**An ODD Tale.**

So, this ODD fellow, all by himself, walks into a bar. A bar you might remember?

“So, what’ll you have?” says the bartender.

“What I want is just a double — of me!” replies the ODD with a little smirk on his face.

“Come on, give me a break, fella!” barks the drink sloshier. “A double of what?”

“A double of me! Look, I am simply a one. Quantity one. I want you to double me. Duplicate me. Replicate me. Reiterate me. Just make two self-similar, identical ones to make a double. An EVEN!” scratched out the lonely one. “And can you serve them up on the table just behind me?”

“Double it is — coming up! Two identical ones — a pair — and right over there!” beeps the bartend. “And what, might I dare ask, are the two ones — better known as the first EVENS — going to have?”

ODD one leans over the bar, kicks back his hat, and wanes, “Well now, I don’t suppose you would wage a guess, now would you? How about this: each of the replicate ones naturally wants to order up duplicates — doubles — of themselves, and wouldn’t you know it, they would like you to serve them at the table behind them!”

“Well, I’ll be dipped!” exclaims the barkeep, choking back the tears of astonishment. You see, he had been through this number journey before and found then — and apparently now once again — that the simplest arrangement of numbers — of quantities — can lead to profound, unimaginable richness. Such was the case with the Mersenne Primes and Perfect Numbers culminating in the Mersenne Prime Squares (MPS).

“So each successive table will have duplicates — doubles — of each of the ones from the table before?” he sobbed.

“That’s right,” said the ODD one, “a whole sea of identical ones grouped into successive groups of EVENS!

“I get it! I get it!” shrieked the bartend. “Its a fractal. Indeed, a whole series of self-similar, redundant, reiterative ones doubling to twos and the twos double to fours, the fours to eights, the eights to sixteens, and on and on! This must go on to infinity! And right here in my own bar!”

“Well, my friend, hold on to your hat, because we are just getting started. That’s just the half of it. Just as every EVEN two is made up of two identical ODD ones (me) as they expand out forming a butterfly-like pattern — The Butterfly Fractal 1 — when we add them all up at each table, we find that every successive table is simple double the total EVENS from the table before: (1)—2–4–8–16–32–64–128–256–512–1024–…” says ODD.

“And now for some of that magic! If you add me (1) to the first table( of 2) you get 3, then add that total to the next table (of 4) to get 7, that total to the next table (of 8) you get 15, and that to the next (of 16) to get 31, and so on. This is called a “running sum, ∑” and gives us that richness we want. You see, the 3–7–31 are the first 3 Mersenne Primes!”

“Holy Rollie, Mother of God, down on your knees,… what’s next, the Perfect Numbers?” howled the barkeep.

“Well, a simple “cool” would do, but now that you mentioned it, yes we also can get the Perfect Numbers (PN) that go with each of the 51 known Mersenne Primes (Mp)!” one replied.

“Cool,” replied the barkeep nonchalantly.

“If you do a limited running sum (∑) of the table sums of EVENS, it equals the Perfect Number. A couple of examples: 1. ∑ 2–4 = 6 the PN of Mp 3; and 2. ∑ 4-8-16 = 28 the PN of Mp 7.

“And,” continued the ODD one, “All the parameters of the MPS — x—x² —y—y² —z—z² —xz—yz—xy — can be found by similarly simple means directly from this very simple doubling of the quantity one! Now that is simply amazing!”

“Phew!” exhaled the bartend, “and we haven’t even left the 2Dimensional plane!”

“Well, hold on to your pants!” said the ODD one. “The running sum (∑) of the cube of the ODDs will give each and every Perfect Number (>6), too! 1³ + 3³ = 1 + 27 = PN 28 and 1³ + 3³ + 5³ + 7³ = PN 496, yet each of these 3D versions can be simply rotated back into the 2D plane, confirming the priority of the simple plane geometry in establishing the structural backbone, itself tied to the linear, 1D form of the line.”

To wit the bartend added, “It seems, quite simply, that every ONE counts!”

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IMAGES

Butterfly Fractal, 2^{n}

front, acrylic on recycled aluminum scrap, 39" x 48" x 1", 2022

Butterfly Fractal, 2^{n}

back, acrylic on recycled aluminum scrap, 39" x 48" x 1", 2022

Butterfly Fractal, 2^{n}

wall-relief, acrylic on recycled aluminum scrap, 39" x 48" x 1", 2022

Butterfly Fractal 1

animated gif

Butterfly Fractal 1

movie

SUMMARY

For centuries, Mersenne Primes were considered from the perspective of simple, plain number — number without dimension. When, instead, one treats the number value in a linear manner, as in the side “length” of a square, the Mersenne Prime Square is born.

As we have also known for many years, every Mersenne Prime has an associated Perfect Number — a number that is always EVEN.

