The Solution to the Primes

Transcription

1 The Solution to the Primes it is a mystery into which the human mind will never penetrate. Leonhard Euler ( ) The equations Defining the prime number sequence James P. Moore

2 The Solution to the Primes Lahren James Publishing Spruce Street. Cambridge, Ontario. N1R 4K4 Copyright 2014 by James. P. Moore All rights reserved No part of this publication may be reproduced or used in any form or by any means without the express written permission of the publisher. ISBN: For Catalogue in publication data contact Library and Archives Canada 1

3 Contents Primus 3 The Equations of the Primes 4 The Twin Primes 11 Prime Number Density 13 Prime Number Proof 14 The Goldbach Conjectures 20 Proof of the Strong Goldbach Conjecture 21 Proof of the Weak Goldbach Conjecture 29 References 39 2

4 "Among all mysteries, that of the prime numbers is undoubtedly the most ancient and most resistant." - G. Tenenbaum and M. Mendés France, from The Prime Numbers and Their Distribution (AMS 2000) Primus The Latin word primus, meaning first in importance was used over two thousand years ago to describe a unique set of numbers the primes. Since then, the primes have experienced the full spectrum of consideration as mathematicians investigate their properties and security companies utilize them with all due imperative to protect our information in the digital age. The prime number sequence is: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, a sequence beginning with the number 2 and continuing to infinity. A prime number has the unique characteristic of having only positive, whole number divisors of 1 and itself. The number 7 has no other positive, whole number divisors other than 1 and 7. In contrast, the number 20 has positive, whole number divisors 1, 2, 4, 5, 10 and 20 and thus is not prime. Over two thousand years ago, Euclid wrote about what today is called the Twin Prime Conjecture a theorem stating that there are an infinite number of Twin Primes, or pairs of prime numbers differing by 2. For example, 3 and 5, 5 and 7, 11 and 13, and 17 and 19 are twin primes. As numbers get larger, prime numbers become less frequent and Twin Primes become even less so. This and other curious traits of the prime number sequence continue to be a mystery. For instance, the Strong Goldbach Conjecture, first described by Christian Goldbach in 1742 suggests that every even number greater than or equal to 4 can be created by adding two prime numbers. The Weak Goldbach Conjecture states that every odd number greater than 5 can be created by the addition of three primes. All three conjectures, and others like them, are a mystery. This paper will begin with a description of the equations defining the prime number sequence (including the newly termed Sabot Function) followed by a formal deductive proof of the prime equations. Using these equations, we will also show the Twin Prime and the Goldbach conjectures to be true. It is important to note that these equations are not representative of a prime sieve. Indeed, what separates them apart is their ability to fully characterize the nuances of the prime sequence and, when applied, solve the aforementioned mysteries. 3

5 The Equations of the Primes We begin with the set of all natural numbers as shown above each being represented by a Dirac Impulse function Schematic representation of the Dirac impulse function by a line surmounted by an arrow. The height of the arrow is usually used to specify the value of any multiplicative constant, which will give the area under the function; the integral area of the impulse having a value of 1. The Dirac impulse function, or δ(t-kt s ) function, is a generalized function on the real number line that is zero everywhere except at t equal to zero, with an integral of one over the entire real line. The delta function is sometimes thought of as an infinitely high, infinitely thin spike at the origin, with an area magnitude of one (1) under the spike physically representing an idealized point mass or point charge. It was introduced by theoretical physicist Paul Dirac. Dirac explicitly spoke of infinitely great values of his integrand. In the context of signal processing it is often referred to as the unit impulse symbol (or function). 1 The height of the arrow is usually used to specify the value of a multiplicative unity constant 1, which gives the area under the function as well. 2 Examining the graph on the next page we note the first three natural numbers in red:

6 1 is not considered a prime number. 2 is a prime by definition given that it has no positive divisors other than one and itself. 3 is a prime number for the same reason. All even numbers other than 2 as shown above in blue and continuing on infinitely have the positive divisor of 2 and are thus not prime numbers leaving us with the following set of numbers: Considering the prime number 3, all odd numbers that have a positive divisor of 3 are not prime numbers. If we also eliminate these numbers from the natural number it leaves us with the following infinite set of numbers: Of all the numbers from two to infinity, clearly half are not candidates for being prime numbers since they are even numbers with a positive, whole divisor of 2. Moreover, one third of all numbers are not candidates for being prime numbers as a result of having a positive, whole divisor of 3. Once these are removed from the natural number line, only odd numbers or odd numbers that are not prime, remain. When we eliminate the primes 2 and 3 and consider the numbers that remain: we observe that the sequence is very close to the remaining prime number sequence: 5,7,11,13, 5

7 Expanding the number line: we find the exceptions to the prime sequence being 25, 35, 49, 55, 65, 77, continuing on infinitely and seemingly without pattern. Examining the numbers in the graph below we note that, with the exception of the number 35, there is a periodic relationship with prime numbers starting at 5; the period being the value six (6). For instance, 5+6=11, 5+12=17, 5+18=23, etc. giving us the infinite sequence: 5,11,17,23,29,35,41, This relationship is described best by the following Dirac impulse function: δ(t - (5 + 6x)T s ): for x=0 to infinity Equation 1 Similarly, when we begin with the prime number 7 we observe the same period 6 through subsequent prime numbers: 7+6=13, 7+12=19, 7+18=25, etc. giving us the infinite sequence: 7,13,19,25,31,37,43, This relationship is described by the following Dirac impulse function: δ(t - (7 + 6x)T s ): for x =0 to infinity Equation 2 6

8 Graphically, it may help if we overlay the two equations as both Dirac and sinusoidal representations reflecting their frequency natures: Given that no prime other than 2 is even and no prime other than 3 has a positive divisor of 3, it is clear that these two equations produce every prime number from 5 through infinity (although they also produce non-prime odd numbers). The two equations: δ(t - (5 + 6x)T s ): for x=0 to infinity Equation 1 δ(t - (7 + 6x)T s ): for x =0 to infinity Equation 2 or stated more simply: (5 + 6x) for x= whole number values from 0 to infinity (7 + 6x) for x= whole number values from 0 to infinity generate every possible prime number from 5 through to infinity and are the fundamental equations of the prime number sequence. The question remains, however, why the non-prime number exceptions 25, 35, 49, 55, 65, 77, 85, 91, 95, etc. as highlighted in the graph: To answer this, we must examine the pattern within these numbers. If we look at the numbers 25, 55 and 85, the value of thirty separates them. Looking at the numbers 35, 65, and 95, thirty again separates them. Continuing on, we see the following pattern in the non-prime numbers created by the two Dirac impulse functions: and, 25, 55, 85, 115, 145, 175, 205, 235, etc. 35, 65, 95, 125, 155, 185, 215, etc. 7

