Let's repeat a key concept covered in Part II and III:

The Perfect Numbers may be expressed as the sequential running sums (**∑**) of the cubes of the sequential ODD numbers as 1³ + 3³ + 5³ ... and we have the genesis of the * Butterfly Fractal~3~ (see Appendix)*.

In the * Butterfly Fractal~1~* and

This gave the **x** and **z**=**Mp** directly and the **y**, **xz**=**PN** and **xy**=**CR** indirectly by calculation.

Now, in the * Butterfly Fractal~3~ (see Appendix)*, we find the

The * Butterfly Fractal~4~*, based on the ∑ of the ODDs, is of course the basis of the

In talking about the * Butterfly Fractal~1~*,

On the PD:

All “x”

All “x²”

All “y²”

All “z²”

Rule: On PD, the square of any PD value — (2ⁿ)² — is found at the exponent value, the # of STEPS:

e.i. 16² = (2⁴)² = 2⁸ =256 = 4 STEPS down PD 16–>256

e.i. 1024² = (2¹⁰)² = 2²⁰ =1048576 = 10 STEPS down PD 1024–>1048576

e.i. 2048² = (2¹¹ )² = 2²² = 4194304 = 11 STEPS down PD 2048–>4194304

Note that while the PD does not contain ALL the 2ⁿ values — e.i. 2³ = 8 — it does contain ALL the 2ⁿ values that are “Active (BLUE Dot above)” — relevant — to the Mps as well as those “Inactive (I)” values that act like placeholders for the Mp sequence, i.e. multiples of 4x (ALL “x” values can be found by sequential multiplication by 4), like 256. These “I” are present as the PD contains every “other” 2ⁿ value starting with 2² — 2² - 2⁴ - 2⁶ - 2⁸ - 2¹⁰ -2¹² …in other words, only those exponential powers of 2 with EVEN exponents. Naturally, the square of any ODD exponent 2ⁿ — like (2⁷)² = an EVEN and will be on the PD, as 2¹⁴.

As stated above, this makes the PD the container for all MPS. ...

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The BF Matrix as seen diagramatically as an outwardly spiraling whirling squares where z^{2}+z=CRnext and CR=2PN. A closer look shows that 4CR+1= z^{2}, e.i. 4(4032)+1=16129.And 2PN-z= z^{2}, e.i. 2(8128)-127=16129. Thus z^{2}=4CR+1=2PN-z. Solving for PN, first 2PN=4CR+1+z, then PN= 2CR+(1+z)/2. Knowing the z+1/2=x, we then have PN=2CR+x. Let's test this: 2(4032)+64=8128. This preview is a little ahead of ourselves, but it does show where things are going! More on this later.

Making BF Matrix-1

Making BF Matrix-2: The product of successive pairs, in BLUE.

Making BF Matrix-3: The product of successive pairs, stating with 2x3, in RED.

Making BF Matrix-4

Making BF Matrix-5

Making BF Matrix-6: The BLACK BF-Summation numbers fill in the x and y axis.

Making BF Matrix-7

Making BF Matrix-8: That's it!

Making BF Matrix-9

Making BF Matrix-10: Follow the path: The BF pattern on the x and y axis has its diagonals filled in by the products of the pair sequences.

Making BF Matrix-11

Making BF Matrix-12: Four Quadrants.

Making BF Matrix-13: The PN diagonal x 2 = CRnext.

Making BF Matrix-14: (RED) PN diagonal = Double the (BLUE) CR diagonal + the (YELLOW) "x" values.

Making BF Matrix-15: Lower(RED) PN diagonal = 4x the Upper (RED) PN diagonal + the (YELLOW) "x" values.

Making BF Matrix-16: ALL the shaded (Quandrants A and D)--BLUE and RED diagonals--are ÷24 after first subtracting out the (YELLOW) "x" values. Except 2.

Making BF Matrix-17: ALL the Active PN and CR values are found below the horizontal center line (except 2,2,6) in the C and D quadrants.

Making BF Matrix-18: ALL the Active PN,CR, Mp=z, x, x^2, y values are found below--and include--the horizontal and lower center lines in the C and D quadrants (except 2,2,6) .The D quadrant--that includes the center lines--houses ALL the Mp=z, PN=xz, and "x" and "x^2" values (except 2,6). Likewise, the C quadrant houses the CR and y Active values. Active values are those within the "containers" that are actually Mersenne Primes/Perfect Numbers.

Making BF Matrix NEXT

BF Matrix Conversions-1a

BF Matrix Conversions-1b

BF Matrix Conversions-2a

BF Matrix Conversions-2b

BF Matrix Conversions-3a

BF Matrix Conversions-3b

BF Matrix Conversions-4 animated gif

NEXT: Oceans of Numbers(Part_II): An Artist, Mathematician and Choreographer ...

LINKS:

Mersenne Prime Squares (Part I) the Introductory white paper.

Mersenne Prime Squares (Part II) 3 Simple Intros.

Mersenne Prime Squares (Part III) the Advanced white paper.

The MPS Project The Mersenne Prime Squares Project.

MathspeedST: TPISC Media Center

Artist Link in iTunes Apple Books Store: Reginald Brooks

Back to Part I of the BIM-Goldbach_Conjecture.

Back to Part III of the BIM-Goldbach_Conjecture.

BACK: ---> PRIMES Index on a separate White Paper BACK: ---> Periodic Table Of PRIMES (PTOP) and the Goldbach Conjecture on a separate White Paper (REFERENCES found here.) BACK: ---> Periodic Table Of PRIMES (PTOP) - Goldbach Conjecture ebook on a separate White Paper BACK: ---> Simple Path BIM to PRIMES on a separate White Paper BACK: ---> PRIMES vs NO-PRIMES on a separate White Paper BACK: ---> TPISC_IV: Details_BIM+PTs+PRIMES on a separate White Paper BACK: ---> PRIME GAPS on a separate White PaperReginald Brooks

Brooks Design

Portland, OR

brooksdesign-ps.net

Art Theory 101 / White Papers Index