# Proof strategy based on dashes

Prerequisite:

The proof strategy set out here starts from one of the assumptions of the Hypothesis H.1 (Hypothesis of existence of Goldbach pairs based on dashed lines), i.e. the fact that the dashed line $T_k = (p_1, p_2,\ldots,p_k)$ hasthe first $k$ prime numbers as its components, so $p_1$, which is the first component, is always the number 2. Its direct consequence is the following property:

Dashes that precede and follow a space

In a linear dashed line whose components are the first prime numbers, in the columns that immediately precede or follow a space there is always a dash in correspondence with the first component of the dashed line.

The proof of the property is based on the fact that the first component of the dashed line is 2:

• If a column is even, it is divisible by 2, so it certainly contains a dash in correspondence with the first row (by definition of dash);
• Thus, spaces must necessarily be in odd columns;
• Immediately before (or after) every odd number, there is always an even number;
• Thus, immediately before (or after) each space there is an even number;
• Thus, immediately before (or after) each space there is a column that contains a dash in the first row.

Based on this assumption, we can rewrite the original Goldbach equation

$$p + q = 2n$$

using the function $\mathrm{t\_value}$:

Let’s start from the original equation, posing the constraint that the first two addends $p$ and $q$ are spaces and belong to the first row:

$p + q = 2n$

The components of the dashed line $T = (p_1, p_2, ... p_k)$ are the first $k$ prime numbers, so $p_1 = 2$. The 2 present in the expression $2n$, therefore, is none other than the first component of the dashed line, so Goldbach’s equation can be rewritten as:

$p + q = p_1 n$

We add and subtract 1 in the left member:

$p - 1 + q + 1 = p_1 n$

Finally, we collect:

$(p - 1) + (q + 1) = p_1 n$

Since $p$ and $q$ are both spaces, due to the previous property there is surely a dash in the row of 2 (i.e. the first one) corresponding to the column $p - 1$ (immediately preceding the space $p$), and there is another at column $q + 1$ (immediately following the space $q$). Then, since in general $\mathrm{t\_value}(w)$ indicates the column of the $w$-th dash, there will exist two positive integers $x$ and $y$ such that $\mathrm{t\_value}(x) = p - 1$ and $\mathrm{t\_value}(y) = q + 1$, so the previous equality can be rewritten as:

$\mathrm{t\_value}(x) + \mathrm{t\_value}(y) = p_1 n$

with the following constraints (the last two are obvious, being $p_1 = 2$, but are indicated for clarity):

1. the $x$-th and $y$-th dash must be on the first row;
2. the $x$-th dash must be in a column that precedes a space;
3. the $y$-th dash must occur in a column following a space.

The two original addends $p$ and $q$ can be obtained from these relationships:

$$p = \mathrm{t\_value}(x) + 1$$
$$q = \mathrm{t\_value}(y) - 1$$

From these observations, the Hypothesis H.1 (Hypothesis of existence of Goldbach pairs based on dashed lines) can be rewritten as follows:

Hypothesis of existence of Goldbach pairs based on dashes

Let $2n \gt 4$ be an even number, and let $k$ be the validity order. Then two positive integers $x$ and $y$ exist such that, with reference to the dashed line $T_k$:

1. the $x$-th and $y$-th dash are both on the first row;
2. $\mathrm{t\_value}(x)$ immediately precedes a space;
3. $\mathrm{t\_value}(y)$ immediately follows a space;
4. $\mathrm{t\_value}(x) + \mathrm{t\_value}(y) = p_1 n$;
5. $\mathrm{t\_value}(x) + 1$ and $\mathrm{t\_value}(y) - 1$ are both within the validity interval. Figure 1: Proof strategy based on dashes. The x-th and the y-th dash, respectively, immediately precede and follow two spaces (green), which are within the validity interval (yellow), and the sum of these spaces is 2n.

