Been researching factor pairs, Pythagorean triples, prime factorization, hexagonal numbers, various rules, etc.

Maybe read this page for some ideas:

https:/ /findthefactors.com/tag/centered-triangular-number/

Couple example snips:

Prime factorization: 976 = 2 × 2 × 2 × 2 × 61, which can be written 976 = 2⁴ × 61

The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 976 has exactly 10 factors.

Or the 901 example, 901 is the sum of two squares two different ways:

30² + 1² = 901

26² + 15² = 901

901 is the hypotenuse of FOUR Pythagorean triples:

60-899-901, calculated from 2(30)(1), 30² – 1², 30² + 1²

424-795-901, which is (8-15-17) times 53

451-780-901, calculated from 26² – 15², 2(26)(15), 26² + 15²

476-765-901, which is 17 times (28-45-53)

Two of those were primitives. That can only happen because ALL of 901’s prime factors are Pythagorean triple hypotenuses.

901 is a composite number.

Prime factorization: 901 = 17 × 53

The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 901 has exactly 4 factors.

Factors of 901: 1, 17, 53, 901

Factor pairs: 901 = 1 × 901 or 17 × 53

901 has no square factors that allow its square root to be simplified. √901 ≈ 30.016662

694 is a composite number.

Prime factorization: 694 = 2 x 347

The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 694 has exactly 4 factors.

>>4065

oooh, thanks VA! It's about Time (something VQC has said a few Times). Perhaps with the market fuckery, it IS time, now finally! VQC has been holding and tweeting, and I think this may be the case!

Also took a turn down the biblical / gematria route, and Star of David stuff.

example:

http:/ /www.whatabeginning.com/Misc/Tess2/P.htm

and (seems less exact, more 'fishing' for numbers, but could provide some inspiration)

Structural Numbers of the Bible

https:/ /www.biblewheel.com/Collaboration/Paardekooper2003_09_09.php

>>4067

Yes VA, the Melange here. Been into this for a while. Mostly reading academic papers, but tonight dumping more of the 'image' type stuff, as sensed your call for some new tangents to think along.

>>4031

Spent a good while looking at your outputs MA. Vahddy interdasting!!

Examples of some basic rules:

A number with three or more factors is a

composite number

Divisibility rules can be used to factor a number and to test the primality of a number. Some divisibility rules:

• All numbers are divisible by 1

• Any even number is divisible by 2. A number is divisible by 2 if it ends with a 0, 2, 4, 6, or 8.

• A number is divisible by 3 if the sum of its digits is divisible by 3.

• A number is divisible by 4 if the last two digits (tens and ones) are divisible by 4.

• A number is divisible by 5 if it ends in a 5 or 0.

• A number is divisible by 6 if it is also divisible by 2 and 3 (see these tests above).

• To test a number for divisibility by 7

o Take the last digit in a number.

o Double and subtract the last digit in your number from the rest of the digits.

o Repeat the process for larger numbers.

• A number is divisible by 8 if the last 3 digits are divisible by 8.

• A number is divisible by 9 if the sum of its digits are divisible by 9

• A number is divisible by 10 if it ends in a zero.

A prime factorization is a factor string expressing a number as the product of only prime factors. Every number has exactly one prime factorization. This prime factorization can be written using exponents if any of

its prime factors appear more than once in the string.

>>4069

Hey PMA, your work has been killer!! You and Teach are really inspiring one another. Nice to follow along with you fine anons.

Has this been brought up? Star of David Theorem. (Given RootODavid and all).

http:/ /mathworld.wolfram.com/StarofDavidTheorem.html

Generalizations to Large Hexagons of The Star of David Theorem with Respect to Gcd

https:/ /link.springer.com/chapter/10.1007/978-94-011-5020-0_4?no-access=true

The Star of David theorem with respect to the greatest common divisor states that GCD(A 1, A 3,A 5) = GCD(A 2, A 4,A 6) for the six coefficients A 1,A 2,A 3,A 4,A 5 and A 6 in order surrounding any element X in Pascal’s triangle. This property was discovered by Gould [4] and first proved by Hillman and Hoggatt [5]. Subsequently a number of alternative proofs and various interesting generalizations have been presented.

https:/ /7daysand8nights.wordpress.com/2010/02/26/star-of-david-theorem/

This blog has same info. Enjoyed this bit:

According to Wolfram’s Mathworld, the Star of David Theorem was first stated by H. W. Gould in 1972, and there were several generalization in the years immediately following. Apprently the association with Pascal’s triangle wasn’t noticed until 6 years ago, however, by B. Butterworth in this article (which is originally about using Pascal’s triangle to illustrate the song “The Twelve Days of Christmas”).

https:/ /threesixty360.wordpress.com/tag/pascals-triangle/

Learning about different types of numbers.

https:/ /www.fq.math.ca/Scanned/9-2/bauer.pdf

Numbers that are both Triangular and Square - their Triangular Roots and Square Roots. R. L. BAUER. (3 pg pdf, 1971)

This popped up including 'recursive' in search.

And, patterns in these triangle numbers. An email exchange with a mathematician regarding The Triangular Squares:

The purpose of this note is not Pell’s equation, but rather to give a more elementary solution, once we propose that the triangular squares obey a linear recurrence. The recurrence I gave needs only two consecutive values, say 0 and 1, to produce all the rest.

http:/ /www.gosper.org/triangsq.pdf

That all stemmed from this guy Hans, and researching Pascal's triangle. It's where I stumbled on the Star of David. He has an interesting perspective.

http:/ /hans.wyrdweb.eu/about-the-number-nine/

About Pascal’s Triangle

When a number represents a Geometric Structure it is called a Figurative Number.

