# Maximum distance between spaces: experimental results

Prerequisite:

One of the strategies that we have developed to try to prove Goldbach’s Conjecture is the one based on spaces. In particular, within the context of this strategy, it’s important to introduce concept of maximum distance between consecutive spaces in a double dashed line. To deepen this concept, we began to study it for single dashed lines, as a first phase, before moving on to double one. Thanks to the maximum space distance calculator we have calculated the maximum distance between consecutive spaces of the dashed lines we have called $T_k$, used in our proof strategies, i.e. those ones having as components the first $k$ consecutive prime numbers. The results, which we’ll present below, showed surprising regularities.

## Maximum distance between consecutive spaces in dashed lines $T_k$

In the following table we’ll report the results we obtained using the maximum space distance calculator for dashed lines of type $T_k$:

$k$ $T_k$ Maximum distance between consecutive spaces First pair of consecutive spaces at maximum distance
2 $(2,3)$ 4 1, 5
3 $(2,3,5)$ 6 1, 7
4 $(2,3,5,7)$ 10 1, 11
5 $(2,3,5,7,11)$ 14 113, 127
6 $(2,3,5,7,11,13)$ 22 9939, 9461
7 $(2,3,5,7,11,13,17)$ 26 217127, 217153
8 $(2,3,5,7,11,13,17,19)$ 34 60043, 60077
9 $(2,3,5,7,11,13,17,19,23)$ 40 20332471, 20332511
10 $(2,3,5,7,11,13,17,19,23,29)$ 46 417086647, 417086693
11 $(2,3,5,7,11,13,17,19,23,29,31)$ 58 74959204291, 74959204349

We noticed that in all cases, except for $k = 9$, the maximum distance is equal to twice the penultimate component of the dashed line, i.e. $2 \cdot p_{k-1}$. For example, for $k = 4$ the maximum distance is 10 which is equal to $2 \cdot 5 = 2 \cdot p_3 = 2 \cdot p_{k-1}$. The case of $k = 9$ is an exception because according to this rule the maximum distance should be 38, while it is 40, i.e. $2 \cdot (p_{k-1} + 1)$. Despite this, there seems to be a very strong trend, so there may be a theoretical foundation, and that therefore the phenomenon can be explained in some way.

Regarding the case of $k = 9$, another aspect we noticed, which perhaps has to do with the anomaly found, is that this is the only case in which there are more than one pairs of spaces at the maximum distance in the first half of the first period of the dashed line, i.e. in the interval $\left[0, \frac{p_1 p_2 \ldots p_k}{2} - 1 \right]$. In fact, in all other cases the reported pair of spaces is the only one found in this interval, and then by symmetry there is another one in the interval $\left[\frac{p_1 p_2 \ldots p_k}{2}, p_1 p_2 \ldots p_k - 1 \right]$. There could also be the case of a pair of spaces straddling the middle of the first period, i.e. such that the first space is in the first interval and the second space is in the second interval, but for $k \gt 2$ this case has never occurred. For $k = 9$, on the other hand, the program has found six pairs of spaces at the maximum distance in the first half of the first period (and, by symmetry, there are as many in the second half): specifically, they are (20332471, 20332511); (24686821, 24686861); (36068191, 36068231); (65767861, 65767901); (82370089, 82370129); (97689751, 97689791).

If you ran the program for $k \gt 11$, or if you feel you have a good theoretical explanation of the program’s results, we would be happy to receive your contribution.