If this hypothesis were true, then the truth of Goldbach’s conjecture would be a direct consequence of it, given that:

• If 1., 2., 3. were true, $p = \mathrm{t\_value}(x) + 1$ and $q = \mathrm{t\_value}(y) - 1$ would be spaces;
• If also 5. were true, $p$ and $q$ would be primes, due to Property T.1 (Spaces and prime numbers);
• 4. is the rewritten Goldbach’s equation; thus, if it were also satisfied, the sum of $p$ and $q$ would be $2n$.

## Possible steps to be followed to prove Hypothesis H.1.T

We observe that the Hypothesis H.1.T has two quite strong assumptions:

• It examines the dashed line $T_k$, which has the first primes as its components (which allowed us to write the number $2n$ of Goldbach’s conjecture as $p_1 n$);
• Constrains the two spaces to be within the validity interval (so we had to set $2n \gt 4$, otherwise we wouldn’t be able to find solutions).

We can think of widening these constraints, generalizing the Hypothesis as follows:

Hypothesis of existence of Goldbach pairs based on dashes

Let $n \gt 0$ be an integer, and $T = (n_1, n_2, \ldots, n_k)$ a linear dashed line of order $k \geq 1$, whose components are prime numbers (not necessarily consecutive). Then, there exist two positive integers $x$, $y$ such that, with reference to the dashed line $T$:

1. the $x$-th and $y$-th dash are both on the first row;
2. $\mathrm{t\_value(x)}$ immediately precedes a space;
3. $\mathrm{t\_value(y)}$ immediately follows a space;
4. $\mathrm{t\_value(x)} + \mathrm{t\_value(y)} = n_1 n$.

In this new Hypothesis we have replaced the dashed line $T_k$, having the first prime numbers as components, with a more generic dashed line $T$ always having prime numbers as components, but not necessarily the first $k$. Consequently, in the equation $\mathrm{t\_value(x)} + \mathrm{t\_value(y)} = p_1 n$ of the previous hypothesis, $p_1$ is been replaced by $n_1$, keeping the fact that the number that multiplies $n$ is the first component of the dashed line, but this component is no longer the first prime number $p_1 = 2$, but a generic prime number $n_1$, which depends on the considered dashed line $T$.
We also removed the validity interval constraint, primarily because this concept is defined only for $T_k$ dashed lines, but also because, by removing this constraint, we can focus on an important aspect of the problem, all anything but trivial: find two spaces of a dashed line whose sum is a fixed number, multiple of the first component of the dashed line itself.

One could try to prove the Hypothesis H.1.T (Hypothesis of existence of Goldbach pairs based on dashes) step by step, starting with the Hypothesis H.1.T.A, as follows:

• Prove the Hypothesis H.1.T.A Hypothesis of existence of Goldbach pairs based on dashes) for second-order dashed lines (the proof for the first order it is quite trivial, being a direct consequence of the fact that a first order dashed line is always an alternated sequence of a dash and some spaces);
• Prove it for third order dashed lines;
• Analyze and compare the proofs obtained in the previous steps, to obtain, by induction, a valid proof for any dashed line of any order (thus also including all dashed lines of type $T_k$);
• Finally, introduce the hypothesis on the validity interval, thus proving the Hypothesis H.1.T.

Currently the investigation is underway on the first issue, and the relative progress is exposed in a dedicated page: Study about the existence of complementary space pairs based on second order dashed lines

The last issue may seem very difficult, but this will actually depend on the results of the previous ones. It is important, in fact, to prove the Hypothesis H.1.T.A in a constructive way, i.e. prove not only that $x$ and $y$ exist, but also, as accurately as possible, which are these numbers, and consequently which are the corresponding $\mathrm{t\_value(x)}$ and $\mathrm{t\_value(y)}$. In this way, when the condition on the validity interval of Hypothesis H.1.T is introduced, most of the work will have been done, because the spaces $p = \mathrm{t\_value}(x) + 1$ and $q = \mathrm{t\_value}(y) - 1$ will have already been localized somehow; it will remain to be verified only if this localization will be enough, to be sure that they are included in the validity interval, i.e. that they are prime.