# Characterization of spaces

Prerequisites:

One of the open problems of dashed line theory is “characterizing” spaces, i.e. looking for a criterion that tells us when a certain column of a dashed line is a space and when it is not. Certainly, a first characterization is the definition of space itself, but in our research we have noticed that it is sometimes useful to find alternative criteria. For example, an idea may be to characterize the spaces on the basis of the dashes found on the column that precedes or follows the space. This is particularly useful when studying Goldbach’s conjecture since, in all of our proof strategies, we always use dashed lines the first component of which is 2 (this is true either in $T_k$ dashed lines or in the factorization dashed line of $2n$), and all the spaces of these dashed lines, being odd, always precede and follow a dash relating to component 2, in the sense that the columns on the right and left of the space both contain a dash relating to this component (i.e. usually corresponding to the first row). So, in these dashed lines, the concepts of space, of space following a dash on the first row, and of space preceding a dash on the first row, are equivalent. Therefore, to characterize the spaces, we will deal with characterizing the last two concepts, formally defined as follows:

Spaces preceding/following a dash

Let $T$ be a dashed line, $s$ a space of $T$ and $t$ a dash of $T$.
We say that $s$ precedes $t$ (or that $t$ follows $s$) if $t$ has value $s + 1$.
We say that $s$ follows $t$ (or that $t$ precedes $s$) if $t$ has value $s - 1$.

In order to obtain the characterizations corresponding to this definition, for the spaces preceding a dash, considering that we are interested in the dashes of the first row, we can proceed this way:

1. We determine for which $x$ the $x$-th dash belongs to the first line and a space precedes it;
2. We determine the value of the column $v = \mathrm{t\_value}(x)$ of this dash;
3. Hence, for the first case of Definition L.C.1. (setting $x = t$), we have that $x$ has value $v = s + 1$, then our space is $s = v - 1$;
4. Once defined the criterion characterizing our values of $x$, then, we will also find the one characterizing the corresponding $v - 1 = \mathrm{t\_value}(x) - 1$, i.e. precisely the spaces preceding a dash.

The reasoning relating to the spaces following a dash is specular:

1. We determine for which $x$ the $x$-th dash belongs to the first line and a space follows it;
2. Like before, we determine the value of the column $v = \mathrm{t\_value}(x)$ of this dash;
3. Hence, for the second case of Definition L.C.1. (setting $x = t$), we have that $x$ has value $v = s - 1$, then our space is $s = v + 1$;
4. Once defined the criterion characterizing our values of $x$, then, we will also find the one characterizing the corresponding $v + 1 = \mathrm{t\_value}(x) + 1$, i.e. precisely the spaces following a dash.
As already noted, only in a dashed line with a component equal to 2, the characterization of the spaces that precede or follow a dash of a specific row can be equivalent to the characterization of all the spaces: this happens when choosing as row just the one of the component 2. The characterizations that derive from the Definition L.C.1 can however be applied to any linear dashed line; however, in the case of dashed lines that do not have 2 as a component, you will not get a characterization of all spaces, but only a subset of them. For example, in the first order dashed line $(4)$, all spaces of the type $2 + 4h$, for each integer $h \geq 0$, do not follow or precede a dash, therefore do not fall within Definition L.C.1.

## Characterization of the spaces of a first order dashed line

The criterion for determining when the $x$-th column of a dashed line of the first order is a space obtained from its own structure: a dashed line of the first order, in fact, is always formed by the repetition of a dash followed by a number of spaces. Let’s take for example the dashed line $T = (n_1) = (3)$, which for clarity is shown up to column 9, with numbered dashes and the spaces highlighted:

0 1 2 3 4 5 6 7 8 9
3     1     2     3

To apply the procedure we established at the beginning, we must first establish when the $x$-th dash belongs to the first line and follows a space (then we will deal with those dashes preceding a space). Since the dashed line is of the first order, all the dashes belong to the first row (which in this case is the only one); it therefore remains to be determined which of them follow a space.
To arrive at the characterization, it is sufficient to observe that, in a dashed line of this type, all the dashes except that of column 0 follow a space; therefore:

1. The $x$ such that the $x$-th dash belongs to the first line and follows a space are all those that satisfy the condition $x \gt 0$;
2. The value of the column $v = \mathrm{t\_value}(x)$ of the corresponding dashes is $n_1 \cdot x$, for the Proposition T.1 (Linear first order $\mathrm{t}$ and $\mathrm{t\_value}$ functions);
3. Then, the space we are looking for is $n_1 \cdot x - 1$.

