PTOP & Goldbach Conjecture

Gather the EVENS Steps are P2-A A=half E

Reginald Brooks

©2019, Reginald Brooks. All rights reserved.

Abstract: 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): 1 The Inverse Square Law (ISL) as laid out in the BIM; 2 PTs — and most especially PPTs — as laid out on the BIM; 3 The PRIME numbers — PRIMES — as laid out on the BIM. The BIM is the FIXED GRID numerical array of the ISL. 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. Extracting those PRIME-Pair sets (PPsets) for each given EVEN forms the basis for the PTOP — Periodic Table Of PRIMES. 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...

Introduction While the (strong) Goldbach Conjecture has been verified up to 4x10**18, it remains unproven. A number of attempts have demonstrated substantial, provocative and often beautiful patterns and graphics, none have proven the conjecture. Proof of the conjecture must not rely solely on the notion that extension of a pattern to infinity will automatically remain valid. No, instead, a proof must, in its very nature, reveal something new about the distribution and behavior of PRIMES that it is absolutely inevitable that such pattern extension will automatically remain valid. The proof is in the pudding! Proof offered herein is just such a proof. It offers very new insights, graphical tables and algebraic geometry visualizations into the distribution and behavior of PRIMES. In doing so, the Proof of the Euler Strong form of the Goldbach Conjecture becomes a natural outcome of revealing the stealthy hidden Number Pattern Sequence (NPS) of the PRIMES.

STATEMENT: Layout & Essentials Proof of the Goldbach Conjecture (strong form, ≥6)

1. Natural (n), Whole Integer Numbers (WIN) — 0,1,2,3,…infinity — form horizontal and vertical Axis of a simple matrix grid. 2. The square os such WINs — n**2=1**2=1, 2**2=4, 3**2=9,…infinity — forms the central Diagonal of said grid — dividing it into two bilaterally symmetric triangular halves. 3. Every Inner Grid (IG) cell within is simply the difference (∆) between its horizontal and vertical Diagonal intercept values. They extend to infinity. The Diagonal WINs form the base of a 90° R-angled isosceles triangle with said IG cell value at the apex. 4. Every IG cell within is also the product of two Axis WINs (Either horizontal or vertical, not both), that form the base of a 90° R-angled isosceles triangle with said IG cell value at the apex. 5. The complete matrix grid extends to infinity and is referred to as the BIM (BBS-ISL Matrix).The BIM forms — and informs — a ubiquitous map (algebraic geometry) to: ⁃ The Inverse Square Law (ISL); ⁃ The Pythagorean Triples (PT); ⁃ The PRIMES (stealthily hidden, but revealed by NPS.

6. The 1st Diagonal that runs parallel to either side of the main Prime Diagonal (PD, not of PRIME numbers, but primary), is composed of the ODD WINs: 1,3,5,…infinity. 7. If we add +1 to each value, that 1st Diagonal now becomes a sequence of ALL the EVEN WINs (≥4): 4, 6, 8,…infinity. 8. Select ANY EVEN WIN and plot a line straight back to its Axis WIN — that Axis WIN = EVEN/2 = core Axis #. 9. Upon that same Axis, PRIME Pair sets (PPsets) — whose sum (∑) equals the EVEN WIN (on the 1st Diagonal) — will be found that form the base of 90° R-angled isosceles triangle(s) whose apex lie(s) on that straight line between the EVEN and its 1/2 Axis WINs. PPsets with identical PRIMES = 1/2 Axis value. 10. The proof that every EVEN WIN has ≥1 PPsets can be seen in the Periodic Table Of PRIMES (PTOP) that stealthily informs the BIM of how each and every EVEN WIN is geometrically related to one or more PPsets. 11. These PPsets are NOT randomly contributing their ∑s to equal the EVEN WINs, rather they come about as the consequence of a strict NPS: the sequential — combining, linking, concatenation — addition of the PRIMES Sequence (PS) — 3,5,7,11,13,17,19,…—to a base PS — 3,5,7,11,13,17,19,….

12. The NPS of this addition forms the PTOP: for each vertical PS — the 1st PRIME remains constant (3), the 2nd PRIME sequentially advances one (1) PS WIN — is matched diagonally with the 2nd PS, but now the 1st PRIME sequentially advances, while the 2nd PRIME remains constant within a given PPset. 13. This matching addition of the 2nd PS at the bifurcation point of the common 2nd PRIMES, forms the zig-zag diagonal PPset Trails that are the hallmark of the PTOP. 14. For every subsequent vertical PPset match, the Trail increase by one (1) PPset. 15. The rate of such PPset Trail growth far exceeds the PRIME Gap rate. 16. The zig-zag diagonal PPset Trails combine horizontally on the PTOP to give the ∑# of PPsets whose ∑s = The EVEN WIN. 17. More than simply proving the Goldbach Conjecture, the PTOP hidden within the BIM reveals a new NPS connection of the PRIMES: PRIMES + PRIMES = 90° R-angle isosceles triangles. 18. The entire BIM, including the ISL—Pythagorean Triples—and, PRIMES, is based on 90° R-triangles!

19. Similar to how the grid cell values of the Axis, PD and IG of the Pythagorean Triples reveal additional, intimate connections within the BIM, so too do the PPsets: the 1st PRIME values of each set points to the # of STEPS from the PD that intersects the given EVEN WIN (Axis**2), on a straight line path back to its Axis, at the apex where its other PPset member intersects — this is no mere coincidence — and that apex is, of curse, the 90° R-angle isosceles triangle that results. For example: EVEN = 24, Axis = 12, PD = 144, # of STEPS from PD towards Axis = 5 and 19 for the 5+19 PPset and 7 and 17 for the 7+17 PPset, and 11 and 13 for the 11+13 PPset that each forms the EVEN 24. 20. As the bifurcation concatenation of the PS — 3,5,7,11,13,17,19,..— with the same base PS — 3,5,7,11,13,17,19,…— of those EVEN WIN — that when “3” is subtracted, the remainder is a next-in-the-sequence PRIME — remains one of a similar split with the Pythagorean Triples: for every “Primitive” parent PT, there are multiple “Non-Primitive” child PTs and it is the PPTs (Primitive Pythagorean Triples) that ultimately form the inter-connectedness of ALL PTs back to the original PPT — the 3-4-5. With the PRIM...

For EVEN 128, 128/2=64=coreAxis. PPset members symmetrical.

PRIMES Index can be found here: http://www.brooksdesign-ps.net/Reginald_Brooks/Code/Html/MSST/TPISC/TPISC_IV_Details/PRIMES_index.html

“PTOP: Periodic Table Of PRIMES & Goldbach Conjecture” is also available as an ebook in Apple Books within iTunes. http://books.apple.com/us/book/id1491237017