ODD-PRIME FORMULA AND THE COMPLETE
PROOFS OF GOLDBACH, POLIGNAC, AND TWIN
PRIME CONJECTURES.pdf

Danny V Calcaben

► To cite this version:

Danny V Calcaben. ODD-PRIME FORMULA AND THE COMPLETE PROOFS OF GOLDBACH, POLIGNAC, AND TWIN PRIME CONJECTURES.pdf. Journal of Number Theory, In press, 10.6084/m9 figshare.26772172. hal-04937134

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# Odd-Prime Formula and the Complete Proofs of the Goldbach, Polignac, and Twin Prime Conjectures

Danny V. Calcaben

This research paper presented the development of a formula for odd-prime numbers and showed its powerful application in proving longstanding conjectures in number theory. The method involved the development of a formula for odd-composite numbers and investigation of its properties to establish a connection with primes. Key results of this study include the derivation of an odd-prime formula which directly helped provide complete proofs to longstanding problems in prime number theory: the Goldbach, Polignac, and Twin Prime Conjectures. It offers a significant insight that made a big contribution in providing solutions to these problems which remained unsolved for centuries.

Keywords: Prime numbers, odd-prime formula, odd-composite formula, prime number conjectures, Goldbach Conjecture, Polignac Conjecture, Twin Prime Conjecture, mathematical proofs.

## 1. Introduction

Prime numbers and their properties are central topics to the branch of mathematics known as Number Theory. Primes are natural numbers greater than 1 that can be divided only by 1 and themselves. This definition of primes is so simple that even a person who does not have much background in mathematics can understand, yet it is also extremely difficult to analyze as there seems to be no sort of pattern governing their sequence. This is probably the reason why there had been no formula ever developed that always produces prime numbers. There are some formulas which, although they generate many primes, they do not encompass all of them. Such examples of those formulas include the quadratic polynomial function, $n^2 - n + 41$ of Leonard Euler as well as the $(2^n - 1)$ function of Marin Mersenne also known as the Mersenne primes.

The first record of extensive studies made about prime numbers dates back to the time of the ancient Greeks between 500 BC to 300 BC when Euclid first proved that there exist infinitely many prime numbers. It inspired future studies about primes and significant developments were made over the centuries such as the Sieve of Eratosthenes and other modern computational methods. However, as studies on the theory of numbers progressed, more questions which have rooted in the complexity of primes' distribution were also raised. Among these problems were conjectures that until now, remained unresolved. They are the Goldbach, Polignac and Twin Prime Conjectures. Considered to be some of the greatest and longest unsolved problems in mathematics, these conjectures represent a big research gap in the field. The quest of mathematicians to deeply understand primes is not just purely academic in nature because it also has a profound implication to other fields such as cryptography and computer science. The security of digital information lies heavily from the fact that primes' properties and distribution are not yet fully understood.

This study bridges the gap by presenting a new formula for prime numbers that serves as a very powerful tool in addressing the Goldbach, Polignac, and Twin Prime Conjectures. It also provides us with a deeper understanding of the primes' distribution that will pave the way for future researches on these mysterious numbers.

This paper provides new mathematical results that were built on well-known conjectures like those proposed by Goldbach and Polignac. The references are solely used for
historical context and general mathematical definitions. The novel concepts and results in this paper are original contributions by the author.

## 1.1. Definitions

**1.1.1. Prime Numbers.** A natural number greater than 1 which can be divided evenly only by 1 and itself is a prime number. Examples are 2, 3, 5, 7, 11, 13, 17, ... etc.

**1.1.2. Composite Numbers.** A natural number greater than 1 that can be divided evenly by other smaller numbers aside from 1 and itself is a composite number. It has at least three positive divisors and can be expressed as a product of smaller prime numbers. Examples include 4, 6, 8, 9, 10, 12, 14, 15, ... etc.

**1.1.3. Odd-Composite Numbers.** An odd counting number greater than 1 that is not a prime number is called an odd-composite number. Examples include 9, 15, 21, 25, 35, 45, 49, ... etc.