9 These two specific sets of numbers (elimination values) are formed from the following equations: 5*( 5 + 6x) for all whole numbers x = 0,1,2. and 5*(7 + 6x) for all whole numbers x = 0,1,2. Note 5+6x and 7+6x are the same equations that created all of the numbers deemed Possible Primes Pp with initial values starting at 5*5=25 when x=0 with a period 5*6=30 in the first set of numbers and an initial value of 5*7=35 for x=0 for the second set of numbers with a period of 5*6=30: 25, 55, 85, 115, 145, 175, 205, 235, 35, 65, 95, 125, 155, 185, 215, 245, Strikingly, for every possible prime number Pp, the relationships: and S 5 (t) =δ(t - Pp(5 + 6x)T s ) for x=0 to infinity Equation 3 S 7 (t) =δ(t - Pp(7 + 6x)T s ) for x=0 to infinity Equation 4 or stated more simply: Pp*(5 + 6x) for x= whole number values from 0 to infinity Pp*(7 + 6x) for x= whole number values from 0 to infinity create sequences of numbers that, when eliminated from the initial Possible Prime number sequence, remove all non-prime (composite) numbers, leaving only the prime number sequence. 8

10 Examples: Pp = 5: Starting point is 5*5 = 25, period is 5*6=30 giving 25, 55, 85, 115, 145, 175, etc. Pp = 5: Starting point is 5*7 = 35, period is 5*6=30 giving 35, 65, 95, 125, 155, 185, etc. Pp = 7: Starting point is 7*5 = 35, period is 7*6=42 giving 35, 77, 119, 161, 203, 245, etc. Pp = 7: Starting point is 7*7 = 49, period is 7*6=42 giving 49, 91, 133, 175, 217, 259, etc. Pp = 11: Starting point is 11*5 = 55, period is 11*6=66 giving 55, 121, 187, 253, etc. Pp = 11: Starting point is 11*7 = 77, period is 11*6=66 giving 77, 143, 209, 275, etc. Pp = 13: Starting point is 13*5 = 65, period is 13*6=78 giving 65, 143, 221, 299, etc. Pp = 13: Starting point is 13*7 = 91, period is 13*6=78 giving 91, 169, 247, 325, etc. Pp = 17: Starting point is 17*5 = 85, period is 17*6=102 giving 85, 187, 289, 391, 493, etc. Pp = 17: Starting point is 17*7 = 119, period is 17*6=102 giving 119, 221, 323, 425, 527, etc. Pp = 19: Starting point is 19*5 = 95, period is 19*6=114 giving 95, 209, 323, 437,551,665, etc. Pp = 19: Starting point is 19*7 = 133, period is 19*6=114 giving 133, 247, 361, 475, 589, etc. In essence, each possible prime number Pp sends out two elimination signals (of unique discrete frequencies) eliminating all non-prime numbers and ensuring the uniqueness of the primes. This process can be seen in the following graphs: 9

11 Possible Primes Pp (t - (5+6x)Ts) (t - (7+6x)Ts) Elimination Values from Sabot Functions (t - 5*(5+6x)Ts) Starting point = 5x5 = 25 with period 5x6 = (t - 5*(7+6x)Ts) Starting point = 5x7 = 35 with period 5x6 = (t - 7*(5+6x)Ts) Starting point = 7x5 = 35 with period 7x6 = (t - 7*(7+6x)Ts) Starting point = 7x7 = 49 with period 7x6 = (t - 11*(5+6x)Ts) Starting point =11x5 = 55 with period 11x6 = (t - 11*(7+6x)Ts) Starting point = 11x7 = 77 with period 11x6 = (t - 13*(5+6x)Ts) Starting point = 13x5 = 65 with period 13x6 = (t - 13*(7+6x)Ts) Starting point = 13x7 = 91 with period 13x6 = Potential Prime Numbers Overlay Elimination Values on Potential Primes Prime Numbers Remain After Elimination Values Removed

12 Equations for Prime Numbers and The Sabot Function Possible Primes: Pp= δ(t - (5 + 6x)T s ) and δ(t - (7 + 6x)T s ) Elimination values: S 5 (t) =δ(t - Pp(5 + 6x)T s ) and S 7 (t) =δ(t - Pp(7 + 6x)T s ) Elimination value start points: Elimination value period: Pp*5 and Pp*7 Pp*6 Graphically, the elimination process is shown below: The functions S 5 (t) =δ(t - Pp(5 + 6x)T s ) and S 7 (t) =δ(t - Pp(7 + 6x)T s ) are called Sabot functions (pronounced Saybow) with each potential prime number issuing two, elimination signals which eliminate all non-prime numbers generated by 5+6x and 7+6x. The Twin Primes Having determined the equations generating the prime number sequence, we are now able to turn our attention to the Twin Prime Conjecture. One fascinating aspect of prime numbers is that, on occasion, primes are separated by 2 (5,7 and 11,13 for instance). The reason for the Twin Primes, their intermittent locations, and the fact that as numbers increase the number of Twin Primes decrease will now be examined. The Twin Primes originate from the two equations: Pp= δ(t - (5 + 6x)T s ) and δ(t - (7 + 6x)T s ) with values extending infinitely from their initial values of 5 and 7. 11

13 Consider the graph below showing only the numbers generated by Pp= δ(t - (5 + 6x)T s ): Now consider the graph below showing only the numbers generated by Pp= δ(t - (7 + 6x)T s ): When we overlay 7+6x on 5+6x we obtain the origins of the Twin Primes from 5 through to infinity: Thus we have shown the origins of the Twin Primes and demonstrated that, because the equations: Pp= δ(t - (5 + 6x)T s ) and δ(t - (7 + 6x)T s ) generate an infinite number of twin values separated by 2 from 5 and 7 to infinity respectively, logically we can conclude there are an infinite number of Twin Primes. The question regarding their intermittent behavior can be demonstrated using the Sabot functions. As discussed, these functions generate the elimination values leaving only the prime number sequence: 12

14 For each potential prime number two elimination signals are generated by the Sabot functions each having a period of Pp*6. For this reason, there is an infinite accumulation of elimination signals acting upon (thinning out) the potential primes leaving fewer primes (as numbers get larger) and, for the same reason, fewer Twin Primes. Prime Number Density Examining the table below, there is only one elimination value applicable for the nine Possible Prime numbers, less than and including 29, and yet for the Possible Prime number 205 (the 68 th Possible Prime), there are 26 elimination values for numbers less than 400 (the 133 rd Possible Prime), there are 62. Number of Possible Primes Possible Prime Less than and Including Number of Elimination Values Continuing on in this manner, it becomes clear why the prime number density follows the equation: D N = P(N) N and explains the reason why the density of prime numbers decrease as numbers increase. 13