From which we obtain the actual characterization:

Spaces of a first order dashed line preceding a dash

All and only the spaces of a first order dashed line $T = (n_1)$ preceding a dash are given by the formula

$n_1 \cdot x - 1$

where $x \gt 0$.

Similarly, we obtain the characterization of the spaces that follow a dash, with the only difference being that in this case $x \geq 0$, because also the dash of column 0 precedes a space:

Spaces of a first order dashed line following a dash

All and only the spaces of a first order dashed line $T = (n_1)$ following a dash are given by the formula

$n_1 \cdot x + 1$

where $x \geq 0$.

To overcome the difference in the constraint on the $x$ between the Propositions L.C.1 and L.C.2, one can think of theorising dashed lines that also include negative values of the dashes. This was not yet necessary for our research, as the prime numbers are by definition positive, but it could be in the future. In any case it is an open point of dashed line theory.

## Characterization of the spaces of a second order dashed line

Similarly to how it was done for the first order, we can proceed with the dashed lines of the second, even if the question becomes more complex, because the rows become two. The starting point is the Theorem T.2 (Computation of the $x$-th dash row in a second order linear dashed line), which says when the $x$-th dash belongs to the first line of a second order dashed line $T = (n_1, n_2)$:

$$\mathrm{t}_T(x) \in T \Leftrightarrow (n_1 x - 1) \mathrm{\ mod\ } (n_1 + n_2) \lt n_2$$

where it is understood that the modulus can start from a minimum of zero, so this equivalence can be rewritten as follows:

$$\mathrm{t}_T(x) \in T \Leftrightarrow 0 \leq (n_1 x - 1) \mathrm{\ mod\ } (n_1 + n_2) \lt n_2 \tag{1}$$

Since we are interested in not all the dashes of the first row, but only those that follow a space, we must exclude those dashes that do not follow a space, that is, the ones that follow a column containing at least one dash. To do this, first of all we can observe that in (1), the expression $(n_1 x - 1) \mathrm{\ mod\ } (n_1 + n_2)$ due to Lemma T.1 (Equality between $(n_1 x - 1) \mathrm{\ mod\ } (n_1 + n_2) + 1$ and $n_1 x \mathrm{\ mod^{\star}\ } (n_1 + n_2)$) can be rewritten as $n_1 x \mathrm{\ mod^{\star}\ } (n_1 + n_2) - 1$:

$$\mathrm{t}_T(x) \in T \Leftrightarrow 0 \leq n_1 x \mathrm{\ mod^{\star}\ } (n_1 + n_2) - 1 \lt n_2$$

Adding 1 to all members of inequality, this formula in turn can be rewritten as follows:

$$\mathrm{t}_T(x) \in T \Leftrightarrow 1 \leq n_1 x \mathrm{\ mod^{\star}\ } (n_1 + n_2) \leq n_2 \tag{2}$$

But due to Theorem T.3 (Difference between the value of a dash of row $i$ and the previous of row $j$, in a second order linear dashed line)), if the $x$-th dash of $T$ belongs to row 1, then the expression $n_1 x \mathrm{\ mod^{\star}\ } (n_1 + n_2)$ represents the difference between its value and that of the previous dash of row 2. So, assuming that $\mathrm{t}_T(x) \in T$, the inequality $1 \leq n_1 x \mathrm{\ mod^{\star}\ } (n_1 + n_2) \leq n_2$ which follows from (2), does not provide any additional information, because it simply asserts that this difference can vary between a minimum of 1 and a maximum of $n_2$, which are all possible values (the value $n_2$ is obtained when in the column of $\mathrm{t}_T(x)$ there is also a dash in the second row; as the ordering of dashes has been defined, in this case the difference is not 0 but $n_2$). But we can modify this inequality in order to require that the $x$-th dash follow a space, the only action required is setting that the difference with the value of the previous dash of the other row is greater than 1, i.e.:

$$n_1 x \mathrm{\ mod^{\star}\ } (n_1 + n_2) \gt 1 \tag{3}$$

Instead, in order to require that the $x$-th dash precedes a space, we must set that the difference with the next dash of the other row is not 1, i.e., for the Corollary of Theorem T.3 (Difference between the value of a dash of row $i$ and the next of row $j$, in a second order linear dashed line):

$$n_2 - n_1 x \mathrm{\ mod^{\star}\ } (n_1 + n_2) \neq 1$$

that is

$$n_1 x \mathrm{\ mod^{\star}\ } (n_1 + n_2) \neq n_2 - 1 \tag{4}$$

Combining the formulas (2), (3) and (4), and using the known formula for calculating $\mathrm{t\_value}$ provided by the Corollary of Theorem T.8 (Formula for calculation of linear second order function $\mathrm{t\_value}$ for first row), we can then assert the following Propositions, together to some respective Corollaries which follow due to Lemma T.1:

