Prerequisite:
As we stated in the previous post, our intent is to find the conditions under which a generic double dashed line T^{(r_1, r_2, \ldots, r_k)} has spaces within a given range.
A way to prove Hypothesis H.1 (second form) is as follows.
While we previously focused on the minimum of the interval (1, 2n), let us now focus on its width. Seeing it as a real interval, we can say that its width is smaller than 2n - 1; it is smaller by an infinitesimal amount, being the interval interval open, however the width is strictly smaller than 2n - 1. Let us now consider the spaces of the double dashed line of the cited Hypothesis; in particular let’s consider the pairs of consecutive spaces: the first with the second, the second with the third, and so on. We calculate the differences (or distances, a term we prefer because it suggests a graphic representation) between these spaces. We will therefore have infinite distances: the second space minus the first, the third minus the second, and so on: it is an infinite sequence of integers. This sequence is limited, because there cannot exist two consecutive spaces arbitrarily distant (this derives from the fact that double dashed lines, like single ones, are periodic; thus, if s is a space, so is s + L, where L is the length of the period, so the space following s cannot be more than L columns far from it). But a limited sequence of integers admits a maximum, for which the following Definition is allowed:
Maximum distance between consecutive spaces of a dashed line
Let T be a dashed line. The maximum distance between two consecutive spaces of T will be indicated with the symbol \mathrm{MDS}(T).
Now, returning to Hypothesis H.1 (second form), suppose that \mathrm{MDS}\left(T_k^ {(0, r_2, \ldots, r_k)}\right) is less than 2n - 1. Then, if 1 is a space, certainly the interval (1, 2n) would contain another space of the dashed line, at least the second one. Indeed, if the second space were out of interval, it would be greater than or equal to 2n, so its distance from the previous space, which is 1, would be at least 2n - 1, in contrast to the assumption that \mathrm{MDS}\left(T_k^{(0, r_2, \ldots, r_k)}\right) \lt 2n - 1. This would allow to prove Hypothesis H.1 (second form) for all numbers 2n such that 1 is a space of the dashed line T_k^{(0, r_2, \ldots, r_k) }, that is, it is an element of a Goldbach pair for that number; thus Hypothesis H.1 would be proved for all numbers 2n of the form 1 + p, with p prime.
What if 1 isn’t a space? In this case it is necessary to prove that the first space is less than 2n. This can be done in at least two ways:
- Either with the method based on the concept of maximum distance between consecutive spaces, setting x = 1 instead of 2.
- Or, in proving that \mathrm{MDS}\left(T_k^{(0, r_2, \ldots, r_k)}\right) \lt 2n - 1, also include a theoretical “space number 0”. This might look like a trick, but if we consider the known expressions of \mathrm{t\_space}(x) for first or second order linear dashed lines and we try to evaluate them for x = 0, it turns out that, whatever the starting dashed line is, \mathrm{t\_space}(0) = -1, however counterintuitive it may be. The known formulas for calculating the function \mathrm{t\_space} for linear dashed lines have been derived for x \gt 0, but in the end they are mathematical functions which can also be evaluated for other values of x, such as 0, even if in doing so they obviously no longer calculate \mathrm{t\_space}, but some more general function that should be defined. The same could be true for double dashed lines: perhaps, after proving that \mathrm{MDS}\left(T_k^{(0, r_2, \ldots, r_k)}\right) \lt 2n - 1 starting from the “classical” definition of maximum distance, we could realize that the same proof would also hold for an “extended” definition of maximum distance, in which the succession of distances begins with the distance between the first space and a theoretical negative “space number 0” (will it be -1 like for linear dashed lines? The question is open). At this point, if the inequality \mathrm{MDS}\left(T_k^{(0, r_2, \ldots, r_k)}\right) \lt 2n - 1 also holds for this extended concept maximum distance, being \mathrm{t\_space}(0) negative, we would have \mathrm{t\_space}(1) \lt \mathrm{t\_space}(1) - \mathrm{t\_space}(0) \leq \mathrm{MDS}\left(T_k^{(0, r_2, \ldots, r_k)}\right) \lt 2n - 1, so \mathrm{t\_space}(1) would be less than 2n, as we want to prove.
