## 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.

**Download:** **PDF**

### Submission history

[v1] 8 Mar 2010

[v2] 11 Mar 2010

[v3] 12 May 2010

**Unique-IP document downloads:** 304 times

**Add your own feedback and questions here:**

*You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.*

*
*