Gather the EVENS
Steps are P2-A
Here’s the thing. Amongst a myriad of other connections, there exists an intimate connection between three number systems on the BBS-ISL Matrix (BIM):
The Inverse Square Law (ISL) as laid out in the BIM;
PTs — and most especially PPTs — as laid out on the BIM;
The PRIME numbers — PRIMES — as laid out on the BIM.
The BIM is the FIXED GRID numerical array of the ISL.
PTs are the Pythagorean Triples and PPTs are the Primitive Pythagorean Triples. They have been extensively covered in the TPISC (The Pythagorean-Inverse Square Connection) series as: TPISC I Basics, TPISC II Advanced, TPISC III Clarity, TPISC IV Details. (See links at bottom.)
The PRIMES vs NO-PRIMES (2019) was covered earlier.
The original MathspeedST (2009-13) work, Brooks (Base) Square (2009-11), that started this journey, was divided in to two sections:
I. TAOST (The Architecture Of SpaceTime);
II. TCAOP (The Conspicuous Absence Of PRIMES).
You see, other than the natural whole numbers that form the BIM Axis’ and the standard 1st Parallel Diagonal (containing ALL the ODDs (≥3), there are NO PRIMES on the BIM.
This is the basis of the PRIMES vs NO-PRIMES work.
Yet, convert that same 1st Parallel Diagonal to ALL EVENS (≥4, by adding 1 to each former ODD), and now the BIM reveals the stealthily hidden PRIMES relationship in forming symmetrical pairs of PRIMES that are ALWAYS equal distance (STEPS) from ANY given EVEN # divided by two.
All this occurs on the BIM Axis. The apex of the 90° Right-angled, isosceles triangle so formed lies on a straight line path from the EVEN/2 to the given EVEN # located on that converted 1st Parallel Diagonal. This relationship is geometrically true and easily seen on the BIM.
Extracting those PRIME-Pair sets (PPsets) for each given EVEN forms the basis for the PTOP — Periodic Table Of PRIMES.
While “hidden” on the BIM, it clearly forms a definitive pattern on the PTOP: for EVERY 3+PRIME PPset that forms the 2nd column on the PTOP — acting like a bifurcation point — a “Trail” of PPsets forms a zig-zagging diagonal pointing down and to the right.
These are ALWAYS — much like a fractal — added with the SAME PRIME Sequence (3,5,7,11,13,17,19,23,29,31,37,...).
When you read a given horizontal line from left to right across the PTOP, you see that each given PPset that was contributed by a PPset Trail, adds up — composes — its respective EVEN. The sum (∑) number of PPsets that form its EVEN is totaled in the last column.
The fractal-like addition of one new, additional PRIME Sequence PPset to each subsequent “Trail” formed results in the overlapping trails growing at a rate that far, far exceeds the growth and incidence of the PRIME Gaps. This ensures that there will always be at least one PPset that will be present to compose ANY EVEN.
The Goldbach Conjecture has been satisfied and a new PRIME pattern has been found.
TPISC Media Resource Center
Yes, a deceptively simple matrix grid square (BIM) can give us:
1. Inverse Square Law (ISL);
2. Pythagorean Triples (PTs);
3. PRIMES (P)- Goldbach Conjecture!
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.
PRIMES vs NO-PRIMES