p \mid m} p = q_1 \cdot \ldots \cdot q_k. This is a divisor of m, because we have:

m = q_1^{m_1} \cdot \ldots \cdot q_k^{m_k} = (q_1 \cdot \ldots \cdot q_k)\left(q_1^{m_1 - 1} \cdot \ldots \cdot q_k^{m_k - k}\right) = P\left(q_1^{m_1 - 1} \cdot \ldots \cdot q_k^{m_k - 1}\right)

where q_1^{m_1 - 1} \cdot \ldots \cdot q_k^{m_k - 1} is an integer number, because m_1, \ldots, m_k are all greater than or equal to 1.

Furthermore, due to their linearity, the factorization dashed lines are symmetrical due to the Property T. 5 (Linear dashed lines are symmetrical). Due to this Property, any set of consecutive columns between two multiples of the length of a period is symmetrical; in particular, for the factorization dashed line of m we can consider the columns from 0 to m, because both are multiples of the length of a period (which is m due to the previous Property). By making this choice, starting from Property T.5, the following property of the factorization dashed lines is obtained:

Symmetry of factorization dashed lines

In the factorization dashed line of a number m, the columns from 0 to m are symmetrical, i.e. the column i coincides with the columnm - i, for each i = 0, \ldots, m.

Number 0 is obviously multiple of m, and the length of a period of the factorization dashed lines is a multiple of m due to Property L.F.1. Then, due to the Property T.5 (Linear dashed lines are symmetrical), the columns from 0 to m are symmetrical. The remaining part of the thesis is obtained by setting, in the cited Property, h := 0 and k := \frac{m}{M}, where M in the statement of the Property it represents the length of a period. In this way we obtain that column h \cdot M + i = 0 \cdot M + i = i coincides with the column k \cdot M - i = m - i, for each 0 \leq i \leq (k - h) \cdot M = k \cdot M = m.

In addition to these characteristics that derive from linearity, the factorization dashed lines have others that instead are specific to dashed lines built in this way. One is the following:

Prime spaces on the right side of the factorization dashed line of an even number

In the factorization dashed line of an even number 2n, all prime number p such that n \lt p \lt 2n are also spaces.

A prime factor of 2n is evidently 2. The other prime factors, therefore, must be divisors of n, therefore they are certainly less than or equal to n. So if a prime number p were greater than n, it could not be a prime factor of 2n, that is, it could not coincide with a component of the its factorzation dashed line. But, in any linear dashed line, a prime number p that does not coincide with any of the components of the dashed line is a space. This is proved by applying the definition of space to p: being a prime number, the only positive integers for which p is divisible are itself and 1, but 1, by definition, is not a dashed line component, while p is not, because it has been said that p does not coincide with any component of the dashed line. Hence, no divisor of p is a component of the dashed line, i.e. p is a space.

A further property is the following:

Small spaces in a factorization dashed line

In a factorization dashed line having q_1, q_2, \ldots q_k as its components, all spaces greater than 1 and less than q_k, either are prime numbers or have as prime divisors only prime numbers less than or equal to q_k which are not components of the dashed line.

Let s be a space greater than 1 and less than q_k. If s is prime, the thesis is trivially true. The remaining case to be proved is the one in which s is not prime, which can be proved by absurd: we start by denying the thesis, i.e. by stating that s has at least one prime divisor p:

  1. which is also a component of the dashed line;
  2. or which is greater than q_k.

But, if the 1. were true, s would be a value of some dash in the row of the component p (by definition of dash), so it would not be a space.

If 2. were true, since p \gt q_k, also s, which is a multiple of it, would be greater than q_k.

In both cases there is a contradiction with the starting hypothesis, so the thesis is true.

If with Dashed line viewer we view the dashed line of number 34, choosing options Prime divisors of n and Highlight columns which are spaces, we see its minimum component is q_1 = 2, its maximum one is q_2 = 17, and the spaces greater than 1 and less than q_2 are the following:

  • 3, 5, 7, 11 and 13, which are prime numbers;
  • 9 and 15, which instead are composite, and their prime factors are 3 and 5, which are between q_1 and q_2 bue atrn’t components of the dashed line.

Factorization dashed lines of type (2, p)

The factorization dashed lines which are of interest for studying Goldbach conjecture are the ones built starting from an even number 2n. A very simple particular case occurs when the factorization dashed line obtained is of the second order. This means that 2n has only two distinct prime divisors, one of which evidently is 2, while the other can be any prime p \gt 2. The factorization dashed line obtained is therefore (2, p) (this does not necessarily mean that n = p, because for example the same factorization dashed line is obtained if n = 2p^2).

Going into more detail of this type of dashed lines, we immediately notice that it is quite simple to find a criterion to understand if a column is a space:

Spaces of a factorization dashed line of type (2, p)

In a factorization dashed line of the type (2, p), with p \gt 2 prime, all the columns s which, at the same time, are odd and are not divisible by p, they are also spaces, and vice versa.

The proof of this property derives from the definition of space itself: the spaces are all and only those columns of the dashed line that are not divisible by any component of the dashed line. In our case, the components are 2 and p, so a column s is a space if and only if it is odd (i.e. it is not divisible by 2) and is not divisible by p.

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