Spaces of a second order dashed line preceding a dash on the first row

All and only the spaces of a second order dashed line $T=(n_1, n_2)$ preceding a dash of component $n_1$ are given by the formula

$n_1 \biggl \lceil \cfrac{n_2 \cdot x + 1}{n_1 + n_2} \biggr \rceil - 1$

where $x$ is such that

$$n_1 x \mathrm{\ mod^{\star}\ } (n_1 + n_2) \in P_T(1)$$

and

$$P_T(1) := \{2, \ldots, n_2\}$$

Spaces of a second order dashed line preceding a dash of the first row, second form

All and only the spaces of a second order dashed line $T=(n_1, n_2)$ preceding a dash of component $n_1$ are given by the formula

$n_1 \biggl \lceil \cfrac{n_2 \cdot x + 1}{n_1 + n_2} \biggr \rceil - 1$

where $x$ is such that

$$(n_1 x - 1) \mathrm{\ mod\ } (n_1 + n_2) \in \{1, \ldots, n_2 - 1\}$$

Spaces of a second order dashed line following a dash on the first row

All and only the spaces of a second order dashed line $T=(n_1, n_2)$ following a dash of component $n_1$ are given by the formula

$n_1 \biggl \lceil \cfrac{n_2 \cdot x + 1}{n_1 + n_2} \biggr \rceil + 1$

where $x$ is such that

$$n_1 x \mathrm{\ mod^{\star}\ } (n_1 + n_2) \in S_T(1)$$

and

$$S_T(1) := \{1, \ldots, n_2 - 2, n_2\}$$

Spaces of a second order dashed line following a dash on the first row, second form

All and only the spaces of a second order dashed line $T=(n_1, n_2)$ following a dash of component $n_1$ are given by the formula

$n_1 \biggl \lceil \cfrac{n_2 \cdot x + 1}{n_1 + n_2} \biggr \rceil + 1$

where $x$ is such that

$$(n_1 x - 1) \mathrm{\ mod\ } (n_1 + n_2) \in \{0, \ldots, n_2 - 3, n_2 - 1\}$$
It could be easily generalized, finding a characterization for the spaces preceding/following a dash of a generic line $i$, but for the sake of brevity we neglect this aspect because, as it has been said, for our purposes only the first line is of interest.
Let’s take for example the dashed line $(n_1, n_2) = (2, 3)$, where the number of the $x$-th dash has been indicated inside its cell, and the spaces have been highlighted:

0 1 2 3 4 5 6
2   1   3   4
3     2     5

For example, let’s apply the Corollary of Proposition L.C.3 to determine the spaces preceding a dash of the first row. They are given by the formula

$2 \biggl \lceil \cfrac{3 \cdot x + 1}{2 + 3} \biggr \rceil - 1 = 2 \biggl \lceil \cfrac{3 \cdot x + 1}{5} \biggr \rceil - 1$

where $x$ is such that

$$0 \lt (2 x - 1) \mathrm{\ mod\ } (2 + 3) \lt 3$$

i.e., after performing the intermediate calculations:

$$0 \lt (2 x - 1) \mathrm{\ mod\ } 5 \lt 3$$

Let’s check which $x$ are dashes of the first row following a space:

 $x$ Remainder of division $(2x - 1)/5$ Does $x$ respect the characterization? 1 1 Yes 2 3 No 3 0 No 4 2 Yes 5 4 No

So, dashes 1 and 4 are the only two that belong to the first row and follow a space, and that’s what we expected, as it can be verified by observing the table. The corresponding spaces are:

$2 \biggl \lceil \cfrac{3 \cdot 1 + 1}{5} \biggr \rceil - 1 = 2 \biggl \lceil \cfrac{3 + 1}{5} \biggr \rceil - 1 = 2 \biggl \lceil \cfrac{4}{5} \biggr \rceil - 1 = 2 \cdot 1 - 1 = 1$
$2 \biggl \lceil \cfrac{3 \cdot 4 + 1}{5} \biggr \rceil - 1 = 2 \biggl \lceil \cfrac{12 + 1}{5} \biggr \rceil - 1 = 2 \biggl \lceil \cfrac{13}{5} \biggr \rceil - 1 = 2 \cdot 3 - 1 = 5$

The spaces obtained are the correct ones, as it can be verified from the table.

