Prime numbers pyramids

Pierre

The Cosmic Force
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session981031 said:
Q: Is there any formula, or any thing about prime numbers
that makes it easier to find them... anything about them that is
unique?
A: Pyramidal.
Q: Pyramid relationships would help one find prime numbers?
A: Graph.
So like some of you, I have been trying to create some numbers pyramids.

Interesting pyramids are certainly the ones where prime numbers arrange geometrically according to some harmonious patterns (I didn't find any single one yet)

I guess that it would be amazingly time consuming to check all possible number pyramids so I have tried to identify the main parameters :

Pyramid shape :
* square layers (1 square, 4 squares, 9 square,...)
* triangle layers (1 square, 3 squares, 7 squares,...)
* ...

Layers "alignment"
* centers
* 1 same corner for each layer
* ...

Numbers used in the whole pyramid
* all number
* only prime numbers
* only odd numbers
* 0 included / excluded
* ...

Numbers arrangement in layer
* spiraling from inside / spiraling from outside
* line 1 row 1, line 1 row 2,...
* Magic squares
* only using the external square / the squares visible from the top
* ...

Location of the first number of the layer
* center
* one corner
* square below the last number square in previous layer
* ...

Any thought ?

PS : A computer program displaying on one side the numbers lists (all, odd, prime,...) and on the other side the different pyramid 3D matrixes in which numbers can be dragged would reduce dramatically time consumption.
 
My current favorite pyramid related to this is the "sedenion" pyramid at the end of this post (and my avatar). No one really understands sedenions well yet, the following seems to be the current state of the art:

Tony Smith said:
Robert de Marrais says in his paper math.GM/0011260: "... We know that surprises will keep on coming at least up to the 2^8-ions, due to the 8-cyclical structure of all Clifford Algebras. And we know at least a little about what such surprises will entail: as can already be seen in low-dimensioned Clifford Algebras including non-real units which are square roots of positive one, numbers whose squares and even higher-order powers are 0 will appear. The 8-cycle implies an iterable, hence ever-compoundable pattern, implying in its turn cranking out of numbers which are 2^Nth roots of 0, approaching as a limit-case an analog of the "Argand diagram" whose infinitude of roots form a "loop" of some sort. If we can work with this it all, it could only be by having as backdrop some sort of geometrical environment with an infinite number of symmetries . . . suggesting the "loop" resides on some sort of negative-curvature surface. For the incomparably stable soliton waves which are deployed within such negatively curved arenas also are just about the only concrete wave-forms which meet the "infinite symmetries" (usually interpreted as "infinite number of conservation laws") requirement. ...".

I agree with Robert de Marrais's view of periodicity of the 2^N-ions, and it seems likely to me that the 2^8-ions might be regarded as a basic building block of number theory and group theory, just as I see the 2^8-dim Clifford algebra Cl(8) as a basic building block of physics in the D4-D5-E6-E7-E8 VoDou Physics model.

I conjecture further that these links might be usable to establish a relationship between the Riemann zeta function and quantum theory.

...since for 16-ions and larger you have interesting zero-divisor "sleeper-cell" substructures, could they be useful with respect to computational systems, perhaps doing things like forming loops that might let the computational system "adjust itself" and/or "teach itself"?

Robert de Marrais commented: "what is truly interesting is this: zero-divisor systems are, ironically, PRESERVERS of associative order! Specifically, each of the four "sails" on a box-kite can be represented (on an isomorphic box-kite diagram, in fact!) as a system of four interconnected Quaternion copies: write each vertex as a pairing of one uppercase and one lowercase letter (with the 'generator' of the given 2^n-ions being the divider of the two: e.g., with the Sedenions, g = the index-8 imaginary, and the pure Sedenions of index > 8 are "uppercase," with the Octonions thereby being written with "lowercase" letters)... each sail can be seen as an ensemble of 5 Quaternion copies (the 4 associative triplets each are completed by the real unit, and the "sterile" zero-divisor-free triplet of generator, strut constant, and their XOR makes 5). Viewing things in closest-packing-pattern style, we have 5 interacting "unit quaternion" algebras -- with the interactions entailing (1, u), where 'u' is the shared non-real unit. Interestingly, this gives a nice way to think about the Tibetan Book of the Dead's "58 angry demons and 42 happy Buddhas," 100 in all = 5 * 16 + 2*10 = 100 distinct units in the interlinked 5-fold "unit quaternion" ensemble. So one first sees the "42 Assessors," then zooms in one one of the 7 isomorphic box-kites (which, as with all isomorphies, can be seen as identical at some higher level); then, one zooms in further on the "second box-kite" which has its struts defined by upper vs. lower case letters, and the triple zigzag analog being the "all lowercase" sail..."
http://www.valdostamuseum.org/hamsmith/seden.gif
 
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