Introduction

Audience

Curious, very familiar with cartesian coordinates and linear equations.
Want to know more about the structure of numbers beyond prime decomposition

Terminology – keep it simple.

Minimum 6k+1 rather than 1mod6
Coprime: Focus on positive integers while discussing coprimes. without losing generality of the results.

Quality

Notes, rough draft, need for completion, can improve
Precision only if the idea doesn’t get clouded by abstraction
Tradeoff between show a lot and prove a lot – show a lot
Avoid too ambitious approach (qualify as solid math) to get out ideas

Wiki

Initially 1 way communication,
if interesting, I can often provide more detail
open to collaboration after reviews
appreciate Knuth emphasis on notation (attempt to use std)

Goal

Characterize 2D patterns: not simple planar groups and not fractals. Possible semigroups.
Additive structures subject to coprime rules. Linear additive for growth. Additive parent for fill.
Part math, part search for perspective on structure
- Question understanding of chaos, butterfly effect
  - Probabilities, dense (Graham, Knuth, & Patashnik, 1989) line source leads to discrete paths

No.	Rule	Proof Summary
1.	aꓕb ↔ bꓕa	Coprime relation is symmetric: gcd(a,b) = gcd(b,a).
2.	aꓕb ↔ aꓕ-b	gcd(a,b) = gcd(a,-b).
3.	aꓕb → {a,b,a+b} are pairwise coprime symbolically: aꓕb ↔ aꓕ(a+b) aꓕb ↔ bꓕ(a+b) aꓕ(a+b) ↔ bꓕ(a+b)	The sum of 2 coprimes is coprime with each of its addends. Proof by contradiction: If 2 terms share a factor so must the third. (distributive law)
4.	aꓕb → aꓕ(ka+b)	By induction on k: true for k=1, that is, aꓕb → aꓕ(a+b) (3.) aꓕ(ka+b) → aꓕ(a + (ka+b)) (3.) = aꓕ(a(k+1) + b) □
5.	aꓕb, kꓕb ↔ kaꓕb	The unique prime decomposition of ka includes only prime factors found in a or k and, therefore, not found in b.
6.	kaꓕb ↔ ka+b ꓕb	by (3.)
7.	aꓕb ↔ aꓕ(a-b)	aꓕb ↔ aꓕ-b (2.) aꓕ-b ↔ aꓕ(a-b) (3.) aꓕb ↔ aꓕ(a-b) Law of Syllogism Note: In 3. thru 6. “b” can be replaced by “-b”. (2.)

aꓕb ↔ (a-b) ꓕb

Rule Comments:

4.	aꓕb → aꓕ(ka+b)	Coprime Translation Symmetry (translate b by ka)
7.	aꓕb ↔ aꓕ(a-b)	Coprime Reflection Symmetry (reflect b to a-b)

Note that if 2 terms are not coprime then the distributive law implies that the sum is not coprime with either term. So clearly, the rules could be stated with the condition “if and only if …”.

One and Two Dimensional Coprime Symmetries

Summary: We show that there is an abundance of one dimensional symmetries within the coprime graph. These symmetries are replicated in columns, rows and a variety of slopes in between. The one dimensional symmetries serve as the basis for two dimensional symmetries of the entire graph.

Why: interpret visual patterns in the coprime graph. The lattice points in the coprime graph are geometric objects with their own symmetry induced visual patterns. The symmetries include rotation, reflection, translation, and glide (? E.g.).

Further motivation: It appears that the coprime patterns are related to the generation of a wide range of dynamic systems. The systems are characterized by additive growth processes that connect symmetries. Coprime sequences (paths) relate to linear growth, and to harmonic and geometric series. Also Pathagorean Triples, real quadratic irrationals/continued fractions (Pell’s Equation). Triangular numbers

Observation: It seems that coprime pairs serve in a supporting role in elementary number theory, that is, a coprime binary relation is given as a condition for a theorem rather than treated as an object of interest for its own sake. The fundamental additive property of coprimes is neither mentioned nor proven in most texts.

Also, the above rules can be viewed intuitively in terms of Stern-Brocot Tree traversals.

Outline of Coprimes

Symmetries

Breaking of symmetries (move)
- prime points breaking into more general coprime points (using radical)
- groups (not fractals?) to semigroups (sequences)
- patterns (images) that follow coprime sequences
- Patterns that break into 2 sets of like memebers

Definitions

Shortened Wikipedia definitions
- A common fraction is a rational number written as a/b, where a and b are both integers.
- A common fraction is said to be a proper fraction, if the absolute value of the fraction is strictly less than one—that is, if the fraction is greater than −1 and less than 1.
- Two integers a and b are coprime if 1 is their only positive integer divisor.
Notation:
- Any pair of coprime integers a and b is considered to be a 2 dimensional lattice point represented by cartesian coordinates (a,b).
- a ⊥ b is our notation for a is coprime to b. Similar notation is described in Wikipedia’s Coprime Integers.

