Prerequisite:
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.
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
2 | – | – | – | – | – | – | |||||
5 | – | – | – |
This type of behaviour does not apply to factorization dashed lines, in which the number from which they are built has more importance than the order; this is because, given two natural numbers, their factorizations can be very different even if the two numbers are very close to each other. This can be verified by using the factorizer, for example on the numbers 546 and 548.
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.
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 column m - i, for each i = 0, \ldots, m.
In addition to these characteristics that derive from linearity, the factorization dashed lines have other ones that instead are specific to dashed lines built this way. One property 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 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.
- which is also a component of the dashed line;
- 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.
- 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 but aren’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; one of our proof strategies is precisely based upon them. 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, are also spaces, and vice versa.