PRIMES vs NO-PRIMES: a new approach!

PORTLAND, Ore. - May 23, 2019 - PRLog -- ~~~
Gather the NO-PRIMES
6yx+-y
Remains are the PRIMES

For some, math is the abstraction. For others, math is the reality, and "our reality" is the abstraction. TPISC (The Pythagorean-Inverse Square Connection) reveals an intimate and ubiquitous relationship with the Inverse Square Law - Pythagorean Triples — and now — PRIMES.

The "Math Reality" is that this simple matrix of the Inverse Square Law ALSO provides a graphical means to reveal ALL the NO-PRIMES amongst  ALL the PRIMES — very much like the negative space informing the positive! Or is it the positive space defining the negative?

The PRIMES are considered the "atoms" of the natural numbers. The approach here has been to find (define) them by finding those that are not! And why not, the BIM (BBS-ISL Matrix grid of the Inverse Square Law and  Pythagorean Triples) has now been shown to have an intimate relationship with the PRIMES on several levels, including laying out visually ALL the NO-PRIMES! And all this can also be done easily in a spreadsheet!

Identifying the PRIMES (P) from the NO-PRIMES (NP) from the pool of ODD numbers is a matter of separation, as one defines the other. Amongst a list of all the ODD numbers (≥3), one may reveal ALL the PRIMES (P) simply be identifying ALL the NO-PRIMES (NP). Two new methods: 1.) algebraic and 2.) algebraic geometry — identify ALL the NP from any list of sequential ODD numbers.

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   How can a deceptively simple matrix grid square give us:

1.  Inverse Square Law (ISL);

2.  Pythagorean Triples (PTs);

3.  Primes (P) vs. No-Primes (NP)?

   The BIM (BBS-ISL Matrix) does that!

   What started as the invention/discovery of the ISL matrix — a simple grid of natural Whole Integer Numbers (WIN) — has led to an unforeseen connection between the ISL — PPTs — PRIMES.

   The BIM is an infinitely expanding matrix where every cell within the grid is uniquely occupied by a WIN that is itself simply the difference between its horizontal and vertical intersection values of the main, Prime Diagonal (PD).

   The PD mirror divides the grid into symmetrical halves. The PD is key as it consists solely of the squares of the Axis numbers (1,2,3,...) as A² = PD.

   In fact, drop vertically down from any PD squared number until you intersect with its squared complement and you will have landed on a Primitive Pythagorean Triple (PPT) row, with a² + b² culminating with c² where that row now intercepts horizontally with the PD, giving a² + b² = c².

   Repeating this will identify all the PPTs.

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   But wait! Now divide every Inner Grid (IG) cell value by 24 and you will generate a criss-crossing diamond grid overlay pattern with almost magical properties!

   The BIM÷24 grid will break out the overall matrix into two (2) types of ODD Axis rows:

• those ÷3 that are never PPT, never PRIMES = Non-Active Rows (NON-AR);

• those NOT ÷3 that PPTs and PRIMES are

exclusively found on = Active Row Sets (ARS) or Active Rows (AR) for short. NOTE: a necessary, but not sufficient, condition as some AR have neither.

   This becomes important as the ÷3/NOT ÷3 pattern is key to identifying the Number Pattern Sequence (NPS) of ALL the NO-PRIMES (NP) and that NPS ultimately defines the elusive pattern of the PRIMES as: ALL ODD WIN (≧3) - NP = P.

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   This has culminated — at least at this point in this exciting mathematical journey — in two (2) simple methods for generating ALL the NP:

1. Algebraic: *NP = 6yx ± y  let y = ODD WIN, x = 1,2,3,... (*one must add the exponentials of 3, 3^x, if one does not first eliminate all ÷3 WIN).

2. Algebraic Geometry: where it all began on the BIM by subtracting (eliminating) ALL the **NP that are shape-located directly on the BIM, one can reveal the remaining PRIMES (**here, one must include the ODD Axis² values).

   Both methods nicely dovetail directly on the BIM.

   *NP = 6yx ± y is the simple, algebraic equation that is derived from and summarizes the visualization approach we see directly on the BIM. This dovetailing of the pure algebra approach with the pure geometry of the BIM gives us the algebraic geometry method. Visual thinkers like to "see" the algebra and algebraic geometry at work — play!

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   Now the really interesting part of all of this is all the NPS revealed in the exposure of ALL of the NP is itself simply the converse — inverse — of ALL the PRIMES.

   When visualizing the PRIMES vs. NO-PRIMES on the BIM, it is much easier to "see" the NP as the positive space with all the NPS and the PRIMES as the remaining negative space — a space completely defined by that of its opposite (inverse) NO-PRIMES space as: NP + P = ALL ODD numbers (≧3).

   Yes, a deceptively simple matrix grid square (BIM) can give us:

1.  Inverse Square Law (ISL);

2.  Pythagorean Triples (PTs);

3.  Primes (P) vs. No-Primes (NP)!

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This is an additional supplement to the MathspeedST ebook.

Other titles include The Pythagorean-Inverse Square Connection (TPISC):

TPISC I: Basics
TPISC II: Advanced
TPISC III: Clarity and the Tree of Primitive Pythagorean Triples
TPISC IV: Details, DSEQEC and PRIMES.

Excerpt From: https://itunes.apple.com/us/author/reginald-brooks/id6576... (https://itunes.apple.com/us/author/reginald-brooks/id6576...) (Reginald Brooks.:"PRIMES vs NO-PRIMES" iBooks.)