Have you come across a similar theory to predict prime numbers?
I believe that I have identified a pattern by which prime numbers occur which is in the form: N = A(P,n) + B
…where A is a function of a multi-symmetrical shape (polyhedron) P and B is the adjusting function. The n defines the size (order) of the matrix of the multi-symmetrical shape. The pattern of observations is that A(P,n) leads close to the prime number N (please read an example further down) whereas B steps up/down to N by a small integer.
By a multi-symmetrical shape (P) is meant a matrix structure of tetrahedra, cubes, isocahedra. The function A is a characteristic of the multi-symmetrical geometric shape in eg a tetrahedron matrix of 1, 4, 10, 20, 35, etc tetrahedra inside or a cube matrix of 1, 8, 27, 64, 125 etc cubes inside.
I introduce the theory in this forum to check if similar theories have been up for scrutinization. In my research so far, I have not found anything similar, ie using matrix/grid structures of multi-symmetrical geometric shapes as the basis for finding and defining prime numbers.
First some background to the reasoning applied to take it from an idea to a theory:
- The logarithmic relation between prime numbers (as in the prime number theorem) indicates a relation of something “natural”.
- My MSc degree in engineering physics many years ago specialized in crystalline structures, and this inspired me to explore the relationship between prime numbers and characteristics of matrix/grid structures which are built by tetrahedrons, cubes, isocaeders, etc (ie multi-symmetric geometric shapes).
- The occurrence of prime numbers in nature (built on the theories of physics) such as observations of cicadas appearing in cycles of a prime number of years also pointed at a relationship between prime numbers and something in nature, such molecular structures.
The general form of the pattern described as N = A(P,n) + B includes N-series for each multi-symmetrical shape, eg N[tetrahedron], N[cube], etc. In other words:
- Going through characteristics of tetrahedron based matrices with 1,…, n “layers” leads to one series of prime numbers
- Going through characteristics of cube based matrices with 1,…, n layers leads to another series of prime numbers
- Going through characteristics of other symmetrical shapes leads to other series
…forming separate ‘families’ of prime numbers
So a prime number can be found by a combination of:
a/ A function which represents characteristics of multi-symmetrical geometrical shapes, e.g. a tetrahedron structure
b/ An ‘adjusting’ function which adds or subtracts a certain number of steps/integers from (a)
The number of nodes and connections/lattices between the nodes are part of the theory/hypothesis and what I mean by characteristics of the matrix structures. For tetrahedron shapes the first/simplest matrix has 1 node, the second an additional 3 nodes, the third an additional 6 nodes, the fourth and additional 10 nodes, etc. For cube matrices/grids, the first matrix has 1 node, the second and additional 7 nodes, the third an additional 19 nodes, the fourth an additional 37 nodes, etc. To continue this cube matrix based series described as A(cube,n) = (n+1)(n+1)(n+1) – n x n x n , A equates to 7, 19, 37, 61, 91, 127, 169, 217, 271, 331, 397, 469, 547, 631, 721, 817 and 919, of which all are primes except the semi-primes 91 (7x13), 169 (13x13), 469 (7x67), 721 (7x103) and 817 (19x43).
So in the case of the “ A(cube,n) = (n+1)(n+1)(n+1) – n x n x n ” equation, “n” represents the size, or order, of the cube matrix structure. Accordingly, A(cube,n) can be described using matrix/grid based terminology as the number of additional nodes required to form the next (larger) order of a cube matrix structure. This representation, referring to multi-symmetrical geometric shapes, exemplifies the more over-arching fundamental theory that prime numbers can be defined as a function of geometrical shapes in nature; in this case a cube-based matrix structure.
For those of you who find this theory plausible and worthwhile working further on, I would be happy to share more details of my findings and observations so far. The theory does need further work, eg to specify how the adjusting function B behaves. If you recognise the above hypothesis from before (and I do not mean models like Ulam spirals or Klauber triangle), please share. Cuban Primes would be one "family" of several, and the other families relate to other polyhedra and crystal grit/matrix structures.
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