On the other hand, let’s check whether the mismatched elements do not respect the criterion $0 \lt (n_1 x - 1) \mathrm{\ mod\ } (n_1 + n_2) \lt 3$: for the dash $x = 3$, which does not follow a space, we have

$$(n_1 x - 1) \mathrm{\ mod\ } (n_1 + n_2) = (2 \cdot 3 - 1) \mathrm{\ mod\ } (2 + 3) = 5 \mathrm{\ mod\ } 5 = 0$$

which, in fact, does not match the inequality.

## Characterization of the spaces of a third order dashed line

Using a reasoning almost identical to that just seen for second order dashed lines, it is possible to obtain the characterization of the spaces for the third one; the only aspect that changes consists in the formulas involved.
We always start from the condition that tells us when the $x$-th dash belongs to the first line of a third order dashed line $T = (n_1, n_2, n_3)$, in which $n_1$, $n_2$ and $n_3$ are two by two coprime; such condition is provided by the Theorem T.4 (Computation of the row of the $x$-th dash for a third order linear dashed line with two by two coprime components):

$$\mathrm{t}_T(x) \in T \Leftrightarrow (n_2 + n_3) \cdot n_1 x \mathrm{\ mod\ } N \in R_T(1)$$

where

$$N := n_1 n_2 + n_1 n_3 + n_2 n_3$$

and

$$R_T(1) := \{n_2 a + n_3 b \mid a \in \{1, \ldots, n_3\}, b \in \{1, \ldots, n_2\}\} \tag{5}$$

This is the condition that indicates when a dash belongs to the first line, but we are interested in those ones following a space. To obtain them, it is sufficient to take the elements of a more restricted set of $R_T (1)$, which is as follows:

$$P_T(1) := \{n_2 a + n_3 b \mid a \in \{2, \ldots, n_3\}, b \in \{2, \ldots, n_2\}\} \tag{6}$$

where P means “Preceding”, to indicate that the spaces preceding a dash are being characterized.

Due to the Theorem T.5 (Difference between the value of a dash of a row and the one of the previous dashed of other rows, for a third order linear dashed line), in the hypothesis that the $x$-th dash of $T$, which for brevity we will name $t$, belongs to the first line, the variable $a$ present in the definition of $R_T(1)$, i.e. in the (5), represents the difference between the value of $t$ and that of the previous dash of row 3, instead the $b$ represents the difference between the value of $t$ and that of the previpos dash of row 2. Hence, to express the fact that $t$ follows a space, both $a$ and $b$ must be greater than 1: then, we obtain the equation (6).
Let’s take for example the dashed line $T = (2, 3, 5)$, shown up to the column 10, with the dashes numbered and the spaces highlighted:

0 1 2 3 4 5 6 7 8 9 10
2   1   3   5   7   9
3     2     6     8
5         4         10

The condition that tells us when the $x$-th dash belongs to the first line, replacing the components, is the following:

$$\mathrm{t}_T(x) \in T \Leftrightarrow (3 + 5) \cdot 2 x \mathrm{\ mod\ } N \in R_T(1)$$

where

$$N = 2 \cdot 3 + 2 \cdot 5 + 3 \cdot 5 = 31$$

and

$$R_T(1) = \{3a + 5b \mid a \in \{1, 2, 3, 4, 5\}, b \in \{1, 2, 3\}\} = \newline \{8, 11, 14, 17, 20, 13, 16, 19, 22, 25, 18, 21, 24, 27, 30\}$$

That is, summarizing:

$$\mathrm{t}_T(x) \in T \Leftrightarrow 16x \mathrm{\ mod\ } 31 \in R_T(1)$$

where

$$R_T(1) = \{8, 11, 14, 17, 20, 13, 16, 19, 22, 25, 18, 21, 24, 27, 30\}$$

The set $P_T(1)$, instead, is the following:

$$P_T(1) = \{n_2 a + n_3 b \mid a \in \{2, \ldots, n_3\}, b \in \{2, \ldots, n_2\}\} = \newline \{3a + 5b \mid a \in \{2, 3, 4, 5\}, b \in \{2, 3\}\} = \newline \{16, 19, 22, 25, 21, 24, 27, 30\}$$

For each $x$-th dash, let’s verify if $16x \mathrm{\ mod\ } 31$ belongs to $R_T(1)$ e a $P_T(1)$:

 $x$ $16x \mathrm{\ mod\ } 31$ Is it in $R_T(1)$? Is it in $P_T(1)$? 1 16 Yes Yes 2 1 No No 3 17 Yes No 4 2 No No 5 18 Yes No 6 3 No No 7 19 Yes Yes 8 4 No No 9 20 Yes No 10 5 No No

Then:

• The dashes 1, 3, 5, 7 e 9 correspond to the condition relative to $R_T(1)$, then they are on the first row;
• The dashes 1 e 7 correspond to the condition relative to $P_T(1)$, then they are on the first row and follow a space.

