#pythagorean triples

[rbrooksdesign]

“Butterfly Fractal Sequence-53”, Reginald Brooks

[jadagul]

lacquiparle replied to your post: I wrote up a thing for my upcoming number theory...

I really like this post. I didn’t realize until I thought about it that it also gives a characterization of all the Pythagorean triples, doesn’t it? Not only does a line through (-1,0) and a rational point on x=1 have to hit the unit circle at a rational point, but all lines through (-1,0) and rational points on the unit circle have rational slope. This will force them to hit x=1 at a rational point, so all Pythagorean triples are given by that formula.

Thanks!

And yeah, that’s definitely a thing. I’d say in this case that the (reduced) Pythagorean triples are parametrized by the rationals; I wrote a followup about that here.

[rbrooksdesign]

MPS, #1, 05-01, 2022, Reginald Brooks

∑ = running sum

PN = Perfect Number = xz

Mp = Mersenne Prime = z = 2ᵖ - 1

Mp² = Mersenne Prime Square = MPS = z²

x = 2ᵖ⁻¹ = short side of PN rectangle

y = x-1

OC = ODD Complement rectangle to PN = yz

Mp² = PN + OC = xz + yz = z²

2ⁿ = exponential power of 2, where n = 1,2,3,…, e.i. 2ⁿ = 1–2–4–8–16…

∑ of 2ⁿ = 1–3–7–15–31–…, the difference (∆) = 2–4–8–16–..

…more

[rbrooksdesign]

In “PTOP (Periodic Table Of PRIMES) on the BIM: PART IV: #2” we will be doing just that.

Here is an older graphic. Take all the Inner Grid cell values on the BIM that divide evenly by 24, color them YELLOW-ORANGE, and this is what you get. It showed that the Primitive Pythagorean Triangles (PPTs) always and exclusively fell on the Active ROWs—where the 1st Col. values were either YELLOW or ORANGE. Turns out the SAME is true for the PRIMES. They, too, only land on the Active ROWs. Note: some Active ROWs have neither PPT or PRIME, some have both, and some one but not the other.

We will be using this graphic for the PRIMES and PPsets, later opening it up.

2018-20, Reginald Brooks

PRIMES Index

~~~~~

[rbrooksdesign]

RoughAnimator, 13-29 , “DSEQEC: Double-Slit Experiment, Quantum Entanglement Conjecture” A single photon consists of an entangled pair: matter and anti-matter, that can interfere with itself to form the interference pattern. 2019, Reginald Brooks

[rbrooksdesign]

Periodic Table Of PRIMES (PTOP) and the Goldbach Conjecture, #15. 2019 Reginald Brooks

Here’s the thing...

Extracting those PRIME-Pair sets (PPsets) for each given EVEN forms the basis for the PTOP — Periodic Table Of PRIMES.

While “hidden” on the BIM, it clearly forms a definitive pattern on the PTOP: for EVERY 3+PRIME PPset that forms the 2nd column on the PTOP — acting like a bifurcation point — a “Trail” of PPsets forms a zig-zagging diagonal pointing down and to the right.

These are ALWAYS — much like a fractal — added with the SAME PRIME Sequence (3,5,7,11,13,17,19,23,29,31,37,...).

When you read a given horizontal line from left to right across the PTOP, you see that each given PPset that was contributed by a PPset Trail, adds up — composes — its respective EVEN. The sum (∑) number of PPsets that form its EVEN is totaled in the last column.

The fractal-like addition of one new, additional PRIME Sequence PPset to each subsequent “Trail” formed results in the overlapping trails growing at a rate that far, far exceeds the growth and incidence of the PRIME Gaps. This ensures that there will always be at least one PPset that will be present to compose ANY EVEN.

The Goldbach Conjecture has been satisfied and a new PRIME pattern has been found.

http://www.brooksdesign-ps.net/Reginald_Brooks/Code/Html/MSST/MSST-TPISC_resources/netart19.htm

http://www.brooksdesign-ps.net/Reginald_Brooks/Code/Html/MSST/MSST-TPISC_resources/MSST-TPISC_resources.html

https://vimeo.com/372243091

[rbrooksdesign]

RoughAnimator, 13-11 , “DSEQEC: Double-Slit Experiment, Quantum Entanglement Conjecture” A single photon consists of an entangled pair: matter and anti-matter, that can interfere with itself to form the interference pattern. 2019, Reginald Brooks

[rbrooksdesign]

RoughAnimator, 13-30 , “DSEQEC: Double-Slit Experiment, Quantum Entanglement Conjecture” A single photon consists of an entangled pair: matter and anti-matter, that can interfere with itself to form the interference pattern. 2019, Reginald Brooks