- Even assuming that 1 is a space, we would arrive at an interesting result, namely that there exists a Goldbach pair for every even number of the form 1 + p, with p prime;
- A knowledge of the maximum distance between consecutive spaces in a dashed line would be useful more generally. For example, even just limiting to single dashed lines, given that in the validity interval the spaces coincide with prime numbers (Property T.1), it would be possible to prove the existence of a prime number in a certain interval. We have proved several theorems of this kind, according to which for each positive integer x there is a prime number p in an interval of the type (x, C(x) \cdot x), where C is a function which depends on x. Instead with the present proof strategy it would be possible, in principle, to prove that there exists a prime number p in an interval of the type (x, C(x) + x) (i.e. we would get an “additive” rather than a “multiplicative” theorem).
We can therefore reformulate Hypothesis H.1 (second form) as follows:
Hypothesis of existence of Goldbach pairs based on the maximum distance between spaces
Let 2n \gt 4 be an even number and let k be its validity order. Let r_i := 2n \mathrm{\ mod\ } p_i, for each i = 2, \ldots, k. Then the maximum distance between two consecutive spaces of the double dashed line T_k^{(0, r_2, \ldots, r_k)} is less than 2n - 1.
From a general point of view, we are interested in finding a function M such that, for each double dashed line D, \mathrm{MDS}(D) \leq M. In particular, we would like to find such a function that depends only on the order of D, i.e. on the number of rows of the table that represents it: thus there would be a maximum distance M(1) for all first-order double dashed lines, a maximum distance M(2) for all second-order ones, and so on.
In this respect, a flaw of the H.1.MDS Hypothesis is that the number 2n - 1 depends on n; it would be more convenient to replace 2n - 1 with an expression that depends on k, so that we can more easily study what happens when k varies. This is possible by remembering that k has been defined (Definition L.2) as the smallest integer such that p_{k+1}^2 \gt 2n. This means that p_k^2 \leq 2n (otherwise the relation defining k would also be satisfied by k-1 and therefore k would no longer have the characteristic of being the smallest integer that satisfies it), hence p_k^2 - 1 \leq 2n - 1. Therefore, if it is shown that the maximum distance between two consecutive spaces of the double dashed line T_k^{(0, r_2, \ldots, r_k)} is less than p_k^2 - 1, automatically it will also be less than 2n - 1. Therefore in Hypothesis H.1.MDS we can replace 2n - 1 with p_k^2 - 1, obtaining a new Hypothesis slightly stronger, but more suitable for studying what happens as k varies:
Hypothesis of existence of Goldbach pairs based on the maximum distance between spaces (stronger version)
Let 2n \gt 4 be an even number and let k be its validity order. Let r_i := 2n \mathrm{\ mod\ } p_i, for each i = 2, \ldots, k. Then the maximum distance between two consecutive spaces of the double dashed line T_k^{(0, r_2, \ldots, r_k)} is less than p_k^2 - 1.
Recall that the last two Hypotheses do not always imply Hypothesis H.1 (second form), but they do so only in case 1 is not a space (that is, when all r_i are different from 1). However, as we have seen, this limitation is of relative importance.
There are essentially two approaches to study the maximum distance between two consecutive spaces of a dashed line: the exact calculation and the approximate calculation.
As regards the exact calculation, we have developed a program that provides the result for any linear dashed line or double dashed line (provided you have enough time available: the execution time is proportional to the length of the dashed line period).
Calculator of the maximum distance between spaces
As regards the approximate calculation, we deal with it in a dedicated page:
Upper bound for maximum distance between consecutive spaces