15 Prime Number Proof Theorem For all natural numbers greater than and including 1, the equations: F 5 (x)=(5+6x) and F 7 (x)=(7+6x) define the set of all Possible Prime number values {Pp} greater than and equal to 5 with the prime number values 2 and 3 being prime by definition. Also, the equations newly termed Sabot Functions: S 5 (x)=pp n *(5+6x) and S 7 (x)=pp n *(7+6x) sequentially eliminate all non-prime values within the potential prime number set {Pp} leaving only prime numbers in the prime number set {Pr}. This simple set-based equation then yields the set of all prime numbers {Pr}: {Pr} = {Pp} {S 5 (x)} {S 7 (x)} In addition, the equations F 5 (x) and F 7 (x) create and characterize the Twin Prime numbers observed in the prime number sequence whereas the Sabot Functions {S 5 (x)} and {S 7 (x)} determine and clarify the reason for their intermittent locations in the prime number sequence. Proof Given: The set of natural numbers: {N 1 } = {1,2,3,4,5,6 } The set of all natural, even numbers from 2 to infinity: {E} = {2,4,6,8,10 } Subtracting these sets results in the set of all natural odd numbers greater than and equal to 1: {O} = {1,3,5,7,9 } 14

16 For all whole number values of x:{0,1,2,3,4, } consider the following set of numbers derived from the following function of x: F 3 (x)= 3+6x: equation (1) 3+6*(0) = 3 3+6*(1) = 9 3+6*(2) = 15 etc. resulting in an odd-number set of values: {O 3 } = {3,9,15,21 } having an initial value of 3 and a discrete period of 6. Similarly, for all whole number values of x consider the following set of numbers derived from the following function of x: F 5 (x)=5+6x: equation (2) 5+6*(0)=5 5+6*(1)=11 5+6*(2)=17 etc. resulting in an odd-number set of values: {O 5 } = {5,11,17,23 } having an initial value of 5 and a discrete period of 6. Also, for all whole number values of x consider the following set of numbers derived from the following function of x: F 7 (x)=7+6x: equation (3) 7+6*(0)=7 7+6*(1)=13 7+6*(2)=19 etc. resulting in an odd-number set of values: {O 7 } = {7,13,19,25, } having an initial value of 7 and a discrete period of 6. We observe, by combining the three sets {O 3 } = {3,9,15,21 }, {O 5 } = {5,11,17,23 }, and {O 7 } = {7,13,19,25, } we obtain the entire set of natural, odd numbers greater than and equal to 3: {O dd } = {3,5,7,9,11,13,15,17,19,21,23,25 } 15

17 Consider: By definition, a Prime Number is a natural number greater than 1 having no whole number divisors other than 1 and itself. By this definition, 2 is a prime number as it is only wholly divisible by 1 and itself; 3 is prime for the same reason. All even numbers greater than 2 are wholly divisible by 2 and are thus not prime numbers. All prime numbers are positive, odd numbers. The set of all odd numbers: {O dd } = {3,5,7,9,11,13,15,17,19,21,23,25 } contains only two distinct types of odd numbers. They are either prime numbers or odd numbers wholly divisible by other odd numbers within the odd number set. For example, in the odd number set {O dd }, 5 is odd and is a prime number, 11 is odd and a prime number whereas 21 is odd but is wholly divisible by the odd number 7 and is thus only an odd number. Consider the equations 1 through 3 and the sets derived from them: F 3 (x)=3+6x F 5 (x)=5+6x F 7 (x)=7+6x {O 3 } = {3,9,15,21 } {O 5 } = {5,11,17,23 } {O 7 } = {7,13,19,25 } We observe that for equation 1: F 3 (x)=3+6x {O 3 } = {3,9,15,21 } all values in this set beyond the value of 3 are wholly divisible by 3 and are thus not prime numbers. For example, 9 is evenly divisible by 3, as is 15, as is 21 per the function F 3 (x). Having shown that the set of all odd numbers, {O dd } is comprised of only prime numbers or odd numbers evenly divisible by other odd numbers, if we eliminate the odd numbers contained in the set derived from F 3 what remains is a set of all odd numbers derived from only the two functions F 5 and F 7: F 5 (x)=5+6x F 7 (x)=7+6x {O 5 } = {5,11,17,23 } {O 7 } = {7,13,19,25 } 16

18 Most importantly then, the sets of numbers: {O 5 } and {O 7 } characterized by {5,11,17,23 } and {7,13,19,25 } represent the remaining set of possible prime numbers {Pp} including only prime numbers or odd numbers that are not prime: {Pp}= {O 5 } + {O 7 } and this set,{pp}is only derived from the two functions: F 5 (x)=5+6x F 7 (x)=7+6x It has been demonstrated that any number based on these two equations is odd and can thus be the only factors that will eliminate values in the remaining set of potential primes{pp}; all other numbers (even numbers and those derived from F 3 ) having been eliminated up to this point. Essentially, for all non-prime numbers in the set {Pp}, it has been proven that they all have their origins in the equations: F 5 (x)=5+6x and F 7 (x)=7+6x Any number, in the potential prime set{pp},when multiplied by a value within in the{pp}set is clearly factorable by that value and thus is not a prime number and can be removed from the set as we work toward obtaining only the set of Prime numbers {Pr}. For example, consider the value 5 from the set {Pp}. If any value within the set {Pp} is wholly divisible by 5, then 5 is a factor and the number is not prime. Hence, all numbers within the {Pp} set of the form: 5*(5+6x) or 5*(7+6x) are not prime as they are wholly divisible by the number 5. Similarly considering the next value 7 in the {Pp} set, it is clear that all values within the set of the form: are not prime. Nor values of 7*(5+6x) or 7*(7+6x) 11*(5+6x) or 11*(7+6x) etc. 17

19 To clarify by example, consider the set {Pp} where the first value in the set is 5 and is a prime number by definition yet all values within the set {Pp} having the form: within the {Pp} set are clearly not prime: 5*(5+6x) or 5*(7+6x) (for all whole number values of x) 25,55,85,115,145 and 35,65,95,125,155 since they are odd numbers having a factor of 5 and are wholly divisible. Note that there is a starting point to each of the infinite sequence of odd-but-not-primes shown. When x=0, the starting point of the infinite sequence is 5*(5+6(0)) = 25 with a discrete period of 5*6=30 giving us the infinite sequence 25,55,85, Similarly, considering the second sequence, we find a starting point of 5*(7+6(0)) = 35 with a period of 5*6 for the infinite sequence 35,65,95,125 If we consider another example in the set, for instance 19: 19*(5+6x) and 19*(7+6x) We observe the numbers within the {Pp} set that are wholly divisible by 19 are clearly not prime having starting values of 19*(5+6(0)) = 95 and 19*(7+6(0)) = 133 respectively with discrete periods of 19*6 = 114 for each resulting in odd-but-not-prime numbers in the two infinite sequences: 95,209,323 and 133,247,361 In general form, we find that all numbers in the set {Pp} that are not prime are numbers originating from the following two functions newly termed Sabot Functions: S 5 (x ) = Pp n *(5+6x) equation (4) S 7 (x) = Pp n *(7+6x) equation (5) Where Pp n is a potential prime number within the set {Pp} that sequentially determines all remaining odd-but-not-prime numbers in the set {Pp}. This simple set equation yields the set of all prime numbers {Pr}: {Pr} = {Pp} - {S 5 (x)} - {S 7 (x)} Considering the question of twin primes throughout the infinite prime number sequence, it has been shown that the functions: F 5 (x)=5+6x F 7 (x)=7+6x {O 5 } = {5,11,17,23 } {O 7 } = {7,13,19,25 } 18

