Paradise Trinity Day

[Amigoo]

Re: https://www.mathsisfun.com/numbers/prim ... o-10k.html

A number (n) is not a prime number if ending 0,2,4,5,6, or 8 (except 2,5)
... and a prime multiplied by a prime is not a prime.

Observation: sqrt(n)/4 is a possible divisor limit
(divisors* greater than this value are not required
since "prime/noprime" is known by this time)

* 3,7,9,11,13,17,19,21,23,27,29,31,33,37, ... (ending 1,3,7,9 only)

Samples (proof of concept):
414267891265733 = 1352921 × 306202573
sqrt(n)/4 = 5088392.988..
2345678991265733 = 11596969 × 202266557
sqrt(n)/4 = 12108052.566..

Rod

[Amigoo]

Summary of "hobbyist" analysis ...

Re: https://www.mathsisfun.com/numbers/prim ... o-10k.html

A number (n) is not a prime number if ending 0,2,4,5,6, or 8 (except numbers 2,5)
... and a prime number multiplied by a prime is not a prime.

sqrt(n)/4 is a possible divisor limit (divisors* greater than this value
are not required since "prime/noprime" status is known by this time);
"noprime" status is confirmed when a division has no remainder.

* 3,7,9,11,13,17,19,21,23,27,29,31,33,37, ... (ending 1,3,7,9 only)

Samples (proof of concept):
414267891265733 = 1352921 × 306202573
sqrt(n)/4 = 5088392.988..
2345678991265733 = 11596969 × 202266557
sqrt(n)/4 = 12108052.566..

Prime kNots requires simple computer calculations ... to infinity

Rod

[Amigoo]

Re: Prime kNots analysis ...

A number (n) is not a prime number if ending 0,2,4,5,6, or 8 (except numbers 2,5)
... and a prime number multiplied by a prime is not a prime.

Divisors: 3,7,9,11,13,17,19,21,23,27,29,31,33,37, ... (ending 1,3,7,9 only)

Divisor List not required since divisors increase by 2 after /3
(skip division on divisors ending 5, but keep adding by 2)

Apparently (and because modern computers are so fast),
the number (n) itself is an easy division terminator,
(or a "no remainder" division terminates calculations).

Rod ... ...

[Amigoo]

Observations presented as a question:

Prime kNots analysis: Ignoring "not prime" numbers ending 0,2,4,5,6, or 8 (except numbers 2,5),
does division with no remainder identify a "not prime"? Divisors = 3,7,9,11, .. (ending 1,3,7, or 9).

"kNots" refers both to "Not" (not a prime number) and "knot" (a group of things).

Rod

[Amigoo]

Re: https://www.amazon.com/review/R215GLZMIX6RDZ

How the formula-seekers have been searching for prime numbers.

Rod

[Amigoo]

Another observation about Prime kNots ...

Ignoring "not prime" numbers ending 0,2,4,5,6, or 8 (except numbers 2,5),
a division with no remainder appears to identify a "not prime" number.

Divisors = 3,7,9,11, .. (ending 1,3,7, or 9 and only prime numbers):
3,5,7,11,13,17,19,23,29,31,37,41,.. (9,21,27,33,39,.. are not primes)

... which seems counter-intuitive (searching for what is already known),
but primes collect as the calculations proceed (computers do this easily).

Rod

[Amigoo]

Another observation about Prime kNots ...

a division with no remainder appears to identify a "not prime" number.

Re: https://www.mathsisfun.com/numbers/prim ... -tool.html

Random searching of prime factors hints that a division termination value
(when to stop dividing by prime numbers) is less than square root of dividend
(when divisor value equals or is greater than the number's square root).

Sample: 2122622292224221 = 21617 × 98192269613 (21226.. is not prime)
sqrt(2122622292224221) = 46071925.206..
2122622292224221/2 = 1061311146112110.5

Rod

[Amigoo]

Re: iamQ design (I Am Quadrature)
Updated in: http://aitnaru.org/images/Khristos_Voskrese.pdf

"Quadrature defined by marriage of sqrt(Pi) and sqrt(2)"
and recolored to reveal another "Smile of Pythagoras"

Rod ... ...

[Amigoo]

Re: iamQ design (I Am Quadrature)
now a web page: http://aitnaru.org/homepage/lifeischoice.html

"The long-elusive Quadrature, defined by a Cartesian marriage of sqrt(Pi) and sqrt(2)
and presented with a Smile of Pythagoras - perspective possible only 'outside the box'."

Rod

[Amigoo]

Re: iamQ design (I Am Quadrature)
now a web page: http://aitnaru.org/homepage/lifeischoice.html

Geometers' secret: 1.9130583802711007947403078280203..,
the iPhi Ratio (a nickname*), is confirmed by this geometry
(long side to short side of circle-squaring right triangle)

* iPhi = impossible Phi (re: "circle cannot be squared")

Re: https://en.wikipedia.org/wiki/Phi
about the other Phi, aka "Golden ratio"
= 1.618033988749894848204586834..

