Number unit, Hilbert Space, Quantum Number Theory.

Abstract

Investigate what I percieve to be a Hilbert Space of numbers. I use the concept of a number unit, in analogy to length, area, volume etc. Such that a prime has dimensions of \(p\). Compunds and partitions are visualised.

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Sum of Primes

Some scope for things such as \[2p + 2p = 4p^2 \\ \\ \begin{bmatrix} 1 \\ 0 \end{bmatrix} + \begin{bmatrix} 1 & 0 \end{bmatrix} \to \begin{bmatrix} \begin{bmatrix} 1 & 0 \end{bmatrix} \\ 0 \end{bmatrix} \to \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}\]

But this system appears to break down in many circumstances. For this particular example, one could continue to add \(2p\) and generate \(2^np^n\) like structures of rank \(n\) quite comfortably.

Obviously this is the overlap of product and multiplication, as this is still the dyadic product. something else is needed, some other operation.

Higher Compounds

Now a matrix has two sides, [?]. But when the array is extended to a cube, there will be 6 sides.

If a number has \(3\) prime factors \(\{A,B,C\}\), then there are the six selections, as permutations of a pair from this set. Thus, where the simple rules of left and right operations worked before, now there will be more rules. Let us draw up a nota