Another method to generate this pattern is to take the process in equation 1 and to generate a criterion for wheter a point in the map is filled or not.
By inspection, the integer \(m\) associated with a grey line starting from pixel \((p_x,1)\) is \((p_x +1)/2\), so the grey line originating from the 3^rd pixel along, that is, \((p_x=3)\), relates to the primality of \(2\) etc.
The yellow lines representing the divisors \(n\) are created using the transformation \((2n-1,1)\), and starts from the \(n^{th}\) pixel along at the top left.
The data map is therefore the sum of the discreet points\[M=\sum_{i=0}^{\infty}\sum_{n=1}^{m/2} ( (2n-1)(i+1), i+1) =1\]
If one were to sum across the diagonals (grey lines), any number that only crosses the start (divisor of 1) line and the finish (divisor of itself) is prime.
Another way of expressing this is that for the data matrix \(M\), \(i \in \mathbb{Z}^0\), and \(n \in \mathbb{Z}^+\) \[if \; ∃ \; i,n : ( \; \alpha = (2n-1)(i+1)\;, \;\; \beta = i+1 \;) \\ M_{\alpha \beta} = 1\] otherwise, \[M_{ \alpha \beta} =0\]
For a given number \(m\), elements required to check its primality are \[m=1; \; M_{1 1} \\ m=2; \; M_{2 2} +M_{1 3} \\ m=3; \; M_{3 3} +M_{2 4} +M_{1 5} \\ m=N; \; M_{N N} +M_{N-1 N+1} +M_{N-2 N+2} + ... + M_{1 2N-1}\]
So we create a measure \(P(N)\) to collect these arguments for any \(N\) \[P(N) = \sum_{i=0}^{N-1} M_{N-i N+i}\]
We then have the knowledge that \[\forall \; N \in \mathbb{P}: P(N)=2, \\ \forall \; N \notin \mathbb{P}: P(N) \ne 2\]
This measure \(P(N)\) is the number of divisors of the number \(N\).