primes, lonesone plus primes, married plus primes, married minus primes, ect.

The main advantages I see with these algorithms are:

1. Frequency studies of the following algorithms may answer long standing and new problems.

2. These algorithms allow incremental operations, not requiring knowedge of previous primes.

3. These algorithms allow for an incremental database, so we don't have to reinitiate.

Synopsis of Prime Thoughts:

All primes must belong to the Chenault Series, 2, 3, (6n-1), (6n+1)

All composites which are either (6n-1) or (6n+1) can be found by at least one of 3 algorithms;

n1 = integer; 1, infinity n2 = integer; 2, infinity

CAI 6n1n2 + n1 + n2

CAII 6n1n2 + n1 – n2

CAIII 6n1n2 – n1 – n2

The algorithm which finds an n determines whether (6n -1) or (6n+1) is a composite.

By eliminating all (6n-1) and (6n+1) composites, we obtain all primes.

**twin prime**if an n is not found by any of the 3 algorithms, i.e., (6n-1), (6n+1) is a

**twin prime**.

PRIME THOUGHTS by THOMAS EDWARD CHENAULT

Copyright 2010, All Rights Reserved

Reproduction of Prime Thoughts and, or the associated Segment Sieve computer program is encouraged for non-commercial uses. Students should not incur any financial cost for the reproduction or the knowledge. All other rights are reserved.

Acknowledgements:

Thanks to the Supreme Being for letting me peek into a small corner of his universe.

Dedication:

To my beloved wife of 47 years. She carries me through life’s trials.

Reasons to Read:

Prime Patterns (ancient problem solved by compliment), all primes identified in any span without knowledge of previous primes (never done before), and more. Prime Thoughts defines 4 equations which dictate the occurrence of all prime numbers (never done before), and expresses those 4 equations as one super algorithm (the Chenault Solution), yielding the most successful basic prime number tool since the Sieve of Eratosthenes 2300+ years ago. Associated is my working computer program (a version of the Chenault Segment Sieve) which exercises my theories. One application of the Chenault Solution allows generation of a permutation of 40 multiple tables within a user desired span.

Thelma’s Thoughts: See associated Chenault Segmented Sieve program source code for definitions and results.

About Thomas (Tom) Edward Chenault (Shin-awlt):

My childhood immediate family consisted of my dad (Jeff William Chenault), my mother (Susan Edith Elliott Chenault), and my sister (Vanita Allene Chenault Waid). As a father, my immediate family includes my wife (Thelma Ane Christenson Chenault), my oldest son (Chris Sterling Chenault), my middle son (Steve Carlson Chenault), and my youngest son (Shaun Edward Chenault).

Representing the major outside influence in my life (academia) was Dr. Anthony Gioia (Joy-ya) (my thesis advisor).

I earned a BS and MS from Texas Tech University. I am more than 11 years retired from my 32 year career as a Mathematician at White Sands Missile Range, New Mexico, USA. My “retirement” consists of financial and personnel management with a strong dose of computer programming.

www.sterlingtek.com is the main internet site for Sterlingtek Incorporated, which my family and I own, and operate.

PAGE 2

CREDITS:

(1) The men and women who developed and perfected the established mathematical and computer tools used in both this book and the associated computerized version of the Chenault Segment Sieve.

(2) Euclid, Eratosthenes, and Babylonians for ideas and proofs.

(3) The coding of the computerized Chenault Segment Sieve program is compatible with a 1990 Fortran compiler.

Since several names are mentioned, I will clarify that the contents of both Prime Thoughts and the associated Chenault Segment Sieve computer program are 100% from and by me, Tom Chenault.

Release Reason:

I am 71 years old now, and experiencing a lack of time and energy for this weekend project of the last 5 years. I leave it up to academicians (part and full time) to expand or develop new projects using the Chenault Solution and my other creations.

Thelma’s Thoughts: Prime Thoughts presents theory and methods. The computerized segment sieve program offers a test platform for the theories and methods.

TABLE OF CONTENTS

CHAPTER 1 OBJECTIVE AND METHODS Page 5

CHAPTER 2 EDITH’S EDICTS Page 5

(Nomenclature and Basic Development)

Jeff’s Jargon - Section I Page 5

HYPOTHESIS I - Chenault Series Page 6

HYPOTHESIS II Unsorted cn(Chenault Numbers) Page 6

HYPOTHESIS III Sorted cn(Chenault Numbers) Page 6

HYPOTHESIS IV Compromised pcn Page 7

TABLE OF CONTENTS (C0NT) PAGE 3

CHAPTER 3 CHRIS’S CRONICLES Page 7

HYPOTHESIS V Compromised mcn Page 8

HYPOTHESIS VI Euclid’s infinite Primes Page 8

HYPOTHESIS VII EUCLID-CHENAULT infinite MP Page 8

Jeff’s Jargon - Section II Page 8

CHAPTER 4 STEVE’S STANZA Page 9

(Chenault Solution Foundation)

Hypothesis VIII - infinite number of tpc Page 9

Hypothesis IX - Elements of a Twin primes set Page 9

have unique factors

Jeff’s Jargon (Section III) Page 9

CHAPTER 5 SHAUN’S SHOW Page 10

(algorithm development)

Hypothesis X: A composite cn is a result of Page 10

36n1n2 +-6n1+-6n2+-1.