20 create sequences of numbers that, intermittently, are two digits apart when these sequence are combined to form the set {Pp}: 5,7,11,13,17,19,23,25, hence the origin of twin primes in the prime number sequence. It has also been proven the reason the twin primes are intermittent throughout the prime number sequence is due to the Sabot Functions S 5 (x) and S 7 (x) eliminating the odd-but-not-prime numbers throughout the {Pp} set. The first two numbers, 25 and 35 are shown as an example: 5,7,11,13,17,19,23,25,29,31,35,37 As the Sabot Functions eliminate all odd-but-not-prime numbers in the set {Pp} we are left with only the infinite prime number sequence with 2 and 3 having been demonstrated as prime numbers: 2,3,5,7,11,13,17,19,23,29,31,37 QED with all due humility James Moore April 22,

21 The Goldbach Conjectures On June 7, 1742 the German mathematician Christian Goldbach wrote a letter to Leonhard Euler (letter XLIII) [1] in which he proposed the following conjecture: Every integer which can be written as the sum of two primes, can also be written as the sum of as many primes as one wishes, until all terms are units. Euler replied in a letter dated 30 June 1742, and reminded Goldbach of an earlier conversation they had (" so Ew vormals mit mir communicirt haben ), in which Goldbach remarked his original conjecture followed from the following statement: Every even integer greater than 2 can be written as the sum of two primes, which is also a conjecture of Goldbach. In the letter dated 30 June 1742, Euler stated: Dass ein jeder numerus par eine summa duorum primorum sey, halte ich für ein ganz gewisses theorema, ungeachtet ich dasselbe nicht demonstriren kann. ( every even integer is a sum of two primes. I regard this as a completely certain theorem, although I cannot prove it. [2] Letter from Christian Goldbach to Leonhard Euler dated on 7. June 1742 (Latin-German). Correspondance mathématique et physique de quelques célèbres géomètres du XVIIIème siècle (Band 1), St.-Pétersbourg 1843, S Goldbach's third version (equivalent to the two other versions) is the form in which the conjecture is typically acknowledged as the Strong Goldbach conjecture, to distinguish it from the Weak conjecture which states that all odd numbers greater than 7 are the sum of three odd primes. [3] The next two sections use the equations that generate the prime number sequence to show that the Strong and Weak Goldbach conjectures are true. 20

22 Theorem 1 The Strong Goldbach Conjecture Proof Given: The set of natural numbers: Every even integer greater than 2 can be written as the sum of two primes. {N 1 } = {1,2,3,4,5,6 } For natural values of x:{1,2,3,4, } consider the following set of numbers derived from the following function of x: F 2 (x) = 2x: {2,4,6,8,10 } equation (1) resulting in an even-numbered set of values: {O 2 } = {2,4,6,8 }. For integer values of x:{0,1,2,3,4, } consider the following set of numbers derived from the following function of x: F 3 (x) = 3+6x: equation (2) 3+6*(0)) = 3 3+6*(1) = 9 3+6*(2) = 15 etc. resulting in an odd-number set of values: {O 3 } = {3,9,15,21 } having an initial value of 3 and a discrete period of 6. Similarly, for integer values of x:{0,1,2,3,4, } consider the following set of numbers derived from the following function of x: F 5 (x) =5+6x: equation (3) 5+6*(0))=5 5+6*(1)=11 5+6*(2)=17 etc. resulting in an odd-number set of values: {O 5 } = {5,11,17,23 } having an initial value of 5 and a discrete period of 6. Also, for integer values of x:{0,1,2,3,4, } consider the following set of numbers derived from the following function of x: F 7 (x) =7+6x: equation (4) 21

23 7+6*(0))=7 7+6*(1)=13 7+6*(2)=19 etc. resulting in an odd-number set of values: {O 7 } = {7,13,19,25, } having an initial value of 7 and a discrete period of 6. We observe that, by combining the four sets: {O 2 } = {2,4,6,8 } {O 3 } = {3,9,15,21 } {O 5 } = {5,11,17,23 } {O 7 } = {7,13,19,25, } we obtain the entire set of natural numbers greater than and equal to 2: Consider: {2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17 } By definition, a Prime Number is a natural number greater than 1 having no positive divisors other than 1 and itself. By this definition, 2 is a prime number as it is only divisible by 1 and itself. Also, 3 is prime by definition. All even numbers greater than 2 have positive divisors other than 1 and themselves and are thus not prime numbers. All prime numbers are positive, odd numbers. The set of all odd numbers: {O dd } = {3,5,7,9,11,13,15,17,19,21,23,25 } Contains only two distinct types of odd numbers: They are either prime numbers or odd numbers having positive divisors from within the odd number set and are not prime. For example, in the odd number set {O dd }, 5 is odd and is a prime number, 11 is odd and is a prime number whereas 21 is odd but has positive divisors 3 and 7 and is thus only an odd number and not prime. Consider equations 2 through 4 and the sets that are derived from them: F 3 (x)=3+6x F 5 (x)=5+6x F 7 (x)=7+6x {O 3 } = {3,9,15,21 } {O 5 } = {5,11,17,23 } {O 7 } = {7,13,19,25 } We observe specifically that for equation 2: F 3 (x)=3+6x {O 3 } = {3,9,15,21 } all values in this set beyond the value of 3 have a positive divisor of 3 and thus are not prime numbers. For example, 9 has the positive divisor 3, as does 15, as does 21 etc. Therefore 3 is the only prime derived from 22

24 this equation. Logically, when we eliminate the odd numbers contained in the set derived from F 3 what remains is the set of all odd numbers derived from the two functions F 5 and F 7: F 5 (x)=5+6x F 7 (x)=7+6x {O 5 } = {5,11,17,23 } {O 7 } = {7,13,19,25 } These two equations generate all odd numbers greater than 3 that are either prime or odd and not prime. Most importantly then, the sets of numbers: {O 5 } and {O 7 } characterized by {5,11,17,23 } and {7,13,19,25 } represent the set of possible prime numbers {Pp} including only prime numbers or odd numbers that are not prime: {Pp}= {O 5 } + {O 7 } and this set,{pp}is derived from the two functions: F 5 (x)=5+6x F 7 (x)=7+6x It has been demonstrated that any number based on these two equations is odd and can be the only factors that will eliminate non-prime values in the remaining set of potential primes{pp}; all other numbers (evens and those derived from F 3 ) having been eliminated up to this point. For all non-prime numbers in the set {Pp}, it is clear that all divisors of the non-prime odd numbers all have their origins in the equations: F 5 (x)=5+6x and F 7 (x)=7+6x Now, consider the results of the functions producing prime numbers {P} exclusive of the odd numbers they also produce (see Prime Number Proof page 14 for a detailed description of the Sabot Functions that generate the primes based on the equations 5 + 6x and 7 + 6x): F 2 (x)=2x {P 2 } = {2} F 3 (x)=3+6x F 5 (x)=5+6x F 7 (x)=7+6x {P 3 } = {3},15,21 } {P 5 } = {5,11,17,23,29,41,47,53,59,71 } {P 7 } = {7,13,19,31,37,43,61,67,73,79 } The values in these sets {P 5 }and {P 7 } are only the prime numbers generated from these two equations. To prove the Strong Goldbach conjecture, we must investigate the sets of numbers generated from the addition of two primes derived from the following: F 2 (1): = 2, the only prime generated from the function F 2 (x) F 3 (1): = 3, the only prime generated from the function F 3 (x) F 5 (x)=5+6x: {P 5 } = {5,11,17,23,29,41,47, } the prime numbers generated from F 5 (x). F 7 (x)=7+6x: {P 7 } = {7,13,19,31,37,43, } the prime numbers generated from F 7 (x). There are a limited number of cases to consider when adding two prime numbers: 23