From the coprime definition (1,0), (0,1) and (1,1) are coprime points but (4,4) is not a coprime point. Although we can often consider a coprime to be equivalent to a proper fraction (or its reciprocal), we recognize that (1,0) and (1,1) = 1/0 and 1/1 are not proper fractions.
- - Rules breakdown for the special case (1,1).
  - Reflection (Rule 7) fails but ?
  - Translation (Rule 4) holds.

Coprime Scope

It can be shown that the proposed rules generate patterns in each cartesian graph quadrant that differ only in sign from an adjacent quadrant. The quadrant is simply a reflection of its adjacent neighbors. So without loss of generality, we limit our graphs to points (a,b) with non-negative integer coordinates.

Notation:

a ⊥ b

a ⊥ b {\displaystyle a\perp b} Coprime Formation rules:

Shown above

Coprime symmetries

1D (basic)
- Period n
  - Reflection by midpoint forming palindrome (Rule 7)
  - Translation by n (Rule 4)
- Identical Columns
  - Applies definition of Radical(n)
  - By example, Radical(2ⁱx3^j) = 6, where 2ⁱx3^j = 6, 12, 18, 24, (not 30), 36, …
  - So columns 6, 12, 18, 24, (not 30), 36, … are identical
- Reflection symmetry
  - Reflection of columns to rows (Rule 1)

2D (with slopes from the origin) (fundamental region)
- Slope = 0 to 1
  - Line of reflection at x=y
  - by Rule 1 wrt coprime symmetry (x,y) is symmetric with (y,x)
    - below the line: Column n contains ordered pairs (k,n), k < n
    - above the line: Row n contains the corresponding inverses (n,k)
    - so the graph could be folded along the y=x line
- Slope = 0 to 1/2
  - Column reflections at y=x/2
    - By 1D description of palindromes above, each column is reflected along the midpoint line at y=x/2
    - So the graph up to Slope 1 could be generated by Rule 7 given the points on or below the line at y=x/2
- Slope = k to k+1
  - By 1D above, each column n (starting at slope =0) is translated by kn between slope k and k+1
    - Between slope = 1 and slope = 2
  - So the 2D graph from slope = 0 to 1 can be generated by reversing the above translation from any slope k to k+1. And the slope=1/2 reflection can be reversed at ½ or k+1/2.
From a geometric viewpoint, a triangular region bounded by a given column n and slopes k and k+1 passing through the origin forms an equal area triangle for all k . Clearly, each such triangle surrounds the same number of points.

Small Prime Symmetry

Summary: When presented as a special case of coprimes, small primes have more structure/symmetry than I was aware of.

Assume the Radical(n) = a, then
- Column Reflection causes each point below the midpoint of a to be paired with one an equal distance above the midpoint. Apply Rule 7: a ꓕ b ↔ a ꓕ (a-b).
- Column Translation causes each point b, where b < a, to be translated by any multiple of a, that is b → ka + b. Apply Rule 4: a ꓕ b → a ꓕ (ka+b).

Radical(2ⁱ) =2, where 2ⁱ = 2, 4, 8, 16, …
- The smallest composite number excluding a factor of 2 is 3² = 9
- 1 ꓕ 2 since the gcd (1, 2) = 1.
- 2 is not coprime to 2 since gcd (2, 2) = 2.
Let 2ⁱ = 8
- The points for 8 can be found by translating the first 2 points.
- By Translation: 2 ꓕ b → 2 ꓕ (k*2+b)
  - 2 ꓕ 1 → 2 ꓕ 3, 2 ꓕ 5, 2 ꓕ 7
  - 3, 5 and 7 are primes since each is both coprime to 2 and less than 9 and hence, cannot be composite.

Radical(2ⁱx3^j) = 6, where 2ⁱx3^j = 6, 12, 18, 24, 36, …
- The smallest composite number excluding factors of 2 or 3 is 5² = 25
- Consider the first 3 points up to and including the midpoint of 6:
  - 1 ꓕ 6 since the gcd (1, 6) = 1.
  - 2ꓕ ̸ 6, 3ꓕ ̸ 6 since 2 and 3 are given as factors of 6
- By Reflection: 6 ꓕ 1 ↔ 6 ꓕ (6-b)
  - 6 ꓕ 1 ↔ 6 ꓕ 5
Let 2ⁱx3^j = 24
- The points for 24 can be found by translating the first 6 points
- By Translation: 6 ꓕ b → 6 ꓕ (k*6+b)
  - 6 ꓕ 1 → 6 ꓕ 7, 6 ꓕ 13, 6 ꓕ 19
  - 6 ꓕ 5 → 6 ꓕ 11, 6 ꓕ 17, 6 ꓕ 23
- {5, 7, 11, 13, 17, 19, 23} must be primes since each is both coprime to 6 and less than 25 and hence, cannot be composite.