Hypothesis XI: A ctppp is a result of Page 10

satisfaction by the Chenault

Solution (6n1n2+-n1+-n2)vv.

Introduction of the Chenault Algorithims Page 10

(in Thelma’s Thoughts)

Capabilities of the Chenault Segment Sieve Page 11

Jeff’s Jargon (Section IV) Page 11

TABLE OF CONTENTS (C0NT) PAGE 4

Hypothesis XII: All (6n + 1) Page 11

can be identified by common voids in

Chenault Algorithms I and III

Hypothesis XIII: All (6n - 1) Page 11

can be identified by voids in

Chenault Algorithm II.

Hypothesis XIV: All twin primes Page 12

be identified by common voids in

Chenault Algorithms I, II, and III.

CHAPTER 6 Tom’s Triumph Segment Sieve Logic Page 12

A. General Description. Page 12

A1. Common voids of CAI, CAII, CAIII. Page 12

A2. Meaning of void of CAII. Page 12

A3. Common voids of CAI, CAII, CAIII. Page 12

B. Calculation steps. Page 13

B1. Get user span. Page 13

B2. Get calculated span. Page 13

B2a. Get lower calculated limit. Page 13

B2b. Get upper calculated limit. Page 13

C. Identify composites in Chenault Series Page 13

within calculated span.

D. 40 options of the Segment Sieve program Pages 13-20

CHAPTER 1 OBJECTIVE AND METHODS PAGE 5

Objective: Encourage research by creating basic mathematics tools enabling

easier, faster, and precise prime, twin prime, and composite calculations.

Methods: 1. Establish the Chenault Series.

2. Establish Chenault Numbers. Chenault Numbers are

members of the Chenault Series.

3. Establish the Chenault Solution

4. Establish the Chenault Algorithms.

(which govern the distribution of primes.

5. Establish the Chenault Segment Sieve (an exercise of the Chenault

Solution) which identifies composite Chenault Numbers in a user

defined span, and by elimination, all primes in that user defined span.

Importantly, the calculation does not require knowledge of any primes

not in the span.

CHAPTER 2 EDITH’S EDICTS

(Nomenclature and Basic Development)

Jeff’s Jargon: (Section I)

Note 1: All numbers, operations, and symbols refer to the standard integer scale (is) unless subscripted vv

(see Vanita’s Vector, Jeff’s Jargon:(Section 3, Page 10).

Note 2: Throughout this composition, the variable n = a positive integer, 1, thru infinity.

A: A prime number is a positive integer number, greater than

1; such that 1 and itself are the only integers which

divide the prime number yielding a positive integer with a zero remainder.

B: The Chenault series consists of: 2, 3, and (6n + 1), (6n – 1).

C: cn = Chenault number = an element of the Chenault series.

D: mp = (6n-1) primes = "minus primes".

E: pp = (6n+1) primes = "plus primes".

F: mc = (6n-1) composite numbers = "minus composites".

G: pc = (6n+1) composite numbers = "plus composites".

H: mcn = (minus Chenault number) = any mp or mc.

I: pcn = (plus Chenault number) = any pp or pc.

J: unsorted cn’s = 2 and 3.

K: sorted cn’s = all (6n+1) and all (6n-1)

CHAPTER 2 EDITH’S EDICTS (cont) PAGE 6

(Nomenclature and Basic Development)

Hypothesis I: Prime numbers are a subset of the Chenault series.

Proof: By definition The Chenault series is: 2, 3, and (6n+1) and (6n-1).

A positive integer number which is 0(mod 6), 2(mod 6), or 4(mod 6) is divisible by 2.

A positive integer which is 3(mod 6) is divisible by 3.

Therefore, all primes besides 2 and 3 have to be either 1(mod 6) or 5(mod 6).

Thelma’s Thoughts: This hypothesis inspired by the Babylonian’s base 60. All primes besides 2 and 3 are sorted cn’s.

Hypothesis II: Unsorted cn’s cannot be factors of any cn except themselves.

Proof: By definition: an unsorted cn is either 2 or 3.

By definition: a cn must be 2, 3, mcn, or pcn.