## 1.2 Proved Theorems

**1.2.1. Fundamental Theorem of Arithmetic (Euclid, 300 BCE).** states that every natural number greater than 1 can be uniquely expressed as a product of one or more prime numbers. For instance, 100 can be expressed as $5 \times 5 \times 2 \times 2$ (or $2 \times 2 \times 5 \times 5$). Because of this theorem, 1 is not considered a prime number. If it is, then the set of prime factors would not be unique. For example, $100 = 1 \times 2 \times 2 \times 5 \times 5 = 1 \times 1 \times 2 \times 2 \times 5 \times 5$.

**1.2.2. Euclid's Theorem (300 BCE).** states that there are infinitely many prime numbers.

## 2. Deriving a Formula for Odd-Composite Numbers

We first show the sequences formed by listing down the odd-composite numbers.

<table><tr><td>9</td><td>15</td><td>21</td><td>27</td><td>33</td><td>39</td><td>45</td><td>51</td><td>57</td><td>63</td><td>69</td><td>75</td><td>81</td><td>.</td><td>.</td><td>.</td></tr><tr><td>25</td><td>35</td><td>45</td><td>55</td><td>65</td><td>75</td><td>85</td><td>95</td><td>105</td><td>115</td><td>125</td><td>135</td><td>145</td><td>.</td><td>.</td><td>.</td></tr><tr><td>49</td><td>63</td><td>77</td><td>91</td><td>105</td><td>119</td><td>133</td><td>147</td><td>161</td><td>175</td><td>189</td><td>203</td><td>217</td><td>.</td><td>.</td><td>.</td></tr><tr><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td></tr><tr><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td></tr><tr><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td></tr></table>

As we can see, each row of numbers above is an arithmetic sequence with a first term that is a square of an odd number greater than or equal to 3. We can represent these odd numbers using the expression $2n + 1$, where $n$ is a natural number. Each term in the sequence after the first can be obtained by adding twice the given odd number to the preceding term. For example, the first term of the sequence in the first row is the square of 3, which is 9. To get to the second term, we add twice 3 to the first term to have $9 + 2(3) = 15$. To find the third term, we add twice 3 to the second term, resulting in $15 + 2(3) = 21$. And the pattern continues infinitely. Similarly, the first term of the second row is 25 which is the square of 5. The second term is $25 + 2(5) = 35$, the third term is $35 + 2(5) = 45$, and so on. This pattern applies to all rows. To find the $m$th term of the arithmetic sequence in each row above, we use the formula,
$$ 
\begin{array}{l}
a_m = a_1 + (m-1)(d) \\
\text{where,} \\
\begin{array}{l}
a_m = \text{mth term} \\
m = \text{the number of terms} \\
a_1 = \text{first term} = (2n + 1)^2 \\
d = \text{common difference} = 2(2n + 1)
\end{array}
\end{array}
$$

Finding for the mth term of each sequence above,

$$ 
\begin{aligned}
a_m &= (2n + 1)^2 + (m - 1)[(2)(2n + 1)] \\
a_m &= (2n + 1)^2 + 2(2n + 1)(m - 1)
\end{aligned}
$$

Factoring out $2n + 1$ we get,

$$a_m = (2n + 1)[(2n + 1) + 2(m - 1)].$$

On the other hand, if we factor each odd-composite number to see a pattern in their sequence, we can represent each as a product of two smaller odd numbers greater than 1. Each row starts with the square of an odd number greater than or equal to 3. We can see that the second factor in the succeeding pairs after the first increases by 2 to get to the next one. Meanwhile, the first pair of factors in the second row is a square of 5 which is 25. 5 is the next odd number after 3, and like the first row, the second factor in the succeeding pairs after the first increases by 2 to get to the next one. The pattern goes infinitely. See the sequences below.

