Proof strategy based on dashes

Prerequisite:

The proof strategy which will be exposed here starts from one of the premises of Hypothesis H.1 (Hypothesis of existence of Goldbach pairs based on dashed lines), i.e. the fact that, if the dashed line T_k = (p_1, p_2,\ldots,p_k) has the first k prime numbers as its components, then p_1, which is its first component, is always the number 2. A direct consequence is this property:

Dashed preceding and following a space

In a linear dashed line with the first prime numbers as its components, in the columns immediately preceding or following a space, there is always a dash corresponding to the first dashed line component.

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

  • If a column is even, it’s divisible by 2, then it certainly contains a dash corresponding with the first row (by definition of dash);
  • Then, the spaces must be in odd columns;
  • Immediately before (or after) every odd number, there’s always an even one;
  • Then, immediately before (or after) every space there’s an even number;
  • Then, immediately before (or after) every space there’s a column containing a dash in the first row.

Based on this premise, original Goldbach’s equation

p + q = 2n

can be solved using \mathrm{t\_value} function:


Let’s start from original equation, imposing the bound that the two prime addends p and q are spaces belonging to the first row:

p + q = 2n

The components of dashed line T = (p_1, p_2, ... p_k) are the first k prime numbers, then p_1 = 2. So, the 2 appearing in the expression 2n is none other than the first component of the dashed line, then Goldbach’s equation can be rewritten like this:

p + q = p_1 n

Let’s add and subtract 1 to the term on the left:

p + 1 + q - 1 = p_1 n

Finally, let’s recollect:

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

Since p and q are both spaces, as a consequence of previous property, certainly a dash exists in the p_1 = 2 row (the first) in correspondence with the column p + 1 (immediately following the space p), and another exists in correspondence with column q - 1 (immediately preceding the space q). Then, since (by definition) \mathrm{t\_value}(w) indicates the column of w-th dash, two positive integers exist, 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 bounds (last two are obvious, since p_1 = 2, but they are shown for more clarity):

  1. x-th and y-th dashes must be on the first row;
  2. x-th dash must correspond with a column following a space;
  3. y-th dash must correspond with a column preceding a space.

Original addends p and q can be obtained from the following relations:

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 like this:

Hypothesis of existence of Goldbach’s pairs based on dashes

Let 2n \gt 4 be an even number, and let k be the validity order related to it. Then, two positive integers exist, x and y, which, referring to dashed line T_k:

  1. x-th and y-th dashes both are on first row;
  2. \mathrm{t\_value}(x) immediately follows a space;
  3. \mathrm{t\_value}(y) pimmediately precedes 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 between the validity interval.
Figure 1: proof strategy based on dashes. The x-th and the y-th dash respectively precede and follow some spaces (in green), which are between the validity interval (in yellow), and their sum is 2n.

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

  • If assertions 1., 2., 3. where true, then p = \mathrm{t\_value}(x) - 1 and q = \mathrm{t\_value}(y) + 1 would be spaces;
  • If also assertion 5. was true, p and q would be prime numbers, for Property T.1 (Spaces and prime numbers);
  • Assertion 4. is Goldbach’s equation rewritten in another form; then, if this was satisfied too, the sum of p and q would be 2n.

To be able to prove this hypothesis, we can proceed by steps. A possible observation is that it has two very strong premises:

  • It takes in exam the dashed line T_k which has the first prime numbers as its components (which has allowed us writing the 2n of Goldbach’s conjecture as p_1 n);
  • It binds the two spaces within the validity interval (then we had to set 2n \gt 4, otherwise no solution can be found).

We can think of relaxing these constraints, generalizing the previous hypothesis like this:

Hypothesis of existence of pairs of complementary spaces based on dashes

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

  1. x-th and y-th dashes are both on the first row;
  2. \mathrm{t\_value(x)} immediately follows a space;
  3. \mathrm{t\_value(y)} immediately precedes 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 k prime numbers as its components, with a more generic dashed line T still having prime numbers as its components, but not necessarily the first k. Consequently, in the equation \mathrm{t\_value(x)} + \mathrm{t\_value(y)} = p_1 n of previous hypothesis, p_1 has been replaced by n_1, keeping the fact that the number multiplicating n is the first component of the dashed line, but this component no longer is the first prime number p_1 = 2, but a generic prime number n_1, depending on the dashed line T considered.
Moreover, we have eliminated the constraint on the validity interval, first of all because this concept is defined only for dashed lines T_k, but also because, by removing this constraint, we can concentrate on an important aspect of the problem, which is far from trivial: finding two spaces of a dashed line whose sum is a fixed number, multiple of the first component of the dashed line itself.

With this premise, Hypothesis H.1.T (Hypothesis of existence of Goldbach’s pairs based on dashes) becomes a specific case of previous one; from this, in fact, we can obtain Hypothesis H.1.T changing two aspects:

  • setting n \gt 2 and T := T_k, where k is the validity order for 2n;
  • adding the condition that \mathrm{t\_value}(x) - 1 and \mathrm{t\_value}(y) + 1 are both within the validity interval of the dashed line T_k.

Then, the steps needed for proving Hypothesis H.1.T are the following:

  • Prove the Hypothesis H.1.T.A (Hypothesis of existence of pairs of complementary spaces based on dashes) for second order dashed lines (the proof for first order is rather simple, being a direct consequence of the fact that a first order dashed line is always a repeated sequence made by a dash followed by some spaces);
  • Prove it for third order dashed lines;
  • Analyze and compare among them the proofs obtained in previous steps, in order to get a proof which is valid for any dashed line of any order (so, including also all dashed lines of type T_k);
  • At the end, introduce the hypothesis about validity interval, then fully proving Hypothesis H.1.T.

For now, investigation is in progress on first step, and advancements about it are exposed in a dedicated page. When it will be completed, we will be able to deal with following ones, until concluding the global argument, in order to obtain a real Theorem, in the last step.
Last step could look very tough, but, actually, this will depend on the results of previous ones. An important aspect, in fact, is proving Hypothesis H.1.T.A in a constructive way, i.e. not only proving 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)}. By doing this, by adding the condition on validity interval of Hypothesis H.1.T, most of the work will be done, because the solutions will already been located in some way; the only thing to be verified will be only if this location is enough to be sure that at least a solution exist within the validity interval.

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