PTOP: Periodic Table Of PRIMES & the Goldbach Conjecture (Euler's "strong" form)
In 2009-10, a solution to Euler's "strong" form of the Goldbach Conjecture "that every even positive integer greater than or equal to 4 can be written as a sum of two primes" was presented as the BBS-ISL Matrix Rule 169 and 170. This work generated a Periodic Table of Primes (PTOP) in which Prime Pair Sets (PPsets) that sequentially formed the EVEN numbers were laid out. It is highly patterned table.
It turned out this PTOP was actually embedded — albeit hidden — within the BIM (BBS-ISL Matrix) itself as shown in Rule 170.
So why this? Now?
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.
This video is a quick overview of how the BIM version of the PTOP and the standalone PTOP come about. It’s easy, it’s hard, it’s really easy after you get past the initial layout. The layout looks busy and complex. Breaking it down, we see it really is just an easy pattern that repeats.
Amongst the “NEW” info presented here, perhaps the single most important is that the bilateral symmetry so prevalent throughout the BIM continues here with the PRIME Pairs as members of the PPset are ALWAYS bilaterally present on either side of the core Axis value that points to the EVEN # under consideration. And those PPsets form a 90° R-angle isosceles triangle whose apex lies on that very path.
This visualization of the PRIMES satisfying the Goldbach Conjecture leads to images, tables, charts and equations that all validate what you can see right before your own eyes.
These can all be seen on my website: brooksdesign-ps.net/Reginald_Brooks/Code/Html/netarti5.htm
Thanks for viewing!
Reginald Brooks, 2019