Consider that the oceans of all natural numbers—symbols of quantities—selectively wash up as waves against the shores. Within these oceans of numbers are all the whole integer numbers from (0) — 1 — 2 — 3 —…—> INFINITY. All the ODDS, all the EVENS. This includes all the PRIMES, which are all ODD, except for the number 2. It also includes all the doubling of numbers, which always equals EVENS.

The * Mersenne Primes* are a special subset of PRIMES: they are always one less than another subset of numbers, the exponential powers of 2. These ODD subsets are one less than an EVEN subset. Not only that, but for each

On a particular beach, the waves that wash up are ALL a particular subset of EVENS — the EVENS of the exponential power of 2: simply a doubling of ALL powers of 2 as 2^{0}=1, 2¹=2, 2²=4, 2³=8, 2⁴=16, 2⁵=32, 2⁶=64, 2⁷=128, 2⁸=256, 2⁹=512, 2¹⁰=1024, 2¹¹=2048, 2¹²=4096, 2¹³=8192,…

All the waves that we see — from little wavelets to the more dominant wave patterns — are simply various interferences of those powers of 2 described above.

Within those waves — by simply subtracting one (-1) — one finds each of those 51 known * Mersenne Primes*, e.i. 4-1=3, 8-1=7, 32-1=31, 128-1=127, 8192-1=8191,… This subset of the PRIMES seems to resonantly harmonious with the subset of the powers of 2.

It is like each exponential power of 2 wavelet is a container for a * Mersenne Prime*! So if one wants to find all the

Remember that we said that an EVEN * Perfect Number* is ALWAYS paired with a

Also remember that those * Mersenne Primes* — one less than an exponential power of 2 numbers — that this time we add that one back to put us back on the wavelet of 2, and, now divided it by 2, to give an EVEN number that we call “x.” If one multiplies this “x” times the associated

For example, take the * Mersenne Prime* = 7, now add back +1 = 8. Divided 8/2=4=x. Multiply x times the

Algebraically, this is the * Euclid-Euler Theorem* (2ᵖ⁻¹)(2ᵖ -1) =

Geometry shows that what we are doing is taking the “x” as the short side of a rectangle and multiplying it by the long side that is the * Mersenne Prime*, giving us the EVEN rectangular area of the

And yes, if one knows the “x” value, one can always work their way back up to reveal the MPS and the * Mersenne Prime* that it contains. And ALL of this can be revealed by simply doubling the values within the exponential power of two — and subtracting (or, adding) 1.

The real magic comes when one starts to tally the running sums (∑) of these power of 2 EVENS.

2^{0}=1,

2¹=2,

2²=4,

2³=8,

2⁴=16,

2⁵=32,

2⁶=64,

2⁷=128,

2⁸=256,

2⁹=512,

2¹⁰=1024,

2¹¹=2048,

2¹²=4096,

2¹³=8192,…

1+2=3 a * Mersenne Prime* = Mp = z

3+4=7 = z

7+8=15

15+16=31 = z

31+32=63

63+64=127 = z

127+128=255

255+256=511

511+512=1023

1023+1024=2047

2047+2048=4095

4095+4096=8191 = z

As x(z) = * Perfect Number* = xz, and as we learned, (z+1)/2 = x, we see that the “x” value is the EVEN value on each line above. Incidentally, x-1 = y and "y" — an important, ALWAYS ODD value — mimics, yet never becomes, equal to"z," the

Herein, one can begin to see that not only are there remarkable overlaps — like interference patterns — between the EVENS and the special ODDS of the * Mersenne Primes*, and, the doubling of the powers of 2 is embedded like a fractal — the

How?

In the broader sense, a fractal is a form that redundantly, re-iteratively informs a larger form by the successive regeneration of its self-similar form. The 1-2-4-8-16-32-64… fractal form does exactly that.