Consider cn(mod 2); 2 = 0(mod 2), 3 = 1(mod 2), mcn = 1(mod 2), and pcn = 1(mod 2).

Consider cn(mod 3); 2 = 2(mod 3), 3 = 0(mod 3), mcn = 2(mod 3), and pcn = 1 (mod 3).

Therefore, we see that 2 is a factor of only the cn 2, and 3 is a factor of only the cn 3.

Thelma’s Thoughts: The prime factors of sorted cn’s are mp’s and, or pp’s.

Hypothesis III: Multiplication of a sorted cn by another sorted cn results in a sorted cn.

Proof: By definition, a sorted cn must be either (6n+1) or (6n-1). The possible multiplicative interactions of two sorted cn’s are:

E1 pcn * pcn = (6n1+1) * (6n2+1) = 36n1n2 + 6n1 + 6n2 + 1

= 6n3 + 1 = pc = pcn

E2 pcn * mcn = (6n1+1) * (6n2-1) = 36n1n2 - 6n1 + 6n2 ΓÇô 1

= 6n3 - 1 = mc = mcn

E3 mcn * pcn = (6n1+1) * (6n2-1) = 36n1n2 + 6n1 - 6n2 ΓÇô 1

= 6n3 - 1 = mc = mcn

E4 mcn * mcn = (6n1-1) * (6n2-1) = 36n1n2 ΓÇô 6n1 ΓÇô 6n2 + 1

= 6n3 + 1 = pc = pcn

Thelma’s Thoughts: All composites, resulting from multiplication of mp’s and, or pp’s, are sorted cn’s.

CHAPTER 2 EDITH’S EDICTS (cont) PAGE 7

(Nomenclature and Basic Development)

Hypothesis IV: A compromised pcn must result from multiplication of either:

a. an even number of mcn’s, or

b. any number of pcn’s, or

c. any combination of a. and b.

Proof: Hypothesis I shows that prime numbers are a subset of the

Chenault series. By definition they are 2, 3, pcn, or mcn.

Hypothesis II shows that the prime factor(s) of a pcn cannot include either 2 or 3.

Hypothesis III shows that a mcn multiplied by another mcn yields a pcn.

By converting all pairs of mcn’s to pcn’s, we have a string of pcn’s which may include some pcn’s which were

original to the problem.

Hypothesis III also shows that a pcn multiplied by another pcn yields another pcn, allowing the reduction

of all pcn’s to one pcn. Hypothesis III also shows that there are no other multiplications yielding a

compromised pcn.

Thelma’s Thoughts: A pcn’s prime factors include either none or an even number of mp’s.

CHAPTER 3 CHRIS’S CRONICLES

(Significant Proofs and Definitions)

Hypothesis V: A compromised mcn must result from multiplication of either:

1. an odd number of mcn’s, or

2. an odd number of mcn’s and a number of pcn’s.

Proof: Hypothesis I shows that prime numbers are a subset of the

Chenault series. By definition they are 2, 3, pcn, or mcn.

Hypothesis II shows that the prime factor(s) of a mcn cannot include either 2 or 3.

Hypothesis III shows that a mcn multiplied by another mcn yields a pcn.

By converting all pairs of mcn’s to pcn’s, we have a string of pcn’s which may include some pcn’s which were original to the problem.

Hypothesis III also shows that a pcn multiplied by another pcn results in another pcn, allowing the reduction of all pcn’s to one pcn.

A remaining mcn factor would, by Hypothesis III, result in a final result of a mcn since a pcn multiplied by a

mcn yields a mcn. Hypothesis III also demonstrates that a compromised mcn results from only

pcn * mcn or mcn * pcn.

Thelma’s Thoughts: The prime factors of a mc include an odd number of mp’s.

CHAPTER 3 CHRIS’S CRONICLES (cont) PAGE 8

About 300 BC, Euclid proved that there is an infinite number of prime numbers. Euclid’s proof is presented as Hypothesis VI.

Hypothesis VI: There are an infinite number of primes.

Proof: Assume there are a finite number of primes. If we obtain the multiple of all known primes, then add 1,

we get a number which is not a multiple of any known primes and which must be a new prime or must

have prime factor(s) which are new prime(s).

Hypothesis VII: There is an infinite number of mp.

A slight modification of Euclid’s proof is sufficient for the proof of Hypothesis VII.

The Euclid-Chenault proof of an infinite number of mp follows.

Proof: Assume there is a finite number of mp. If we obtain the multiple of all known primes, then subtract 1, we have a number which is not a multiple of any known primes and which therefore must be a new mp or must have one or more new mp factor(s) by Hypothesis V.

Thelma’s Thoughts: The multiple of all known primes is always a

multiple of 6 due to 2 x 3.

