Within the inverse square law
Fractal — Symmetry — Inverse Square Law (ISL). These are not the usual descriptive terms associated with the Prime numbers (PRIMES) Yet that is exactly what best describes the PRIMES!
The PRIMES, when brought together as P1 & P2 members of PRIME Pair Sets (PPsets) demonstrate a robust symmetry and an intimate relationship with the ISL when shown on the BIM (BBS-ISL Matrix grid of the Inverse Square Law). This symmetry is brought out in the geometric relationship between the PPsets and the EVEN numbers (EVENS) that they inform consistent with Euler’s Strong Form of the Goldbach Conjecture.
This come about as the well established PRIMES Sequence (PS) — 3-5-7-11-13-17-19-23-29-31-... acts in a fractal-like manner, i.e., it demonstrates redundant, repetitive and re-iterative behavior in presenting self-similar reflection of itself as it constructs PPsets and PPset “TRAILS.”
The TRAILS are formed from the concatenation of PS’s progressively onto each successive PRIME of a given PS — forming a series of PPsets along the way. The TRAILS can also be seen to be formed directly as individual PPsets on the BIM. Here, each PPset is easily seen to be intersection of 1 PPset member from the Horizontal AXIS and 1 PPset member from the Vertical AXIS, together forming the P1, P2 members. The TRAIL is simply all those PPsets thus formed across a Row — or down a Column — on the bilaterally symmetrical BIM. All this is easily visible on the BIM.
That leads us to our story about the PRIMES—Fractals—Symmetry—and the intimate relationship they have with the ISL as seen on the BIM.
The “PTOP (Periodic Table Of PRIMES) & the Goldbach Conjecture” (2019), also referred to as BIM: Part I, updated and clarified the PTOP, PPsets and their PPset TRAILS that were originally presented in MathspeedST (2010). BIM = BBS-ISL Matrix.
In this work, a major update and refinement has been made.
Each of the three parts — BIM: II, III and IV presents new findings that visibly demonstrate the PRIMES on the BIM.
Each and every PRIME, when treated as part of a PPset (P1, P2), can be found and individually profiled DIRECTLY on the BIM as seen in BIM: Part II.
Plotting all the Lower Diagonal P2 PRIMES on a table and then re-plotting those results back onto the BIM opened up a new vista.
By substituting the AXIS values for each of those P2 PRIMES from the table, one has now formed a new SubMatrix of the BIM with ALL the PPsets in place — BIM: Part III.
In BIM: Part I-III, we have mostly used just half of the bilaterally symmetric BIM to reveal and describe the geometry.
In BIM: Part IV, we now look at the whole BIM with the PPset SubMatrix in place.
Treating these PPsets as objects and counting them within progressively larger square areas — forming what is called “Object AREAS” — a pure ISL Number Pattern Sequence (NPS) is found.
As the PS-Fractal series of each AXIS joins to form the PPsets, their actual numbers — as PPset TRAILS — progressively grows and sums up to quantities with Object AREAS that directly mirror the fundamental ISL NPS: 1—4—9—16—25–…
And while this intimate relationship of the PRIMES to the ISL can not predict the next largest PRIME, it can ABSOLUTELY account for each and every PRIME at, and below, any given PRIME, regardless of size.
While Euler’s Strong Form of the Goldbach Conjecture is proved along the way, the real significance is what we have seen unfolding in BIM: Parts I-IV.
The role of Symmetry and Fractal underlie everything about the PRIMES.
The PRIMES on the BIM is all about how the fractal nature of the PS becomes expressed as symmetry on the BIM as isosceles and equilateral triangles, forming the PPsets that ultimately form ALL the EVEN numbers!
What is marvelous, incredible, mind-blowing in every way, is that this same symmetry—fractal—isosceles/equilateral triangle relationship is found — indeed, is part and parcel — throughout the BIM.
The ISL seems to reflect the most basic and fundamental relationship(s) between quantity and the numbers that account for it.