## Complete Exposition of Non-Primes Generated from a Geometric Revolving Approach by 8x8 Sets of Related Series, and thereby *ad negativo *Exposition of a Systematic Pattern for the Totality of Prime Numbers

**Authors:** Stein E. Johansen

We present a certain *geometrical* interpretation of the natural numbers, where these numbers appear as *joint
products of 5- and 3-multiples* located at specified *positions* in a *revolving chamber*. Numbers without
factors 2, 3 or 5 appear at *eight* such positions, and any prime number larger than 7 manifests at one of
these eight positions after a specified amount of rotations of the chamber. Our approach determines the
sets of rotations constituting primes at the respective eight positions, as the *complements* of the sets
of rotations constituting *non-primes* at the respective eight positions. These sets of rotations constituting
non-primes are exhibited from a *basic 8x8-matrix* of the *mutual products* originating from the eight prime numbers located
at the eight positions in the *original* chamber. This 8x8-matrix is proven to generate *all* non-primes
located at the eight positions in *strict rotation regularities* of the chamber. These regularities are
expressed in relation to the multiple 11^{2} as an anchoring *reference point* and by means of convenient
*translations* between certain classes of multiples. We find the expressions of rotations generating *all*
non-primes located at *same* position in the chamber as a set of *eight related series*. The *total* set of
non-primes located at the eight positions is exposed as *eight* such sets of eight series, and with each
of the series *completely* characterized by *four simple variables* when compared to a reference series
anchored in 11^{2}. This represents a *complete* exposition of non-primes generated by a quite simple mathematical
structure. *Ad negativo* this also represents a *complete* exposition of all *prime numbers* as the union of the
eight complement sets for these eight non-prime sets of eight series.

**Comments:** 41 pages, Submitted to Journal of Calcutta Mathematical Society, Nov 18, 2009.

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### Submission history

[v1] 8 Mar 2010

[v2] 11 Mar 2010

[v3] 12 May 2010

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