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.
The method we’ll describe here is based on the following observation. After all, we already know what the interval in which we want to search for spaces looks like, that is (1, 2n); in particular, we know that the minimum is always 1, and that this minimum is excluded. Therefore, given a generic double dashed line D, the following possibilities can arise:
- 1 is a space. If so, it’s the first space, since 0 definitely isn’t. Returning to the initial notations, we therefore have that \mathrm{t\_space}_D(1) = 1. Then, if there are spaces between 1 and 2n excluded, among them there will certainly be the second one. So in this case we have to prove that \mathrm{t\_space}_D(2) \lt 2n.
- 1 is not a space. In this case, if there are spaces between 1 and 2n excluded, among them there will certainly be the first one, so in this case we have to prove that \mathrm{t\_space}_D(1) \lt 2n. However, let’s re-examine the condition of the previous case: if the second space were smaller than 2n, even more so the first space would be smaller. So, to prove that \mathrm{t\_space}_D(1) \lt 2n, we can prove that \mathrm{t\_space}_D(2) \lt 2n: in this case it is not a necessary condition, but it is sufficient.
As we have seen, in any case the condition \mathrm{t\_space}_D(2) \lt 2n would guarantee the existence of at least one space in the interval (1, 2n). This, if D is the double dashed line of Hypothesis H.1 (second form), would guarantee the truth of the Hypothesis itself, which can therefore be further reformulated:
Hypothesis of existence of Goldbach pairs based on dashed lines (third form)
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 second space of the double dashed line T_k^{(0, r_2, \ldots, r_k)} is less than 2n.
This reformulation takes us back to the initial problem of calculating spaces, but this time in a double dashed line instead of a “single” dashed line. But, as we said, the explicit calculation of the spaces is very difficult already in single dashed lines, let alone double ones! However, in our case we don’t have to compute the second space exactly, we just have to prove that it is less than some fixed number; so even an approximate value would be fine, as long as we can guarantee that the real value does not deviate so much that it exceeds 2n. Generalizing, given a double dashed line D we can make a reasoning of this type:
- Suppose we find a function B(x) such that, for each x, \mathrm{t\_space}_D(x) \lt B(x)
- So, given a constant C, to prove that \mathrm{t\_space}_D(x) \lt C it suffices to prove that B(x) \lt C
The advantage, reasoning this way, would be that we can try to find a function B(x) that is much simpler to express algebraically, than the function \mathrm{t\_space}, making the second point simple. However, the complicated problem would be to find this function, as required by the first point. We thought of dealing with this problem in increasing degrees of difficulty, starting from single dashed lines for increasing orders, to then extend the results to double dashed lines. We have currently found the function B(x) for (single) linear dashed lines up to fourth order (the results will be soon available here).