The formula for calculating $\mathrm{t\_value}(x)$ for a third order dashed line $(n_1, n_2, n_3)$ can be obtained by the Theorem T.11 (Partial solution of the downcast characteristic equation of linear $\mathrm{t}$, from the third to the first order) starting with the calculation of $y$, setting $i = 1$, $j = 2$ and $k = 3$, and finally applying the Proposition T.1 (Linear first order $\mathrm{t}$ and $\mathrm{t\_value}$ functions):

$\mathrm{t\_value}(x) = n_1 \biggl \lceil \cfrac{n_2 \cdot n_3 \cdot x + n_2 + n_3}{n_1 \cdot n_2 + n_1 \cdot n_3 + n_2 \cdot n_3} \biggr \rceil$

However, from the third order on we prefer to free space characterization from the formulas for $\mathrm{t\_value}$, because they are different aspects that in our opinion should be afforded saparately (for example, it should be remembered that the previous formula is valid under proper hypotheses, that are not relevant for the characterization of spaces). So, indicating simply with $\mathrm{t\_value}(x)$ the value of the $x$-th dash, we obtain the following characterization for the spaces preceding a dash of the first row, relative to the third order:

Spaces of a third order dashed line preceding a dash on the first row

All and only the spaces of a third order dashed line $T=(n_1, n_2, n_3)$ preceding a dash of component $n_1$ are given by the formula

$\mathrm{t\_value}_T(x) - 1$

where $x$ is such that

$$(n_2 + n_3) \cdot n_1 x \mathrm{\ mod\ } (n_1 n_2 + n_1 n_3 + n_2 n_3) \in P_T(1)$$

and the set $P_T(1)$ is defined as follows:

$$P_T(1) := \{n_2 a + n_3 b \mid a \in \{2, \ldots, n_3\}, b \in \{2, \ldots, n_2\}\}$$

Furthermore, similarly to how it was done for the second order, the following characterization can be obtained for the spaces that follow a dash of the first line:

Spaces of a third order dashed line following a dash on the first row

All and only the spaces of a third order dashed line $T=(n_1, n_2, n_3)$ which follow a dash of component $n_1$ are given by the formula

$\mathrm{t\_value}_T(x) + 1$

where $x$ is such that

$$(n_2 + n_3) \cdot n_1 x \mathrm{\ mod\ } (n_1 n_2 + n_1 n_3 + n_2 n_3) \in S_T(1)$$

and the set $S_T(1)$ is defined as follows:

$$S_T(1) := \{n_2 a + n_3 b \mid a \in \{1, \ldots, n_3 - 2, n_3\}, b \in \{1, \ldots, n_2 - 2, n_2\}\}$$
It can be observed that the difference between the sets $P_T (1)$ and $S_T(1)$ is that in the former, compared to $R_T (1)$, 2 was excluded from the sets used within the definition of $R_T (1)$, and 2 is the smallest number of such sets, while in the second set, $n_3 - 1$ and $n_2 - 1$ were excluded. This is analogous to what can be observed in the second order, comparing the formulas (3) and (4) with the (2). While the P of $P_T (1)$ stands for “Previous”, the S of $S_T(1)$ stands for “Subsequent”.

For brevity we omit the demonstration of Proposition L.C.6, because the procedure to obtain it is perfectly analogous to the one seen for the second order.

The goal, once the characterizations for the first orders have been found, is generalizing it, that is, determining one characterization which is universally valid for any order. This part of the investigation is still ongoing, and is still waiting for a solution.
It can be assumed, taking for example the Proposition L.C.5, that for the generic order $k$ it is necessary to define the set $P_T (1)$ on the basis of $k-1$ starting sets, each of which element 1 is excluded, but this remains to be proved.
A further observation is that, in the condition of the second order, we see the espression $(n_1 + n_2)$, that, using the elementary symmetric polynomials, can be rewritten as $\sigma_1(n_1, n_2)$; in the condition of the third order we see instead the expression $(n_1 \cdot n_2 + n_1 \cdot n_3 + n_2 \cdot n_3)$, which can be rewritten as $\sigma_2(n_1, n_2, n_3)$. Both expressions, by setting $k \in \{2, 3\}$, can then be rewritten with the only expression $\sigma_{k - 1}(n_1, \cdots, n_k)$: this type of writing allows to easily generalize to higher orders, but also in this case the hypothetical formulas have yet to be proved.