Every possible figurative number is generated by the Triangle of Pascal.

The Fractal Sierpinsky Triangle is the Triangle of Pascal Modulo 2.

The Triangle of Pascal was known long before Pascal (re)discovered it.

It was known in Ancient India as the Meru Prastara and in China as the Yang Hui. Meru Prastara relates the triangel to a Mystical Mountain called Mount Meru. Mount Meru is also implemented in the Sri Yantra.

The Triangle shows the Coefficients of the Function F(X,Y))= (X+Y)**n. If n=0 F(X,Y)=1 and if n=1 F(X,Y)=X+Y so the Coeffcients are (1,1).

Pascals Triangle is a 2-Dimensional System based on the Polynomal (X+Y)N. It is always possible to generalize this structure to Higher Dimensional Levels. 3 Variables ((X+Y+X)N) generate The Pascal Pyramid and n variables (X+Y+Z+….)**N generate The Pascal Simplex.

The rows of the Pascal’s Triangle add up to the power of 2 of the row. So the sum of row 0 is 20 and the sum of row 1 is 21 =2.

The Sum of the rows of the higher n-dimensional versions of the Triangle is nN where n is the Amount of Variables and N the level of expansion. So the Sum of Pascal’s Pyramid (3 variables X,Y,Z) is 3N.

The most interesting property of the Triangle is visible in the Diagonals.

The First Diagonal contains only 1′s. The Ones represent Unique Objects. They are the Points in the Tetraktys.

The Second Diagonal contains the natural numbers. These Numbers are used to Count Objects that are The Same. The Natural Numbers are the Lines that connect the Points. The Natural Numbers are the Sum of the previous Ones.

The Third Diagonal contains the triangular numbers. The Triangular Numbers are the Sum of the previous Natural Numbers.

This pattern repeats itself all the time.

The Fourth Diagonal contains the tetrahedral numbers (Pyramid Numbers) and the Fifth Diagonal, the pentatope numbers.

Fermat stated that Every Positive Integer is a Sum of at most three Triangular numbers, four Square numbers, five Pentagonal numbers, and n n-polygonal numbers.

The Tetrahedron with basic length 4 (summing up to 20) can be looked at as the 3-Dimensional analogue of the Tetraktys.

The Diagonals of the Triangle of Pascal contain every Possible 2-Dimensional Figurative Number (and Structure).

These Numbers are Projections of Higher Dimensional Numbers and Higher Dimensional Structures.

The Higher Dimensional Versions of the Triangle (the Pascal Pyramid, The Pascal Simplex) contain these structures.

The Rows of the Triangle Sum to the Powers of Two (2 Dimensions). These Powers control the Levels of Expansion.

Every 7th step the Fractal Pattern of the Triangle repeats itself on a higher Level.

The Figurative Numbers are the Geometric Shapes that are created by the Lines of the Natural Numbers that are connecting the Points of the One.

Pascal’s Triangle also contains the numbers of the Fibonacci Sequence (“The Golden Spiral“).

When we take the Modulo 9 (the Digital Root of Pythagoras) of the Numbers of Fibonacci a repeating patterns of 24 steps shows itself that can be represented by a Star Tetrahedron or Stella Octangula. The Star Tetrahedron is a Three Dimensional Star of David.

^^ uggh, sorry for blacking out and reddit formatting with that last copypasta.

>>4073 Hey Isee, good to see ya!

Agree, it's interesting, and given it's last thing I do before crashing, in my dreams. I often wake grasping at some concept. This morning it was a series in the e that was just there, if only a bit more lucid of a dream…

So Hans sends me down the path of

Hexagonal numbers, which are a form of pyramidal numbers. Didn't VQC mention pyramids at one point or am I losing it?

https:/ /en.wikipedia.org/wiki/Centered_hexagonal_number

The sum of the first n centered hexagonal numbers is n3. That is, centered hexagonal pyramidal numbers and cubes are the same numbers, but they represent different shapes. Viewed from the opposite perspective, centered hexagonal numbers are differences of two consecutive cubes, so that the centered hexagonal numbers are the gnomon of the cubes. (This can be seen geometrically from the diagram.) In particular, prime centered hexagonal numbers are cuban primes (more new terms! Like pronic numbers, and polygonal numbers, etc.).

To bring this back to Pascal:

When all the odd numbers (numbers not divisible by 2) in Pascal's Triangle are filled in (black) and the rest (the evens) are left blank (white), the recursive Sierpinski Triangle fractal is revealed (see figure at near right), showing yet another pattern in Pascal's Triangle. Other interesting patterns are formed if the elements not divisible by other numbers are filled, especially those indivisible by prime numbers. Go here to download programs that calculate Pascal's Triangle and then use it to create patterns, such as the detailed, right-angle Sierpinski Triangle at the far right.

(http:/ /ptri1.tripod.com/)

Reminds me of CONWAY, and the patterns that grow. When I see this image of the blacked out Pascal triangle and Sierpinski's triangle (attached), and relate to some of the Chain work MA and PMA highlighted, makes me think that maybe that's what we're seeing. A form where the path switches back and forth black to white (chain shifting). And along the edge, it's solid, the ROOT. The rays that shoot off are like the internal edges of the more major features, within the (fractal) structure.