25 Case 1 F 2 (1): = 2 added to F 2 (1): = 2 Case 2 F 2 (1): = 2 added to F 3 (1): = 3 Case 3 F 2 (1): = 2 added to the set of prime numbers generated by F 5 (x)=5+6x:{5,11,17,23,29,41,47, } Case 4 F 2 (1): = 2 added to the set of prime numbers generated by F 7 (x)=7+6x:{7,13,19,31,37,43,61 } Case 5 F 3 (1): = 3 added to F 3 (1): = 3 Case 6 F 3 (1): = 3 added to the set of prime numbers generated by F 5 (x)=5+6x:{5,11,17,23,29,41,47, } Case 7 F 3 (1): = 3 added to the set of prime numbers generated by F 7 (x)=7+6x:{7,13,19,31,37,43,61, } Case 8 The set of prime numbers generated by F 5 (x)=5+6x:{5,11,17,23,29,41,47, } added to the set of prime numbers generated by F 5 (x)=5+6x:{5,11,17,23,29,41,47, } Case 9 The set of prime numbers generated by F 5 (x)=5+6x:{5,11,17,23,29,41,47, } added to the set of prime numbers generated by F 7 (x)=7+6x:{7,13,19,31,37,43,61, } Case 10 The set of prime numbers generated by F 7 (x)=7+6x:{7,13,19,31,37,43,61, } added to the set of prime numbers generated by F 7 (x)=7+6x:{7,13,19,31,37,43,61, } Case 1: F 2 (1) + F 2 (1) = 2+2 = 4 Case 2: F 2 (1) + F 3 (1) = 2+3 = 5. Since the result is an odd number this case is not applicable Case 3: F 2 (1) + F 5 (x) = 2 + {5,11,17,23,29,41,47, } = {7,13,19,25,31,43,49, }. Since the result is a set containing only odd numbers this case is not applicable. 24

26 Case 4: F 2 (1) + F 7 (x) = 2 + {7,13,19,31,37,43,61, } = {9,15,21,33,39,45,63, }. Since the result is a set containing only odd numbers this case is not applicable. Case 5: F 3 (1) + F 3 (1) = = 6 Case 6: F 3 (1) + F 5 (x) = 3 + {5,11,17,23,29,41,47, } = {8,14,20,26,32,44,50, } Case 7: F 3 (1) + F 7 (x) = 3 + {7,13,19,31,37,43,61 } = {10,16,22,34,40,46,64, } Case 8: For this case we begin with the first value in F 5 (x); that being F 5 (1) = 5 and add it to all values in the set of primes generated by F 5 (x): F 5 (x) + F 5 (x) = {5,11,17,23,29,41,47, } + {5,11,17,23,29,41,47, } F 5 (1) + F 5 (x) = 5 + {5,11,17,23,29,41,47, } = {10,16,22,28,34,46,52 } We then continue with the second value in F 5 (x) that being F 5 (2) = 11 and add it to all values in the set of primes generated by F 5 (x): F 5 (x) + F 5 (x) = {5,11,17,23,29,41,47, } + {5,11,17,23,29,41,47, } F 5 (2) + F 5 (x) = 11 + {5,11,17,23,29,41,47, } = {16,22,28,34,40,52,58, } Continuing with the third value in F 5 (x) that being F 5 (3) = 17 and add it to all values in the set of primes generated by F 5 (x): F 5 (x) + F 5 (x) = {5,11,17,23,29,41,47, } + {5,11,17,23,29,41,47, } F 5 (3) + F 5 (x) = 17 + {5,11,17,23,29,41,47, } = {22,28,34,40,46,58, } This process continues to infinity and yet the set of numbers produced reduces to a single set as follows: F 5 (1) + F 5 (x) = 5 + {5,11,17,23,29,41,47, } = {10,16,22,28,34,46,52 } F 5 (2) + F 5 (x) = 11 + {5,11,17,23,29,41,47, } = { 16,22,28,34,40,52,58, } F 5 (3) + F 5 (x) = 17 + {5,11,17,23,29,41,47, } = { 22,28,34,40,46,58, } etc. We note that, due to the periodic nature (discrete period of 6) of the equations producing the primes, the results generated by each successive addition of value and set overlap leaving one resultant set. This is simplified by noting that the set is derived from the addition of the first two values in each set and then the addition of 6 to each subsequent value: {10,16,22,28,34,40,46,52 } 25