Jeff’s Jargon: (Section II)

A: tpc: Twin Prime Candidate = 6n-1, 6n+1 pair for a given n.

B: potential twin prime partner: ptpp

Letting a n6-1 prime be the reference, its potential

twin prime partner is the n6+1 for that n.

Letting a n6+1 prime be the reference, its potential

twin prime partner is the n6-1 for that n.

C: lonesome prime: mp or pp whose ptpp is composite.

D: lonesome plus prime: pp whose ptpp is mc.

E: lonesome minus prime: mp whose ptpp is pc.

F: lonesome plus composite: (6n+1) is composite, while

(6n-1) is prime (same n).

G: lonesome minus composite: (6n+1) is prime, while

(6n-1) is composite (same n).

H: married plus prime: pp of a twin prime pair.

I: married minus prime: mp of a twin prime pair.

J: the composite table: all composite (6n-1) and

all composite (6n+1)

K: the plus composite table all composite (6n+1)

L: the minus composite table all composite (6n-1)

CHAPTER 4 STEVE’S STANZA PAGE 9

(Solution Foundation)

A twin prime consists of two consecutive odd numbers, both of which are prime numbers. About 300 B.C., Euclid proposed the Twin Prime Conjecture which questions, ΓÇ£Are there an infinite number of twin primes?ΓÇ¥.

Thelma’s Thoughts: The consecutive -cn,+cn form a twin prime candidate (tpc).

Hypothesis VIII: There are an infinite number of tpc.

Proof: From hypothesis I, we see that one set of tpc occur at each n. n can be any positive integer and has no finite limit.

Thelma’s Thoughts: Suggested derivative works such as proofs of infinite numbers of pp primes, of lonesome primes, of lonesome plus primes, of lonesome minus primes, of twin primes, of married plus primes, of married minus primes, should be investigated by the reader after a distribution study of the 4 equations which constitute the Chenault solution, and the effects of combining those results.

Hypothesis IX: The prime factor(s) of one member of a tpc, are unique with respect to the prime factor(s) of the other member of that tpc.

Proof: Let k = a prime factor of one member of a tpc. For k to be

a prime factor of the other member of that tpc, k would be

a factor of the absolute difference between the members of

the tpc = 2. So k = 2, but by Hypothesis II, k cannot be 2.

Jeff’s Jargon: (Section III)

A: integer scale = is

B: Vanita’s Vector (VV) = an integer scale that is a replica of integer scale(is).

An integer on Vanita’s Vector corresponds to (is) of that (vv)

integer multipled by 6.

C: vv point = point on Vanita’s Vector. Note 3: Throughout this composition,

m = vv point; n = number on integer scale

D: tppptn = vv point = Twin Prime Pivot Point

= n for an (6n-1),(6n+1) pair of primes.

E: ptpppt = vv point = potential twin prime pivot point

= n for an (6n-1),(6n+1) pair on is.

F: ctpppt = vv point = compromised Pivot Point

= either 6n -1 and, or 6n + 1 is composite.

CHAPTER 5 SHAUN’S SHOW PAGE 10

(algorithm development)

Hypothesis X: A composite cn is a result of 36n1n2 +-6n1+-6n2+-1.

Proof: 2 and 3 are not composite, so consider mcn’s and pcn’s.

From Hypothesis III,

(E1) pcn * pcn = (6n1+1)is*(6n2+1)is = (36n1n2 +6n1 +6n2 +1)is

If n3 = (6n1n2 +n1 +n2)is, we get (6n3 + 1)is = pc = pcn

(E2) pcn * mcn = (6n1+1)is*(6n2-1)is = (36n1n2 ΓÇô6n1 +6n2 -1) is

If n3 = (6n1n2 ΓÇôn1 +n2)is, we get (6n3 ΓÇô 1)is = pc = mcn

(E3) mcn * pcn = (6n1-1)is*(6n2+1)is = (36n1n2 +6n1 -6n2 -1) is

If n3 = (6n1n2 +n1 ΓÇôn2)is, we get (6n3 ΓÇô 1)is = pc = mcn

(E4) mcn * mcn = (6n1-1)is*(6n2-1)is = (36n1n2 ΓÇô6n1 ΓÇô6n2 +1)is

If n3 = (6n1n2 ΓÇôn1 ΓÇôn2)is, we get (6n3 + 1)is = pc = pcn

Hypothesis XI: A ctpppt is a result of satisfaction of the

Chenault Solution (6m1m2 +-m1 +-m2)vv.

Proof:

(F1) E1 conversion from Hypothesis X to VV (by subtracting 1, then dividing by 6), yields (6m1m2 +m1 + m2)vv .

(F2) E2 conversion from Hypothesis X to VV (by adding 1, then dividing by 6), yields (6m1m2 -m1 + m2)vv .