Radical(2ⁱx3^jx5^k) = 30, where 2ⁱx3^jx5^k = 30, 60, 90, 120, 150, …
- The smallest composite number excluding factors of 2,3 or 5 is 7² = 49
- The midpoint of 30 is 15
- The primes up to 15 have been shown to be {2, 3, 5, 7, 11, 13}
- Excluding the given factors of 30 leaves primes {7, 11, 13}
- Adding 1 to the coprime list since gcd(1, 30) = 1 yields {1, 7, 11, 13}
- By Reflection: 30 ꓕ b ↔ 30ꓕ (30-b)
  - So b ϵ {1, 7, 11, 13} → {29, 23, 19, 17}
  - {7, 11, 13, 17, 19, 23, 29} must be primes since each is both coprime to 30 and less than 49 and hence, cannot be composite.
Let 2ⁱx3^jx5^k = 60
- The remaining coprime points for 60 can be found by translating (or alternately reflecting) the first 30 points
- By Translation: 30 ꓕ b → 30 ꓕ (k*30 + b)
  - So b ϵ {1, 7, 11, 13, 17, 19, 23, 29} → {31, 37, 41, 43, 47, 49, 53, 59} generates coprimes to 30, but not necessarily primes. Of course, the numbers less than 49 are prime, 49 is not a prime, 53 and 59 would need further tests.

What have we shown:

Through coprime symmetry operations of reflection and translation, the primes from 3 through 47 can be determined. They are {3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 41, 47}.

The small prime symmetry is broken when the coprime rules are satisfied by a composite (such as 49).

Describe small coprime symmetry breaking.

Note: Use symbols for implies and maps into correctly

Observation:

The reflection of primes to primes gives more structure to primes than I had realized.

The above discussion demonstrated that twin primes must reflect to twin primes until the “smallest composite” constraint is exceeded. For example, given 2ⁱx3^jx5^k = 60, the twin pair (11,13) translates to the twin pair (41, 43), however (11,13) reflects to (47, 49) with 49 being a composite.

The Small prime symmetry topic would be more elegant if 1 were considered to be a prime. For example, pair (1, 3) generates twin prime (5, 7) by translation or reflection with a = 8.

For a coprime column, a reflection appears to be more fundamental than a translation. Starting with a Radical(n) sized pattern, ½ of the pattern can be reflected to produce a whole pattern which can then be translated.

In some meaningful sense, primes can be viewed as a subset of coprimes.

Linear Equations and Coprime ID Symmetries[edit | edit source]

Patterns of coprime pairs can appear along linear equation lines. These patterns map into column x values.

Row 1
- evey integer is coprime to 1, so (n, 1) is a coprime point for all n
  - So eqn y=1 contains only coprime points
  - coprime point (n, 1) → coprime point (n, n-1) (By rule 7 )
  - So eqn y=x-1 contains only coprime points
- coprime point (n, n-1) → coprime point (n-1, n) (By Rule 1)
  - So eqn y=x+1 contains only coprime points
The values on the line y=kx+b translate by y=-2b to the line y=kx-b
- y=3x+2 translates to y=3x-2 by y=-4
For the line y=kx+b, the size of b has not been restricted with respect to k or x, so
- y=x-30 translates to y=x+30

The graphs show geometric interpretations of some of the relations. Additional relations shown in the graphs but not described yet include:

a
b
c

- Y = x -30
- At the column x=30 and 1<=y<=30, the 8 coprime points as shown.
- At y=x-30 8 coprime points share the y values with column x=30
- At y=kx-30 8 coprime points share the y values with y=x-30
Can go patterns to eqns or eqns to patterns
Geometric group symmetries
- Tools
- Generators

I considered discussing mathematically the following (to sound smart) but not going to because it doesn’t seem essential, it may be wrong, it probably is a distraction from the problem I’m interested in.?

coprime palindrome sequence
The fundamental domain relationship to radical(n)
The 1D dihedral lattice group known as Dih(Z) or D∞.
Lattices

General Coprime Relations

Linear Equations and Coprime ID Symmetries[edit | edit source]

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General Coprime Relations

Linear Equations and Coprime ID Symmetries[edit | edit source]

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