1 doubles to 2, 2 double to 4, 4 doubles to 8, and so on. If one plots this pattern out by presenting each quantity as simply 1, 1 - 1, 1-1-1-1,… as represented by any repeating form — a line, a penny, a glass of beer — soon a bilateral, symmetrical pattern emerges with each side mirroring both the other side, and, the overall pattern of both “wings.” Within each wing, each doubling pattern is redundantly, re-iteratively repeated. The “* Butterfly Fractal 1*” is born.

(Note: * Butterfly Fractal 1* refers to this original pattern of the exponential power of two — starting with 2

So how do we get from the Butterfly image to the actual geometry of the MPS as seen on the * BIM (BBS-ISL Matrix)*?

The key is the difference of 1 between the “x” and “y” values. The MPS is composed to two rectangles, the PN and OC. The PN short side = x, the OC short side = y. And y = x-1. That means that OC + 1 + OC short sides = the full side of the MPS. And the full side of the MPS is the Mp = z.

Thus, the MPS = (y+1+y)(z), and as z=x+y, we have MPS=z²= (y+1+y)(z)= (y+1+y)(x+y).

The key is the difference of 1 between “x” and “y” and that 1 is, of course, 1 unit wide by “z” units long. Remember, it lies between the two equal “y” values on one side (TOP) of the MPS, but it extends the full length="z" down the side of the MPS.

Nearly ALL of the information embedded within the “* Butterfly Fractal*” is found right here in this central strip of 1 within the MPS. The pattern remains true for ALL MPS.

In each and every case, the central strip is built vertically as the exponential power of 2. Starting with 1, the next tier is 2, followed by 4, followed by 8, …. and so on depending upon the size of the MPS. The total running sum (∑) always equals “z” thus, as z=x+y, and y=x-1, the central strip of 1xz can be seen to always be composed of x+y=z, with the last, and largest, number in the series is the “x” value and the running sum (∑) of the remaining numbers equals “y.”

Example: MPS=z²=Mp²=7²=49, with z=7. With z=7, we know that (z+1)/2=x=(7+1)/2=4 and y=x-1=4-1=3. If we take the MPS area and divide it into the two rectangular areas of which it is composed we have MPS=PN + OC as 49=28+21, i.e. 7x7=49, 4x7=28 and 3x7=21.

Now let’s apply the “key difference of 1 between the ‘x’ and ‘y’ values.” Change the TOP side of the MPS from x+y to y+1+y. The center strip of 1 — 1x7 — is composed of 1+2+4=7 as 1+2=3=y and 4=x. The last, largest number is always the “x” value and the ∑ of the lesser numbers is the “y.”

The “* Butterfly Fractal 1*” has come to roost in the center strip of 1!

Knowing all this, one can always work backwards from y—>x—>z—>MPS.

One can also take the center strip — starting with 1 at the BOTTOM, rotate it 90° to the LEFT, and place it as a ledger at the TOP of the MPS, i.e. 4–2–1 with x=4 and y=2+1=3. The accounting of this works perfectly.

The x=4 means there are 4 vertical strips of 1x7, as 4x7=28 in our example. Of course, we know that that includes the center strip 1x7, or in other words, we have 3—1x7 strips + 1–1x7 (center strip) + 3–1x7 strips, as 3x7=21 + 1x7=7 + 3x7=21 fills in the MPS with an area of 49.

Having laid out the fractal fingerprint of the * Mersenne Prime-Perfect Number* set as revealed in the MPS, one now has to look at: what are the patterns that can lead to eliminating those exponential power of 2 wavelet containers that do NOT resonate with the subset of

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__NEXT: Details 3: BIM_MPS_Details-II: The Exponential Power of 2 and the Butterfly Fractal~1~__

Helpful LINKS (images, text and math):

Mersenne Prime Squares (Page 69)

Mersenne Prime Squares (Page 70: 3 simple intros)

Mersenne Prime Squares: Advanced (Page 71)

The MPS Project (Mersenne Prime Squares Project) (Page 72)

Oceans of Numbers (Part II) __ Overview__ work, Page 75)

More LINKS:

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