(F3) E3 conversion from Hypothesis X to VV (by adding 1, then dividing by 6), yields (6m1m2 +m1 - m2)vv .

(F4) E4 conversion from Hypothesis X to VV (by subtracting 1, then dividing by 6), yields (6m1m2 -m1 - m2)vv .

Thelma’s Thoughts: Converting from ins to VV means

1) adding 1 to is (in case we are dealing with a mcn as the user defined limiting ins number).

2) Integer dividing by 6.

More Thelma’s Thoughts:

If we arrive at a ctpppt by F1 (Chenault Algorithm I), we

know that the associated pcn is a composite.

If we arrive at a ctpppt by F2, (Chenault Algorithm II.5),we know that the associated mcn is a composite.

If we arrive at a ctpppt by F3 (Chenault Algorithm II), we know that the same result could have been achieved from F2 since the variables are interchangeable.

If we arrive at a ctpppt by F4 (Chenault Algorithm III), we know that the associated pcn is a composite.

CHAPTER 5 SHAUN’S SHOW (cont) PAGE 11

(algorithm development)

More Thelma’s Thoughts: Expressing F1, F2, F3, and F4 together yields the Chenault Solution, (6m1m2 +-m1 +- m2)vv. The Chenault Algorithms I, II, and III yield computation of any compromised point on Vanita’s Vector.

Knowledge of which Chenault Algorithm(s) isolated a composite identifies whether the Chenault Algorithm identified a mc or a pc associated with the compromised point on Vanita’s Vector.

The Chenault Segment Sieve, an exercise of the Chenault Solution, will identify all Vanita’s Vector members which are ctpppt thru any finite span desired.

Jeff’s Jargon: (Section IV)

A: CA1 = Chenault Algorithm I.

B: CA2 = Chenault Algorithm II.

C: CA3 = Chenault Algorithm III.

D: Chenault Segment Sieve = eliminates all ptppp on Vanita’a Vector that satisfy either CA1, CA2, or CA3.

E. Segment Sieve = Chenault Segment Sieve

= an exercise of the Chenault Solution

Hypothesis XII: All (6n + 1) primes in a span can be located by identifying all common voids in the span results from applying Chenault Algorithms I and III

All resulting common voids on Vanita’s Vector are such that ((6 * (common void)) + 1) is a pp.

Proof: on page 12, we see that the only multiplication methods to arrive at a ctppt which is compromised due to hosting a pc are F1 = (6m1m2 +m1 + m2)vv and F4 = (6m1m2 -m1 - m2)vv. F1 is the Chenault Algorithm I. F4 is the Chenault Algorithm III.

Hypothesis XIII: All (6n ΓÇô 1) primes in a span can be located by identifying all voids in the span resulting from applying Chenault Algorithm II.

All resulting voids on Vanita’s Vector are such that ((6 * void) ΓÇô 1) is a mp.

Proof: on page 12, we see that F2 = (6m1m2 -m1 + m2)vv and

F3 = (6m1m2 -m1 + m2)vv are the only ways to arrive at a mc by multiplication. Since the variables m1 and m2 can be interchanged in either equation, F2 and F3 yield identical results. F3 is the Chenault Algorithim II.

CHAPTER 5 SHAUN’S SHOW (cont) PAGE 12

(algorithm development)

Hypothesis XIV: All twin primes ((6n + 1), (6n ΓÇô 1) pairs) in a span can be located by identifying all common voids in the span resulting from applying

Chenault Algorithm I, Chenault Algorithm II, and Chenault Algorithm III.

Proof: Hypothesis XII shows that Chenault Algorithm’s I and III identify all ctppp which have a pc. Hypothesis XIII shows that Chenault Algorithm II identifies all ctppp which have a mc. The result of exercising Chenault Algorithms I, II, and III is that any common void on Vanita’s Vector is such that

twin prime = ((6 * (common void)) - 1), (6 * (common void)) + 1)

CHAPTER 6 Tom’s Triumph ΓÇô Segment Sieve Logic

(Calculation Logic)

The Chenault Segment Sieve is an application of the Chenault Solution which calculates prime numbers within a user defined span without knowledge of previous primes. Knowledge of which of the Chenault Algorithms detected a composite pivot point, allows determination of minus composites and plus composites.

A. Common voids of an n on Vanita’s Vector by Chenault Algorithms I and III indicate a plus prime at (6*n+1).

B. A void at an n by Chenault algorithm II indicates a minus prime at 6*n-1.

C. Common voids in Chenault Algorithms I, II, and III isolate all n's of twin primes,

i. e., (n*6) - 1, (n*6) + 1 constitute a twin prime if CAI, CAII, and CAIII have a

common void at n.

