# Maximum space distance calculator

The Maximum space distance calculator is a program which we developed for computing the value of the maximum distance between two consecutive spaces in a linear dashed line or double dashed line (for a theoretical introduction, see our proof strategy based on spaces and the detailed study Upper bound for maximum distance between consecutive spaces).

The program has a textual interface, it can receive paremeters from command line but it can be also executed without parameters (with a double click): in the latter case it will ask user all necessary parameters, in an interactive fashion.

The results given by the program are correct for any dashed line such that the length of its period does not exceed the maximum signed integer representable with 64 bit ($2^{63} - 1$). But the true limit is the execution time, which depends mainly on the processor speed and is proportional to the period length of the input dashed line.

Type Operating system Link Size Notes
Executable Windows 64 bit MaxDistSpaces_windows_en.exe 1 MB
Executable Linux 64 bit MaxDistSpaces_linux_en 2,9 MB Compiled on kernel 5.15
Source code Multi-platform MaxDistSpaces_en.zip 14,3 KB Progetto Code::Blocks, scritto in C++ utilizzando solo librerie standard

## Results obtained for dashed lines $T_k$

We’ll write in the following table the results we obtained by executing the program for dashed lines of the kind $T_k$, that are dashed lines of order $k$ with the first $k$ prime numbers as components: $(2, 3, 5, \ldots, p_k)$.

$k$ $T_k$ Maximum distance between consecutive spaces First couple 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 noted that in all cases, except $k = 9$, the maximum distance is equal to two times the second to last 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, by the rule above, the maximum distance should be 38, but it’s 40, that is $2 \cdot (p_{k-1} + 1)$. Nevertheless, a very strong trend seems to be present, and it would be nice to justify it in some way.

About the case of $k = 9$, another thing we noted, which may be connected to the observed anomaly, is that it’s the only case in which the program found more than one couple of spaces at 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 the other cases the couple in the table is the only one in that interval, and in addition 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 be also the case of a couple of spaces over the half 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 never happened). For $k = 9$, instead, the program found six couple of spaces at 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 executed the program for $k \gt 11$, or if you feel to have a good theoretical explanation of the results of the program, we would be glad to receive your contribution.