<table><tr><td>3*3</td><td>3*5</td><td>3*7</td><td>3*9</td><td>3*11</td><td>3*13</td><td>3*15</td><td>3*17</td><td>3*19</td><td>3*21</td><td>3*23</td><td>.</td><td>.</td></tr><tr><td>5*5</td><td>5*7</td><td>5*9</td><td>5*11</td><td>5*13</td><td>5*15</td><td>5*17</td><td>5*19</td><td>5*21</td><td>5*23</td><td>5*25</td><td>.</td><td>.</td></tr><tr><td>7*7</td><td>7*9</td><td>7*11</td><td>7*13</td><td>7*15</td><td>7*17</td><td>7*19</td><td>7*21</td><td>7*23</td><td>7*25</td><td>7*27</td><td>.</td><td>.</td></tr><tr><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td></tr><tr><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td></tr><tr><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td></tr></table>

Each pair of factors in the sequences above can be rewritten in the form shown below.

<table><tr><td>3(3)+2(3)(0)</td><td>3(3)+2(3)(1)</td><td>3(3)+2(3)(2)</td><td>3(3)+2(3)(3)</td><td>3(3)+2(3)(4)</td><td>.</td><td>.</td><td>.</td></tr><tr><td>5(5)+2(5)(0)</td><td>5(5)+2(5)(1)</td><td>5(5)+2(5)(2)</td><td>5(5)+2(5)(3)</td><td>5(5)+2(5)(4)</td><td>.</td><td>.</td><td>.</td></tr><tr><td>7(7)+2(7)(0)</td><td>7(7)+2(7)(1)</td><td>7(7)+2(7)(2)</td><td>7(7)+2(7)(3)</td><td>7(7)+2(7)(4)</td><td>.</td><td>.</td><td>.</td></tr><tr><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td></tr><tr><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td></tr><tr><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td></tr></table>
Therefore, we can generalize the sequences described above and derive a general formula for identifying all odd-composite numbers. If we let $C_o$ be any odd-composite number, then,

$$C_o = (2n + 1)(2n + 1) + 2(2n + 1)(m - 1).$$

Factoring out $2n + 1$ we get,

$$C_o = (2n + 1)[(2n + 1) + 2(m - 1)]$$

where,

$C_o$ is an odd-composite number.

$2n + 1$ is an odd number greater than or equal to 3.

$m - 1$ is an element of the set of whole numbers.

$n$ is a natural number.

$m$ is a natural number.

Notice that the formula for the $m$th term of each arithmetic sequence is the same with the formula for determining each odd-composite number. This is because each sequence, in which we are finding the $m$th term, consists exclusively of odd-composite numbers. As a result, every value obtained from this formula will surely be an odd-composite. This similarity ensures the structure and pattern within the sequences. This tells us that the method for identifying odd-composite numbers is consistent and reliable.

## 2.1. The Pair of Factors of Every Odd-Composite Number

Let's look again at the numbers below.

<table><tr><td>3*3</td><td>3*5</td><td>3*7</td><td>3*9</td><td>3*11</td><td>3*13</td><td>3*15</td><td>3*17</td><td>3*19</td><td>3*21</td><td>3*23</td><td>.</td><td>.</td></tr><tr><td>5*5</td><td>5*7</td><td>5*9</td><td>5*11</td><td>5*13</td><td>5*15</td><td>5*17</td><td>5*19</td><td>5*21</td><td>5*23</td><td>5*25</td><td>.</td><td>.</td></tr><tr><td>7*7</td><td>7*9</td><td>7*11</td><td>7*13</td><td>7*15</td><td>7*17</td><td>7*19</td><td>7*21</td><td>7*23</td><td>7*25</td><td>7*27</td><td>.</td><td>.</td></tr><tr><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td></tr><tr><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td></tr><tr><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td></tr><tr><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td><td>.</td></tr></table>

Notice that each term is formed by multiplying a pair of odd numbers in the sequences above. The first term of each sequence above is the square of an odd number $\ge 3$. The following terms involve the initial odd number paired with progressively larger odd numbers. This system ensures that all odd-composite numbers can be generated from these sequences because all possible combinations of pairs of odd factors are taken. Having this condition, we can observe that the pair of factors of every odd-composite number falls into two cases.