Rod

[Amigoo]

EZ squared circle math, incorporating iPhi,
using Pythagorean Theorem: a^2 + b^2 = c^2

Given: For D = 2.0, Side of Circle's Square (SoCS) = sqrt(Pi)

SoCS for D = 2.0 (circle's diameter, right triangle's hypotenuse)
= 1.7724538509055160272981674833411.. (sqrt(Pi), triangle's side b)
therefore, sqrt(Pi) / iPhi = right triangle's side a

1.7724538509055160272981674833411.. (sqrt(Pi), side b)
/ 1.9130583802711007947403078280203.. (iPhi)
= 0.92650275035220848584275966758914.. (side a)

Since a^2 + b^2 = c^2
0.85840734641020676153735661672063.. (side a^2)
+ 3.1415926535897932384626433832795.. (side b^2, Pi)
= 4.0 (side c^2, right triangle's hypotenuse^2)

Thus, sqrt(4.0) = 2.0 (circle's diameter, side c)

Rod ... ...

[Amigoo]

Re: Squarely Entwined design (from 2017)

Double spiral of 2(sqrt(1/Pi)), both having growth factor of 2 per quarter turn,
and revealing association of Pi/2, sqrt(Pi), 2.0 ... as defined by iPhi ratio
1.9130583802711007947403078280203.. (long side to short side).

1.7724538509055160272981674833411.. sqrt(Pi)
/ 1.5707963267948966192313216916398.. Pi/2
= 1.1283791670955125738961589031215..

Rod

[Amigoo]

Re: Py Squares design

Three circles squared and the Pythagorean Theorem (a^2 + b^2 = c^2)
... again begging the question: "Whence transcendental Pi?"

Diameters = Pi/2, sqrt(Pi), 2.0 with right triangle ratios:
1.1283791670955125738961589031215.. (hypotenuse to long side)
1.9130583802711007947403078280203.. (long side to short side)

Rod

[Amigoo]

Re: Py Squares design
"a^2 + b^2 = c^2 as quadrature"

"Been there! Done that!"

Rod

[Amigoo]

Re: Py Squares design
"a^2 + b^2 = c^2 as quadrature
... with Pythagorean precision."

The large circle-squaring right triangle was recolored red
to highlight that each side is a side of a circle's square (SoCS)
(each side has length equal to SoCS).

Thus sayeth Pythagoras: "My Py! My Py!"

Rod ... ...

[Amigoo]

Re: Py Squares design
"a^2 + b^2 = c^2 as quadrature
... with Pythagorean precision."

New scalene busyness in the top circle.
Note how that circle-squaring scalene
identifies the circle's radius.

"Lines and triangles and squares! Oh, Py!"

Rod

[Amigoo]

Re: Py Squares design

"a2 + b2 = c2 as quadrature with Pythagorean precision,
where angles of 777 = 27.597.., 62.403.., 90.0 degrees"

Geometers' secret: acos(sqrt(Pi)/2) = 27.597..
= acos(0.88622692545275801364908374167057..)

"angles of 777" refers to similar '7' shapes in triangles.

Rod

[Amigoo]

Re: Three Coins Cartesian design (from 2008)

In June 2008, curiosity about the still-unsolved Greek geometry challenges*
began with contemplation that three coins could define a trisected angle.
* trisecting an angle, doubling a cube's volume, squaring the circle

Three Coins shows the geometry defining this trisected angle
... but it's not the solution - explore the possibilities

Rod

[Amigoo]

Re: Three Coins Cartesian design (from 2008)

A few lines were added to show that a trisected line*
might help create the points for angle trisection.

* trisecting a line is known geometry

Rod ... ...

[Amigoo]

Re: Three Coins Cartesian design

"Goal to go?"
A geometric skirmish (red lines) "inside the 10 yard line"
hints that the center of the trisection field is predictable:
four similar line segments appear to become six
... therefore divisble by 3!

Rod (up against an "impossible" front line)

[Amigoo]

Re: Three Coins Cartesian design
"Goal to go?"

This "goal post" method confirms the three line segments lengths
and light blue line emphasizes that the midpoint perpendicular lines
must connect at their opposite end to the angle's vertex.
Easier than squaring the circle! Texas 'T' time?

Rod

[Amigoo]

Re: Three Coins Cartesian design
"Goal to go?"

Clue: The straight 3-segment line (red) does indeed define 3 chords
of that respective smaller circle*, and can identify a trisected angle!
How this trisected angle correlates with the original is unknown!
* smaller circle, larger angle

Seems to me, geometric proof may then exist when ...
"if this angle is trisected, then that angle is trisected".

Rod

[Amigoo]

Re: Three Coins Cartesian design
"Goal to go?"

More embellishment to envision "if this angle, then that angle";
truly a geometry journey akin to training for morbus cyclometricus
(aka "sanitas cyclometricus" to those Cartesian believers).

Rod

[Amigoo]

Re: Three Coins Cartesian design
"Goal to go?"

Obviously, a truism of angle trisection:

A 3-segment line that forms similar chords of a circle
defines a trisected angle of that circle with the center
as the vertex of the angle.

Rod

[Amigoo]

Re: Three Coins Cartesian design
"Goal to go?"

A 3-segment line that forms similar chords of a circle
defines a trisected angle of that circle with the vertex
at the center of this circle.

And observe that a circle with a certain diameter
creates the arc that constructs those similar chords.

Rod ... ...