27 Case 9: For this case we begin with the first value in F 5 (x); that being F 5 (1) = 5 and add it to all values in the set of primes generated by F 7 (x). Subsequent values are also shown: F 5 (x) + F 7 (x) = {5,11,17,23,29,41,47, } + {7,13,19,31,37,43,61 } F 5 (1) + F 7 (x) = 5 + {7,13,19,31,37,43,61 } = {12,18,24,36,42,48,66 } F 5 (2) + F 7 (x) = 11 + {7,13,19,31,37,43,61 } = { 18,24,30,42,48,54,72, } F 5 (3) + F 7 (x) = 17 + {7,13,19,31,37,43,61 } = { 24,30,36,48,54,60,78 } etc. We note again that, due to the periodic nature (discrete period of 6) of the equations producing the primes, the results generated by each successive addition of value and set overlap leaving one resultant set. This is simplified by noting that the set is derived from the addition of the first two values in each set and then adding 6 to each subsequent value: {12,18,24,30,36,42,48,54,60,66,72 } Case 10: For this case we begin with the first value in F 7 (x); that being F 7 (1) = 7 and add it to all values in the set of primes generated by F 7 (x). Subsequent values are also shown: F 5 (x) + F 7 (x) = {7,13,19,31,37,43,61 } + {7,13,19,31,37,43,61 } F 7 (1) + F 7 (x) = 7 + {7,13,19,31,37,43,61, } = {14,20,26,38,44,50,68 } F 7 (2) + F 7 (x) = 13 + {7,13,19,31,37,43,61 } = { 20,26,32,44,50,56,74 } F 7 (3) + F 7 (x) = 19 + {7,13,19,31,37,43,61 } = { 26,32,38,50,56,62,80 } etc. We note again that, due to the periodic nature (discrete period of 6) of the equations producing the primes, the results generated by each successive addition of value and set overlap leaving one resultant set. This is simplified by noting that the set is derived from the addition of the first two values in each set and then adding 6 to each subsequent value: {14,20,26,32,38,44,50,56,62,68,74,80 } Observing the values derived from all cases resulting in even numbers: Case 1: F 2 (1) + F 2 (1) = 2+2 = 4 Case 5: F 3 (1) + F 3 (1) = = 6 Case 6: F 3 (1) + F 5 (x) = 3 + {5,11,17,23,29,41,47, } = Case 7: F 3 (1) + F 7 (x) = 3 + {7,13,19,31,37,43,61 } = {8,14,20,26,32,44,50, } {10,16,22,34,40,46,64, } Case 8: F 5 (x) + F 5 (x) = {5,11,17,23,29,41,47, } + {5,11,17,23,29,41,47, } = {10,16,22,28,34,40,46,52, } Case 9: F 5 (x) + F 7 (x) = {5,11,17,23,29,41,47, } + {7,13,19,31,37,43,61 } = {12,18,24,30,36,42,48,54,60,66,72 } Case 10: F 5 (x) + F 7 (x) = {7,13,19,31,37,43,61 } + {7,13,19,31,37,43,61 } = {14,20,26,32,38,44,50,56,62,68,74 } It is noteworthy to consider the table (Figure 1) on the next page and observe that, due to the periodic nature of the equations, there is overlap in generating even numbers from each case (note similar colours for values of even numbers that are the same). For instance there are four ways of generating the value 40 and more ways of generating the value 202. As the values increase, the number of ways of producing an even number increase. It is also noteworthy that the values found in Case 10 are found in Case 6 making Case 10 redundant. As well, the values in Case 7 and Case 8 are identical. 26

28 27 Figure 1. Table of Even Numbers Generated by the Cases

29 Observing the values in Cases 1 through 10, we find all of the even numbers are represented from 4 through to infinity, thus it has been shown that all even numbers greater than or equal to 4 can be derived from the addition of two prime numbers: {2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54 } QED with all due humility James P. Moore June 20,

30 Theorem 2 The Weak Goldbach Conjecture Every odd integer greater than 5 can be written as the sum of three primes. Proof Given: The set of natural numbers: {N 1 } = {1,2,3,4,5,6 } For natural values of x:{1,2,3,4, } consider the following set of numbers derived from the following function of x: F 2 (x) = 2x: {2,4,6,8,10 } equation (1) resulting in an even-numbered set of values: {O 2 } = {2,4,6,8 }. For integer values of x:{0,1,2,3,4, } consider the following set of numbers derived from the following function of x: F 3 (x) = 3+6x: equation (2) 3+6*(0)) = 3 3+6*(1) = 9 3+6*(2) = 15 etc. resulting in an odd-number set of values: {O 3 } = {3,9,15,21 } having an initial value of 3 and a discrete period of 6. Similarly, for integer values of x:{0,1,2,3,4, } consider the following set of numbers derived from the following function of x: F 5 (x) =5+6x: equation (3) 5+6*(0))=5 5+6*(1)=11 5+6*(2)=17 etc. resulting in an odd-number set of values: {O 5 } = {5,11,17,23 } having an initial value of 5 and a discrete period of 6. 29

31 Also, for integer values of x:{0,1,2,3,4, } consider the following set of numbers derived from the following function of x: F 7 (x) =7+6x: equation (4) 7+6*(0))=7 7+6*(1)=13 7+6*(2)=19 etc. resulting in an odd-number set of values: {O 7 } = {7,13,19,25, } having an initial value of 7 and a discrete period of 6. We observe that, by combining the four sets: {O 2 } = {2,4,6,8 } {O 3 } = {3,9,15,21 } {O 5 } = {5,11,17,23 } {O 7 } = {7,13,19,25, } we obtain the entire set of natural numbers greater than and equal to 2: Consider: {2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17 } By definition, a Prime Number is a natural number greater than 1 having no positive divisors other than 1 and itself. By this definition, 2 is a prime number as it is only divisible by 1 and itself. Also, 3 is prime by definition. All even numbers greater than 2 have positive divisors other than 1 and themselves and are thus not prime numbers. All prime numbers are positive, odd numbers. The set of all odd numbers: {O dd } = {3,5,7,9,11,13,15,17,19,21,23,25 } Contains only two distinct types of odd numbers: They are either prime numbers or odd numbers having positive divisors from within the odd number set and are not prime. For example, in the odd number set {O dd }, 5 is odd and is a prime number, 11 is odd and is a prime number whereas 21 is odd but has positive divisors 3 and 7 and is thus only an odd number and not prime. Consider equations 2 through 4 and the sets that are derived from them: F 3 (x)=3+6x F 5 (x)=5+6x F 7 (x)=7+6x {O 3 } = {3,9,15,21 } {O 5 } = {5,11,17,23 } {O 7 } = {7,13,19,25 } 30

32 We observe specifically that for equation 2: F 3 (x)=3+6x {O 3 } = {3,9,15,21 } all values in this set beyond the value of 3 have a positive divisor of 3 and thus are not prime numbers. For example, 9 has the positive divisor 3, as does 15, as does 21 etc. Therefore 3 is the only prime derived from this equation. Logically, when we eliminate the odd numbers contained in the set derived from F 3 what remains is the set of all odd numbers derived from the two functions F 5 and F 7: F 5 (x)=5+6x F 7 (x)=7+6x {O 5 } = {5,11,17,23 } {O 7 } = {7,13,19,25 } These two equations generate all odd numbers greater than 3 that are either prime or odd and not prime. Most importantly then, the sets of numbers: {O 5 } and {O 7 } characterized by {5,11,17,23 } and {7,13,19,25 } represent the set of possible prime numbers {Pp} including only prime numbers or odd numbers that are not prime: {Pp}= {O 5 } + {O 7 } and this set,{pp}is derived from the two functions: F 5 (x)=5+6x F 7 (x)=7+6x It has been demonstrated that any number based on these two equations is odd and can be the only factors that will eliminate non-prime values in the remaining set of potential primes{pp}; all other numbers (evens and those derived from F 3 ) having been eliminated up to this point. For all non-prime numbers in the set {Pp}, it is clear that all divisors of the non-prime odd numbers all have their origins in the equations: F 5 (x)=5+6x and F 7 (x)=7+6x Now, consider the results of the functions producing prime numbers {P} exclusive of the odd numbers they also produce (see The Proof of the Primes for a detailed description of the Sabot Functions that generate the primes based on the equations 5 + 6x and 7 + 6x): F 2 (x)=2x {P 2 } = {2} F 3 (x)=3+6x F 5 (x)=5+6x F 7 (x)=7+6x {P 3 } = {3},15,21 } {P 5 } = {5,11,17,23,29,41,47,53,59,71 } {P 7 } = {7,13,19,31,37,43,61,67,73,79 } 31

