In efforts to find prime numbers one can make use of the fact that 2 is the only even prime. Using the 2d bloack matrix structure, the number map can be contracted into a quarter of the space. Due to the form of the matrices \(a,b,c,d\) they only pertain to information about one of the numbers in the even odd pairs (1,2),(3,4),(5,6) etc...
The top row is the odd number’s divisor information and the bottom the even number’s divisor information. As we ’don’t care’ about the even numbers in a prime search one can allow \[a \to 0 \;\; b \to 0 \;\; d \to 0\]
Now \(c\) is the only matrix that pertains to odd numbers divisability. This creates a simplified map of the number field where an entry in a number row other than the \(2\) or \(I\) bounding the lower traingle counts as an indicator that the odd partner of that row’s pair is not prime. The revised matrix is
\[\begin{matrix} 1,2 \\ 3 \\ 5 \\ 7 \\ 9 \\ 11 \\ 13 \\ 15 \\ 17 \\ 19 \\ 21 \\ 23 \\ 25 \\ 27 \end{matrix} \begin{bmatrix} 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & I & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & . & I & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & . & 0 & I & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & c & . & 0 & I & 0 & 0 & 0 & 0 & 0 & 0 \\ 2 & . & . & 0 & 0 & I & 0 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & . & 0 & 0 & I & 0 & 0 & 0 & 0 \\ 2 & I & c & . & 0 & 0 & 0 & I & 0 & 0 & 0 \\ 2 & . & . & 0 & . & 0 & 0 & 0 & I & 0 & 0 \\ 2 & . & . & 0 & . & 0 & 0 & 0 & 0 & I & 0 \\ 2 & c & 0 & c & 0 & . & 0 & 0 & 0 & 0 & I \\ 2 & . & . & . & 0 & . & 0 & 0 & 0 & 0 & 0 & I \\ 2 & 0 & c & 0 & 0 & 0 & . & 0 & 0 & 0 & 0 & 0 & I \\ 2 & I & 0 & . & c & 0 & . & 0 & 0 & 0 & 0 & 0 & 0 & I \\ \end{bmatrix}\]
This matrix shows that out of the numbers listed on the side only 9, 15, 21, 25 and 27 are not prime. Double prime pairs are simply shown as consecutive rows with no entries between left column and diagonal.
It is an odd property of the original map that in block 2x2 matrix representaion a matrix with a single 1 in the top right corner did not appear. This would need to be searched for in the sequence. But it may be a consequence of the lower triangular symmetry of all matrices involved. If there does exist a generalised product governing these matrices that retains the conventional arithmetic of numbers then that may well conserve the lower triangular-ness of matrices involved as matrix addition,multiplication and kronecker products/sums do.
After the removal of the complicated a.b.d terms. A further compactification of this matrix is possible such that the divisor field takes only a 16th of it’s original space. One can define the block matrices of block matrices \[M_4=\begin{bmatrix} 2 & 0 \\ 2 & I \end{bmatrix} \;\; l=\begin{bmatrix} 2 & 0 \\ 2 & 0 \end{bmatrix} \;\; r=\begin{bmatrix} 2 & c \\ 2 & 0 \end{bmatrix} \\ x=\begin{bmatrix} 0 & 0 \\ c & 0 \end{bmatrix} \;\; y=\begin{bmatrix} 0 & c \\ 0 & 0 \end{bmatrix} \;\; z=\begin{bmatrix} c & 0 \\ 0 & 0 \end{bmatrix}\]
And \(I_4\) takes the place of the two diagonal \(I_2\) matrices. The compacted field is now \[\begin{matrix} 1,2,3 \\ 5,7 \\ 9,11 \\ 13,15 \\ 17,19 \\ 21,23 \\ 25,27 \end{matrix} \begin{bmatrix} 4 & 0 & 0 & 0 & 0 & 0 & 0 \\ l & I & 0 & 0 & 0 & 0 & 0 \\ r & 0 & I & 0 & 0 & 0 & 0 \\ 4 & x & 0 & I & 0 & 0 & 0 \\ l & 0 & 0 & 0 & I & 0 & 0 \\ r & y & 0 & 0 & 0 & I & 0 \\ 4 & z & x & 0 & 0 & 0 & I \end{bmatrix}\]
In fact from this compression one can identify double primes as the rows with no non-zero entry between the left hand column and diagonal! For a primality check, any x in a row indicates that the second of the two odd numbers relating to that row is not prime. A y or z indicates the first is not prime.