Therefore, in our ODD *MPS*, there must be an EVEN *PN* — and there is. It is found by asymmetrically dividing the *MPS* into two rectangles with the EVEN rectangle being the slightly larger of the two. It is the *PN*.

The ODD rectangle is the complement to it, thus being the *OC*.

As the short side of each rectangle — when squared — becomes the embedded Perfect Square of that rectangle, with the remaining complement being the Complement Rectangle, *CR*. Thus the *PN* has its EVEN Perfect Number Square (*PNS*) and the *OC* has its ODD Complement Square (*OCS*).

The *CR*s are identical and always EVEN.

The *PNS* is always EVEN, while the *OCS* always ODD.

A perfectly logical expression of EVEN-ODD interactions gives a natural **Number Pattern Sequence** (**NPS**) to each and every *MPS*.

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- ∑ = running sum
- PN = Perfect Number = xz
- Mp = Mersenne Prime = z = 2ᵖ - 1 = x + y
- Mp² = Mersenne Prime Square = MPS = z²
- x = 2ᵖ⁻¹ = short side of PN rectangle
- y = x-1 = short side of OC rectangle
- Mp² = PN + OC = xz + yz = z²
1_2_4_8_16_32_64_128_256_512_1024_…

—The Butterfly Fractal 1 — defines the Mersenne Prime-Perfect Number relationship.

If you take a Perfect Number (PN) — which is always EVEN — and sequentially divided it by two, it will always land on an ODD number known as a Mersenne Prime (Mp). Euclid — and later, Euler — knew about this and expressed it as the Euclid-Euler Theorem.

In my installation, “The MPS Project,” in Building 5 in July of 2021, I showed visually why this is true as the new geometry extension of this Euclid-Euler Theorem plays out: namely, that squaring the Mp (treated as a “long” side) to give a geometric Square — Mersenne Prime Square (MPS) — contains within itself two Rectangles, one being the PN, the other its ODD Complement (OC). (The PN has a “short” side = x, the OC = y, with y = x-1 and x + y = z = Mp.)

If you take that PN Rectangle and sequentially diminish it by half (divide by two over and over) until it reaches an ODD, that ODD = the Mersenne Prime (Mp, as 1 x the side of that PN Rectangle).

While symmetry prevails, the center defines!

CONCLUSION

The relationship between Mersenne Primes and Perfect Numbers is clarified as a natural consequence of their inherent geometric profile as revealed on the **BIM**.

Of perhaps greater significance is the actual play-out of the ODDS and EVENS, or is it the EVENS and ODDS. In this special case, the powers of 2 give the EVENS a very high profile presence in determining and defining a relationship to the normally more dominating ODD PRIMES.

Is it not interesting that sequentially dividing the EVEN *PN* by 2 will continue for the same number of times equal to its *p-*1 value before reaching an ODD. That ODD, of course, is *z* = *Mp*, i.e. *PN*/*x* = *Mp*, which is not surprising as *PN* = *xMp* = 2ᵖ⁻¹(2ᵖ-1) = *Euclid-Euler Theorem*.

But there is also a new window. Much as with the ***PTOP*** on the **BIM**, if one takes any EVEN and ÷ by 2, one finds symmetrically to either side, two members of a PRIME Pair set (PPset) that sum up to said EVEN; now, something new.

If one takes the approach, but now with the EVENS: to either side of EVEN PNS on the **BIM** Axis, symmetrically there are two EVENS that sum up to said *PN*, and these two EVENS are in fact the *PNS* and *CR*, e.i. *PN* = 28 when located as *PN*/2 = 28/2 = 14 on the **BIM** Axis, symmetrically 2 STEPS to either side are 12 and 16, the *CR* and *PNS*, respectively.

Another, example: *PN* = 496, 496/2 = 248, and 240 and 256 lie symmetrically 8 STEPS to either side of 248 on the **BIM** Axis, with *CR* = 240 and 256 = *PNS* for the 496 *PN*.

One more example: *PN* = 8128, 8128/2 = 4064, and *PNS* = 4096 and *CR* = 4032 lie symmetrical 32 STEPS to 4064, summing up to *PN* = 8128.

In every case, the STEPS are an EVEN, being a power of 2, 2*ⁿ*, specifically 2ᵖ*-*2 = 2ᵖ/4 = *x*/2.

We have much to learn on this relationship between ODD and EVEN numbers. The **BIM-Mersenne Prime - Perfect Number** is, perhaps, just the portal we need to see this vista that lies just beyond!

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REFERENCES

Wikipedia: Mersenne Prime

Wikipedia: Perfect number

Wikipedia: Euclid-Euler theorem

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~. ~~~. ~~~. ~~~. ~~~. _{~}.

White Paper: Mersenne Prime Squares

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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.

Building Five Building Five (Bldg5).

North Pole Studio North Pole Studio.

Ocean of Numbers Overview of it all.

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