33 To prove the Weak Goldbach conjecture, we must investigate the sets of numbers generated from the addition of three primes derived from the following: F 2 (1): = 2, the only prime generated from the function F 2 (x) F 3 (1): = 3, the only prime generated from the function F 3 (x) F 5 (x)=5+6x: {P 5 } = {5,11,17,23,29,41,47, } the prime numbers generated from F 5 (x). 3 F 7 (x)=7+6x: {P 7 } = {7,13,19,31,37,43, } the prime numbers generated from F 7 (x). 4 There are a limited number of cases to consider when adding three prime numbers: Case 1 F 2 (1): = 2 added to F 2 (1): = 2 added to F 2 (1): = 2 Case 2 F 2 (1): = 2 added to F 2 (1): = 2 added to F 3 (1): = 3 Case 3 F 2 (1): = 2 added to F 3 (1): = 3 added to F 3 (1): = 3 Case 4 F 2 (1): = 2 added to F 2 (1): = 2 added to F 5 (x)=5+6x:{5,11,17,23,29,41,47, } Case 5 F 2 (1): = 2 added to F 3 (1): = 3 added to F 5 (x)=5+6x:{5,11,17,23,29,41,47, } Case 6 F 2 (1): = 2 added to F 5 (x)=5+6x:{5,11,17,23,29,41,47, } added to F 5 (x)=5+6x:{5,11,17,23,29,41,47, } Case 7 F 2 (1): = 2 added to F 5 (x)=5+6x:{5,11,17,23,29,41,47, } added to F 7 (x)=7+6x:{7,13,19,31,37,43,61 } Case 8 F 2 (1): = 2 added to F 7 (x)=7+6x:{7,13,19,31,37,43,61 } added to F 7 (x)=7+6x:{7,13,19,31,37,43,61 } Case 9 F 3 (1): = 3 added to F 3 (1): = 3 added to F 3 (1): = 3 3 See The Proof of the Primes for the derivation of the prime numbers from their fundamental equations. 4 See The Proof of the Primes for the derivation of the prime numbers from their fundamental equations. 32

34 Case 10 F 3 (1): = 3 added to F 3 (1): = 3 added to F 5 (x)=5+6x:{5,11,17,23,29,41,47, } Case 11 F 3 (1): = 3 added to F 5 (x)=5+6x:{5,11,17,23,29,41,47, } added to F 5 (x)=5+6x:{5,11,17,23,29,41,47, } Case 12 F 3 (1): = 3 added to F 5 (x)=5+6x:{5,11,17,23,29,41,47, } added to F 7 (x)=7+6x:{7,13,19,31,37,43,61 } Case 13 F 3 (1): = 3 added to F 7 (x)=7+6x:{7,13,19,31,37,43,61 } added to F 7 (x)=7+6x:{7,13,19,31,37,43,61 } Case 14 F 5 (x)=5+6x:{5,11,17,23,29,41,47, } added to F 5 (x)=5+6x:{5,11,17,23,29,41,47, } added to F 5 (x)=5+6x:{5,11,17,23,29,41,47, } Case 15 F 5 (x)=5+6x:{5,11,17,23,29,41,47, } added to F 5 (x)=5+6x:{5,11,17,23,29,41,47, } added to F 7 (x)=7+6x:{7,13,19,31,37,43,61 } Case 16 F 5 (x)=5+6x:{5,11,17,23,29,41,47, } added to F 7 (x)=7+6x:{7,13,19,31,37,43,61 } added to F 7 (x)=7+6x:{7,13,19,31,37,43,61 } Case 17 F 7 (x)=7+6x:{7,13,19,31,37,43,61 } added to F 7 (x)=7+6x:{7,13,19,31,37,43,61 } added to F 7 (x)=7+6x:{7,13,19,31,37,43,61 } Case 1: F 2 (1) + F 2 (1) + F 2 (1) = = 6 Since the result is an even number this case is not applicable Case 2: F 2 (1) + F 2 (1) + F 3 (1) = = 7 Case 3: F 2 (1) + F 3 (1) + F 3 (1) = = 8 Since the result is an even number this case is not applicable Case 4: F 2 (1) + F 2 (1) + F 5 (x) = 2+2+{5,11,17,23,29,41,47, } = {9,15,21,27,33,45,51, } 33

35 Case 5: F 2 (1) + F 3 (1) + F 5 (x) = 2+3+{5,11,17,23,29,41,47, } = {10,16,22,28,34,46,52, } Since the values in this set are all even numbers this case is not applicable Case 6: F 2 (1) + F 5 (x) + F 5 (x) = 2+{5,11,17,23,29,41,47, }+{5,11,17,23,29,41,47, } = {12,18,24,30,36,42, } Since the values in this set are all even numbers this case is not applicable Case 7: F 2 (1) + F 5 (x) + F 7 (x) = 2+{5,11,17,23,29,41,47, }+{7,13,19,31,37,43,61 } = {14,20,26,32,38,44, } Since the values in this set are all even numbers this case is not applicable Case 8: F 2 (1) + F 7 (x) + F 7 (x) = 2+{7,13,19,31,37,43,61 }+{7,13,19,31,37,43,61 } = {16,22,28,34,40,46,, } Since the values in this set are all even numbers this case is not applicable Case 9: F 3 (1) + F 3 (1) + F 3 (1) = = 9 Case 10: F 3 (1) + F 3 (1) + F 5 (x) = 3+3+{5,11,17,23,29,41,47, } = {11,17,23,29,35,47,53, } Case 11: F 3 (1) + F 5 (x) + F 5 (x) = 3+{5,11,17,23,29,41,47, }+{5,11,17,23,29,41,47, } = {13,19,25,31,37,43, } Case 12: F 3 (1) + F 5 (x) + F 7 (x) = 3+{5,11,17,23,29,41,47, }+{7,13,19,31,37,43,61 } = {15,21,27,33,39,45, } Case 13: F 3 (1) + F 7 (x) + F 7 (x) = 3+{7,13,19,31,37,43,61 }+{7,13,19,31,37,43,61 } = {17,23,29,35,41,47 } Case 14: F 5 (x) + F 5 (x) + F 5 (x) = {5,11,17,23,29,41,47, }+{5,11,17,23,29,41,47, }+{5,11,17,23,29,41,47, } = {15,21,27,33,39,45,51 } Case 15: F 5 (x) + F 5 (x) + F 7 (x) = {5,11,17,23,29,41,47, }+{5,11,17,23,29,41,47, }+{7,13,19,31,37,43,61 } = {17,23,29,35,41,47, } 34

