N-dimensional Primenuum

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For \(i \geq 1\), let \(p_i\) be the \(i\)th prime number and \(g_i\) be the \(i\)th prime gap:

\[g_i = p_{i+1} - p_i\]

Notice that the prime gaps are even numbers, unless \(g_1 = 1\) (the gap between 2, the only even prime, and 3). For instance, a gap of 2 corresponds to twin primes (like 3 and 5).

Given an integer \(n \geq 1\) and the set of natural numbers \(\mathbb{N}\), we will work in the \(n\)-dimensional space \(\mathbb{N}^n\). Such a space has \(n\) orthogonal dimensions, that we can represent with direction vectors \(e_1, e_2, \ldots, e_n\) such that, for every \(i\) between \(1\) and \(n\), the components of vector \(e_i\) are all 0 unless that in position \(i\) which is \(1\). For instance, if \(n = 3\), we have:

\[ e_1 = (1, 0, 0) \\ e_2 = (0, 1, 0) \\ e_3 = (0, 0, 1) \]

An n-dimensional Primenuum is a sequence \((x_i)_{i \geq 1}\) of points in \(\mathbb{N}^n\) defined as follows:

1. the starting point \(x_1\) is the origin, that is the vector with all elements equal to 0;
2. for the following \(n\) steps, the next point \(x_{i+1}\) is the point reached starting from the previous point \(x_i\) and moving a distance of \(g_i\) along direction \(e_i\), that is \(x_{i+1} = x_i + g_i \cdot e_i\);
   
3. for the following \(n\) steps, the next point \(x_{i+1}\) is the point reached starting from the previous point \(x_i\) and moving a distance of \(g_i\) along direction \(-e_i\) (the opposite of \(e_i\)), that is \(x_{i+1} = x_i - g_i \cdot e_i\);
   
4. repeat the process from step 2.

For instance, consider the first 8 prime numbers and 7 gaps, and a space with \(n = 3\) dimensions. The primes are:

\[p_i = 2, 3, 5, 7, 11, 13, 17, 19\] The prime gaps are:

\[g_i = 1, 2, 2, 4, 2, 4, 2\] The first 8 points of the Primenuum are:

\[ \begin{array}{lcl} x_1 & = & (0, 0, 0) \\ x_2 & = & (0, 0, 0) + 1 \cdot (1, 0, 0) = (1, 0, 0) \\ x_3 & = & (1, 0, 0) + 2 \cdot (0, 1, 0) = (1, 2, 0) \\ x_4 & = & (1, 2, 0) + 2 \cdot (0, 0, 1) = (1, 2, 2) \\ x_5 & = & (1, 2, 2) - 4 \cdot (1, 0, 0) = (-3, 2, 2) \\ x_6 & = & (-3, 2, 2) - 2 \cdot (0, 1, 0) = (-3, 0, 2) \\ x_7 & = & (-3, 0, 2) - 4 \cdot (0, 0, 1) = (-3, 0, -2) \\ x_8 & = & (-3, 0, -2) + 2 \cdot (1, 0, 0) = (-1, 0, -2) \end{array} \]