CHAPTER 6 Tom’s Triumph ΓÇô Segment Sieve Logic (cont) Page 13

(Calculation Logic)

Calculation Steps:

A. Get user desired span: convert span to vv if needed, since all core calculations of the associated computer process use vv.

B. Get calculated limits. The calculated min ΓÇô max span needs to account for the extreme vv values which can affect the user defined points. This results in the calculated lower limit.

B.1. The calculated lower vv limit needs to include the lowest vv values which can effect the user desired span. The Chenault Algorithm which exerts maximum influence from below the user min, is CA1, 6m(1)m(2)+ m(1) + m(2).

B.2. The calculated upper vv limit needs to include the highest vv values which can effect the user desired span. The Chenault Algorithm which exerts maximum influence on the user max from above, is CA3, 6m(1)m(2)- m(1) - m(2).

C. Indicate all (n6-1) and all (n6+1) in calculated span that are composite by:

C1: Indicate all vv in user span that are satisfied by

Chenault Algorithm I (CA1), 6m(1)m(2) + m(1) + m(2).

C2. Indicate all vv in user span that are satisfied by

Chenault Algorithm II (CA2), 6m(1)m(2) + m(1) - m(2).

C3. Indicate all vv in user span that are satisfied by Chenault Algorithm III (CA3), 6m(1)m(2) - m(1) - m(2).

C4. The method used for calculation may dictate the need for Indicating all vv in user span that are satisfied by ΓÇ£Chenault Algorithm II.5ΓÇ¥, 6m(1)m(2) - m(1) + m(2).

Thelma’s Thoughts: CAII.5 is used if m(1) and m(2) have different limits. See Segment Sieve program for example.

D. The associated sieve program can output any permutation of these 40 options in any span supported by the hardware/software used. I anticipate other applications.

1. The Twin prime table. (is points)

Note: detected by common voids in the output from

6m(1)m(2) + m(1) + m(2) = CA1 and

6m(1)m(2) + m(1) - m(2) = CA2 and

6m(1)m(2) - m(1) - m(2) = CA3

Note: ((common void) * 6) + 1 = pp and

((common void) * 6) - 1 = mp

(is points)

CHAPTER 6 Tom’s Triumph ΓÇô Segment Sieve Logic (cont) PAGE 14

(Calculation Logic)

2. The Twin prime Pivot Point table.

Note: The Married Plus and Married Minus

Pivot Point tables are identical to

(The Twin prime Pivot Point table).

Note: detected by common voids in the output from

6m(1)m(2) + m(1) + m(2) = CA1 and

6m(1)m(2) + m(1) - m(2) = CA2 and

6m(1)m(2) - m(1) - m(2) = CA3

Output: (Vanita’s Vector Points)

3. The Married Plus prime table.

pp that are part of a twin prime pair

Note: Twin primes are detected by common voids in the output from

6m(1)m(2) + m(1) + m(2) = CA1 and

6m(1)m(2) + m(1) - m(2) = CA2 and

6m(1)m(2) - m(1) - m(2) = CA3

Note: ((common void) * 6) + 1 is a married plus prime. (is points)

4. The Married Minus prime table. mp that are part of a twin prime pair

Note: Twin primes are detected by common voids in the output from

6m(1)m(2) + m(1) + m(2) = CA1 and

6m(1)m(2) + m(1) - m(2) = CA2 and

6m(1)m(2) - m(1) - m(2) = CA3

Note: ((common void) * 6) - 1 is a married minus prime. (is points)

5. The prime table (includes primes in (pp, mc), (pc, mp), and (pp, mp))

Note: The pp are detected by common voids

in the output from

6m(1)m(2) + m(1) + m(2) = CA1 and

6m(1)m(2) - m(1) - m(2) = CA3

((common void of CA1 and CA2) * 6) + 1) is a pp.

Note: The mp are detected by voids in the output from

6m(1)m(2) + m(1) - m(2) = CA2

((void of CA2) * 6) - 1) is a mp.

(is points)

CHAPTER 6 Tom’s Triumph ΓÇô Segment Sieve Logic (cont) PAGE 15

(Calculation Logic)

6. The Pivot Point table of primes.

(includes the Pivot Points for pairs: (pp, mc) and (pc, mp) and (pp, mp))

Note: The Pivot Points for pp are detected by common voids in output of

6m(1)m(2) + m(1) + m(2) = CA1 and

6m(1)m(2) - m(1) - m(2) = CA3

Note: The Pivot Points for mp are detected by voids in output of

6m(1)m(2) + m(1) - m(2) = CA2

Output: (Vanita’s Vector Points)

7. The Plus prime table.

Note: detected by common voids in

6m(1)m(2) + m(1) + m(2) = CA1 and

6m(1)m(2) - m(1) - m(2) = CA3

((common void of CA1 and CA3) * 6) + 1 = pp

(is points)