36 Case 16: F 5 (x) + F 7 (x) + F 7 (x) = {5,11,17,23,29,41,47, }+{7,13,19,31,37,43,61 }+{7,13,19,31,37,43,61 } = {19,25,31,37,43,49,55, } Case 17: F 7 (x) + F 7 (x) + F 7 (x) = {7,13,19,31,37,43,61 }+{7,13,19,31,37,43,61 }+{7,13,19,31,37,43,61 } = {21,27,33,39,45,51,57, } Observing the values derived from all cases resulting in odd numbers: Case 2: F 2 (1) + F 2 (1) + F 3 (1) = 7 Case 4: F 2 (1) + F 2 (1) + F 5 (x) = {9,15,21,27,33,45,51, } Case 9: F 3 (1) + F 3 (1) + F 3 (1) = 9 Case 10: F 3 (1) + F 3 (1) + F 5 (x) = {11,17,23,29,35,47,53, } Case 11: F 3 (1) + F 5 (x) + F 5 (x) = {13,19,25,31,37,43, } Case 12: F 3 (1) + F 5 (x) + F 7 (x) = {15,21,27,33,39,45, } Case 13:F 3 (1) + F 7 (x) + F 7 (x) = {17,23,29,35,41,47 } Case 14: F 5 (x) + F 5 (x) + F 5 (x) = {15,21,27,33,39,45,51 } Case 15: F 5 (x) + F 5 (x) + F 7 (x) = {17,23,29,35,41,47, } Case 16: F 5 (x) + F 7 (x) + F 7 (x) = {19,25,31,37,43,49,55, } Case 17: F 7 (x) + F 7 (x) + F 7 (x) = {21,27,33,39,45,51,57, } We note that Cases 12, 14 and 17 (coloured green below) produce redundant values to those in Case 4: Case 2: F 2 (1) + F 2 (1) + F 3 (1) = 7 Case 4: F 2 (1) + F 2 (1) + F 5 (x) = {9,15,21,27,33,45,51, } Case 9: F 3 (1) + F 3 (1) + F 3 (1) = 9 Case 10: F 3 (1) + F 3 (1) + F 5 (x) = {11,17,23,29,35,47,53, } Case 11: F 3 (1) + F 5 (x) + F 5 (x) = {13,19,25,31,37,43, } Case 12: F 3 (1) + F 5 (x) + F 7 (x) = {15,21,27,33,39,45, } Case 13: F 3 (1) + F 7 (x) + F 7 (x) = {17,23,29,35,41,47 } Case 14: F 5 (x) + F 5 (x) + F 5 (x) = {15,21,27,33,39,45,51 } Case 15: F 5 (x) + F 5 (x) + F 7 (x) = {17,23,29,35,41,47, } Case 16: F 5 (x) + F 7 (x) + F 7 (x) = {19,25,31,37,43,49,55, } Case 17: F 7 (x) + F 7 (x) + F 7 (x) = {21,27,33,39,45,51,57, } This leaves the following: Case 2: F 2 (1) + F 2 (1) + F 3 (1) = 7 Case 4: F 2 (1) + F 2 (1) + F 5 (x) = {9,15,21,27,33,45,51, } Case 9: F 3 (1) + F 3 (1) + F 3 (1) = 9 Case 10: F 3 (1) + F 3 (1) + F 5 (x) = {11,17,23,29,35,47,53, } Case 11: F 3 (1) + F 5 (x) + F 5 (x) = {13,19,25,31,37,43, } Case 13: F 3 (1) + F 7 (x) + F 7 (x) = {17,23,29,35,41,47 } Case 15: F 5 (x) + F 5 (x) + F 7 (x) = {17,23,29,35,41,47, } Case 16: F 5 (x) + F 7 (x) + F 7 (x) = {19,25,31,37,43,49,55, } 35

37 We note that in the remaining Cases 13 and 15 produce redundant values to those in Case 10: Case 2: F 2 (1) + F 2 (1) + F 3 (1) = 7 Case 4: F 2 (1) + F 2 (1) + F 5 (x) = {9,15,21,27,33,45,51, } Case 9: F 3 (1) + F 3 (1) + F 3 (1) = 9 Case 10: F 3 (1) + F 3 (1) + F 5 (x) = {11,17,23,29,35,47,53, } Case 11: F 3 (1) + F 5 (x) + F 5 (x) = {13,19,25,31,37,43, } Case 13: F 3 (1) + F 7 (x) + F 7 (x) = {17,23,29,35,41,47 } Case 15: F 5 (x) + F 5 (x) + F 7 (x) = {17,23,29,35,41,47, } Case 16: F 5 (x) + F 7 (x) + F 7 (x) = {19,25,31,37,43,49,55, } This leaves the following: Case 2: F 2 (1) + F 2 (1) + F 3 (1) = 7 Case 4: F 2 (1) + F 2 (1) + F 5 (x) = {9,15,21,27,33,45,51, } Case 9: F 3 (1) + F 3 (1) + F 3 (1) = 9 Case 10: F 3 (1) + F 3 (1) + F 5 (x) = {11,17,23,29,35,47,53, } Case 11: F 3 (1) + F 5 (x) + F 5 (x) = {13,19,25,31,37,43, } Case 16: F 5 (x) + F 7 (x) + F 7 (x) = {19,25,31,37,43,49,55, } We note that Case 4 produces the same value as in Case 9: Case 2: F 2 (1) + F 2 (1) + F 3 (1) = 7 Case 4: F 2 (1) + F 2 (1) + F 5 (x) = {9,15,21,27,33,45,51, } Case 9: F 3 (1) + F 3 (1) + F 3 (1) = 9 Case 10: F 3 (1) + F 3 (1) + F 5 (x) = {11,17,23,29,35,47,53, } Case 11: F 3 (1) + F 5 (x) + F 5 (x) = {13,19,25,31,37,43, } Case 16: F 5 (x) + F 7 (x) + F 7 (x) = {19,25,31,37,43,49,55, } Leaving: Case 2: F 2 (1) + F 2 (1) + F 3 (1) = 7 Case 4: F 2 (1) + F 2 (1) + F 5 (x) = {9,15,21,27,33,45,51, } Case 10: F 3 (1) + F 3 (1) + F 5 (x) = {11,17,23,29,35,47,53, } Case 11: F 3 (1) + F 5 (x) + F 5 (x) = {13,19,25,31,37,43, } Case 16: F 5 (x) + F 7 (x) + F 7 (x) = {19,25,31,37,43,49,55, } As described in the proof of the Strong Goldbach conjecture, due to the periodic nature of the equations, there is continuous overlap in generating odd numbers from each case so that the odd values from 9 to infinity have duplicate values being generated by other Case(s). 36

38 Observing the values in simplified Cases 2, 4, 10, 11, and 16, we find all odd numbers are represented from 7 through to infinity, thus it has been shown that all odd numbers greater than 5 can be derived from the addition of three prime numbers: Case 2: F 2 (1) + F 2 (1) + F 3 (1) = 7 Case 4: F 2 (1) + F 2 (1) + F 5 (x) = {9,15,21,27,33,45,51, } Case 10: F 3 (1) + F 3 (1) + F 5 (x) = {11,17,23,29,35,47,53, } Case 11: F 3 (1) + F 5 (x) + F 5 (x) = {13,19,25,31,37,43, } Case 16: F 5 (x) + F 7 (x) + F 7 (x) = {19,25,31,37,43,49,55, } {7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41 } QED with all due humility James P. Moore June 20,