BIM-MPS Details, Part V

The MPS on the BIM (BIMMPS)

Bringing it all together: Parts I - II - III - IV - V.







Sidenote: before we get started, and in an attempt to stay on track, we briefly glance at some Appendix material. It's one of those little excursions where the deeper you look the further and further away you go, and yet the irony is, if you go far enough you return to where you started! Fitting for the Appendix, but not to be ignored.

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 Butterfly Fractal~2~, we find the MPS parameters based around the EVEN numbers, especially within the exponential power of 2. When we look at their summations, we find it delivers the ODDs we are looking for that inform the MPS directly.

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 MPS parameters based around the ODD numbers, especially the Running Sums (∑) of the ODD numbers sequence of 1,3,5,… giving the MPS and the ODD³ sequence of 1³, 3³, 5³,… giving the PN!

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

In talking about the Butterfly Fractal~1~, Butterfly Fractal~2~, Butterfly Fractal~3~ and Butterfly Fractal~4~, it begs the question: how are these all related? How can the ∑ of the exponential power of 2, of the cubed ODDs and even simply the ODDs by themselves, all be related to each other and, especially, to the Butterfly Fractal~4~ = BIM?


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. ...

Sidenote end.



BIM-MPS Details, Part V

Making the Butterfly Fractal Matrix

These are some of the details of the big parts alluded to in the next section: Oceans of Number (Part II). Sometimes it's good to see some of the source material ahead of the bigger view.


The BF Matrix as seen diagramatically as an outwardly spiraling whirling squares where z2+z=CRnext and CR=2PN. A closer look shows that 4CR+1= z2, e.i. 4(4032)+1=16129.And 2PN-z= z2, e.i. 2(8128)-127=16129. Thus z2=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:

Do remember that the "container" values are included as an integral part of the pattern.


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 ...


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

MathspeedST: eBook (free)

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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 Paper

Reginald Brooks

Brooks Design

Portland, OR

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