8. The Plus prime Pivot Point table.

Note: detected by common voids in

6m(1)m(2) + m(1) + m(2) = CA1 and

6m(1)m(2) - m(1) - m(2) = CA3

Output: (Vanita’s Vector Points)

9. The Minus prime table.

Note: detected by voids in

6m(1)m(2) + m(1) - m(2) = CA2

(void in CA2) * 6) - 1 = mp

(is points)

10. The Minus prime Pivot Point table.

Note: detected by voids in

6m(1)m(2) + m(1) - m(2) = CA2

Output: (Vanita’s Vector Points)

CHAPTER 6 Tom’s Triumph ΓÇô Segment Sieve Logic (cont) PAGE 16

(Calculation Logic)

11. The Lonesome prime table.

Note: (the primes in (pp,mc) and (pc,mp) pairs)

(pp,mc) detected by Pivot Point being a common void in

6m(1)m(2) + m(1) + m(2) = CA1 and

6m(1)m(2) - m(1) - m(2) = CA3 and a

member of 6m(1)m(2) + m(1) - m(2) = CA2

pc,mp) detected by Pivot Point being a void in

6m(1)m(2) + m(1) - m(2) = CA2 and a

member of one (or both) CA1 and CA3

NOTE:((common void of CA1 and CA3) * 6) + 1 = pp

((member of CA2) * 6) - 1 = mc

Output: (Vanita’s Vector Points)

12. The Lonesome prime Pivot Point table.

Pivot Points for (pp, mc) or (pc, mp) pairs

Note: detected by common voids in

6m(1)m(2) + m(1) + m(2) = CA1 and

6m(1)m(2) - m(1) - m(2) = CA3

and Pivot Point being a member of

6m(1)m(2) + m(1) - m(2) = CA2

or

detected by Pivot Point being a member of either (or both)

6m(1)m(2) + m(1) + m(2) = CA1 or

6m(1)m(2) - m(1) - m(2) = CA3

and being a void of

6m(1)m(2) + m(1) - m(2) = CA2

(Vanita’s Vector Points)

13. The Lonesome Plus prime table.

Note: identifies primes from (pp,mc) pairs

(pp,mc) detected by Pivot Point being a common void of:

6m(1)m(2) + m(1) + m(2) = CA1 and

6m(1)m(2) - m(1) - m(2) = CA3 and

Pivot Point being a member of

6m(1)m(2) + m(1) - m(2) = CA2

((common void of CA1 and CA3) * 6) + 1 = pp

((member of CA2) * 6) - 1 = mc (is points)

CHAPTER 6 Tom’s Triumph ΓÇô Segment Sieve Logic (cont) PAGE 17

(Calculation Logic)

14. The Lonesome Plus Pivot Point table.

Note: detected by Pivot Point being a common void in

6m(1)m(2) + m(1) + m(2) = CA1 and

6m(1)m(2) - m(1) - m(2) = CA3 and

Pivot Point being a member of

6m(1)m(2) + m(1) - m(2) = CA2

(Vanita’s Vector Points)

15. The Lonesome Minus prime table.

Note: identifies primes from (pc,mp) pairs

detected by Pivot Point being a member of

6m(1)m(2) + m(1) + m(2) = CA1 or

6m(1)m(2) - m(1) - m(2) = CA3

((member of either (or both) CA1 or CA3) * 6) + 1 = pc

and Pivot Point being a void of

6m(1)m(2) + m(1) - m(2) = CA2

((void of CA2) * 6) - 1 = mp

(is points)

16. The Lonesome Minus Pivot Point table.

Note: detected by Pivot Point being a member of either (or both)

6m(1)m(2) + m(1) + m(2) = CA1 or

6m(1)m(2) - m(1) - m(2) = CA3

and being a void of

6m(1)m(2) + m(1) - m(2) = CA2

(Vanita’s Vector Points)

17. Composite table (all pc and all mc).

Note: pc detected by Pivot Point being a member of either (or both)

6m(1)m(2) + m(1) + m(2) = CA1 or

6m(1)m(2) - m(1) - m(2) = CA3

((member of either (or both) CA1 or CA3) * 6) + 1 = pc

mc detected by Pivot Point being a member of

6m(1)m(2) + m(1) - m(2) = CA2

((member of CA2) * 6) - 1 = mc

(is points)

CHAPTER 6 Tom’s Triumph ΓÇô Segment Sieve Logic (cont) PAGE 18

(Calculation Logic)

18. Pivot Point table for composites (pc's and mc's).

Note: Pivot Point for pc detected by being a member of either (or both)

6m(1)m(2) + m(1) + m(2) = CA1 or

6m(1)m(2) - m(1) - m(2) = CA3

Pivot Point for mc detected by being a member of

6m(1)m(2) + m(1) - m(2) = CA2

(Vanita’s Vector Points)

19. pc table

Note: pc detected by Pivot Point being a member of one or both

6m(1)m(2) + m(1) + m(2) = CA1 or

6m(1)m(2) - m(1) - m(2) = CA3 (or both)

((member of either (or both) CA1 or CA3) * 6) + 1 = pc

(is points)

20. Pivot Point table for pcs.

Note: pc detected by Pivot Point being a member of one or both

6m(1)m(2) + m(1) + m(2) = CA1 or

6m(1)m(2) - m(1) - m(2) = CA3

(Vanita’s Vector Points)

21. mc table

Note: mc detected by Pivot Point being a member of

6m(1)m(2) + m(1) - m(2) = CA2

((member of CA2) * 6) - 1 = mc

(is points)

22. Pivot Point table for mc.

Note: mc detected by Pivot Point being a member of

6m(1)m(2) + m(1) - m(2) = CA2

(Vanita’s Vector Points)

23. The pc table generated by

CA1 = 36m(1)m(2) + 6m(1) + 6m(2) + 1

Note: those pc detected by Pivot Point being a member of

6m(1)m(2) + m(1) + m(2) = CA1

((member of CA1) * 6) + 1 = pc

(is points)

CHAPTER 6 Tom’s Triumph ΓÇô Segment Sieve Logic (cont) PAGE 19

(Calculation Logic)

24. Voids in pc table generated by

(CA1 * 6) + 1

(is points)

25. The Pivot Point table generated by CA1

(Vanita’s Vector Points)

26. Voids in the Pivot Point table generated by

CA1 = 6m(1)m(2) + m(1) + m(2)

(Vanita’s Vector Points)

27. The mc table.

Note: The mc table is detected by a Pivot Point being generated by

CA2.

((member of CA2) * 6) - 1 = mc

(is points)

28. ((n * 6) -1) = mp voids in mc table generated by

36m(1)m(2) +6m(1) -6m(2) -1 = (CA2 * 6) - 1

((void of CA2) * 6) - 1 = mp

(is points)

29. The Pivot Point table for mc's

Note: generated by CA2 = 6m(1)m(2) + m(1) - m(2)

(6 * (member of CA2)) - 1 = mc

(Vanita’s Vector Points)

Note: is identical to the Pivot Point table generated by

6m(1)m(2) - m(1) + m(2)

30. Voids in the Pivot Point table generated by

CA2 = 6m(1)m(2) + m(1) - m(2)

Note: these voids are Pivot Points for mp

(Vanita’s Vector Points)

31. The mc table generated by

36m(1)m(2) - 6m(1) + 6m(2) - 1

Note: identical to 27 output since m(1) and m(2) are interchangeable.

(is points)

32. ((n * 6) -1) Voids (these are mp) in the mc table generated by

36m(1)m(2) - 6m(1) + 6m(2) - 1 (same results as 28)

(is points)

CHAPTER 6 Tom’s Triumph ΓÇô Segment Sieve Logic (cont) PAGE 20

(Calculation Logic)

33. The Pivot Point table generated by

6m(1)m(2) - m(1) + m(2)

(Vanita's Vector Points)

Note: identical to the Pivot Point table generated by

6m(1)m(2) + m(1) - m(2) = CA2 (identical to 29)

34. Voids in the Pivot Point table generated by

6m(1)m(2) - m(1) + m(2) (identical to 30 output)

(Vanita's Vector Points)

35. pc table generated by

36m(1)m(2) - 6m(1) - 6m(2) + 1 = (CA3 * 6) + 1

(is points)

36. ((n*6)+1) voids (possible pp) in pc table generated by

36m(1)m(2) - 6m(1) - 6m(2) + 1

(is points)

37. The Pivot Point table generated by

6m(1)m(2) - m(1) - m(2) = CA3

(Vanita's Vector Points)

38. Voids in the Pivot Point table generated by

6m(1)m(2) - m(1) - m(2) = CA3

(Vanita’s Vector Points)

39. Pivot Point table generated by combining outputs of

6m(1)m(2) + m(1) + m(2) = CA1 and

6m(1)m(2) - m(1) - m(2) = CA3

Note: ((member of either (or both) CA1 or CA3) = (n * 6) + 1 = pc

(Vanita’s Vector Points)

40. voids in Pivot Point table generated by combining

output from 6m(1)m(2) + m(1) + m(2) = CA1 and

6m(1)m(2) - m(1) - m(2) = CA3

Note: ((commom void for CA1 and CA3) * 6) + 1 = pp

(Vanita’s Vector Points)

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