**Case 1:** The factors of an odd-composite number are equal.

In this case, the factors of an odd-composite number are equal. This happens with the first odd-composite number in each sequence. Examples are $9 = 3 \times 3$, $25 = 5 \times 5$, $49 = 7 \times 7$, and so forth.
**Case 2:** The factors of an odd-composite number are distinct.

Here, the factors of an odd-composite number are not equal, there's a smaller factor and a larger factor. For example, we have $15 = 3 \times 5$, $21 = 3 \times 7$, $27 = 3 \times 9$, and so on for the first row. $35 = 5 \times 7$, $45 = 5 \times 9$, $55 = 5 \times 11$ in the second row and this pattern continues infinitely. It is important to note that every odd-composite number after the first term in each sequence is the product of two distinct factors.

With the formula for finding odd-composite numbers and the cases for their pairs of factors, we can now write the method for finding odd-composite numbers into the following theorem.

**Theorem 2.1.1.** For all $m, n \in \mathbb{N}$, every odd-composite number ($C_0$) can be expressed as a square of an odd number ($2n + 1$) added to twice the product of that odd number and a whole number ($m - 1$).

**3. Properties of Odd-Composite Number Theorem,**

$$C_0 = (2n + 1)[(2n + 1) + 2(m - 1)].$$

**Property 3.1.** The value of $(2n + 1)$ increases as the value of $(m - 1)$ decreases and vice versa.

**Proof:** Let $k \in \mathbb{N}$, be the number of $(2n + 1)$ factors of $C_0$ and $y_1, y_2, y_3, \dots, y_k$ be every $(2n + 1)$ factor such that $3 \le 2n + 1 \le \sqrt{C_0}$ where $y_2 > y_1, y_3 > y_2, y_4 > y_3, \dots, y_k > y_{k-1}$.

Also, let $x_1, x_2, x_3, \dots, x_k$ be every whole number $(m - 1) = \frac{C_0 - (2n+1)^2}{2(2n+1)}$ where $x_2 > x_1, x_3 > x_2, x_4 > x_3, \dots, x_k > x_{k-1}$. Then by definition of odd-composite number given by the formula, $C_0 = (2n + 1)[(2n + 1) + 2(m - 1)]$ we can write,

$$C_0 = y[y+2x]$$
or
$$C_0 = y^2 + 2xy$$

Solving for $x$ in terms of increasing values of $y$ we get,

$$\frac{C_0 - y_1^2}{2y_1} = x_k$$

$$\frac{C_0 - y_2^2}{2y_2} = x_{k-1}$$

$$\frac{C_0 - y_{k-1}^2}{2y_{k-1}} = x_2$$

$$\frac{C_0 - y_k^2}{2y_k} = x_1$$
Here we see that for every $y = (2n + 1)$ factor of $C_0$, there exists a unique value of $x = (m - 1)$. This means that if an odd-composite number has $k$ number of $(2n + 1)$ factors, there are also $k$ whole numbers $(m - 1)$ corresponding to those factors. If we plug in the equation each $(2n + 1)$ represented by $(y)$ in increasing order, $(m - 1)$ represented by $(x)$ goes down from the biggest $(x_k)$, down to the smallest $(x_1)$. Hence, the value of $(m - 1)$ increases as the value of $(2n + 1)$ decreases and vice versa.

**Property 3.2.** $(2n + 1)$ is an odd factor of $C_0$ and $3 \le 2n + 1 \le \sqrt{C_0}$.

**Proof:** It is given from Theorem 2.1.1 that $(2n + 1)$ is an odd number being multiplied to $[(2n + 1) + 2(m - 1)]$ and has $C_0$ as a product. Thus, $2n + 1$ is a factor of $C_0$. Since $n \in \mathbb{N}$, the smallest possible value of $2n + 1$ is when $n = 1$. That is $2(1) + 1 = 3$. And as $n$ gets bigger, the value of $2n + 1$ becomes an odd number greater than 3. Property 3.1 asserts that the value of $(2n + 1)$ increases as the value of $(m - 1)$ decreases and vice versa. This means that the largest possible value for $(2n + 1)$ is when the least possible value of $m - 1$ is plugged in the equation. Since $m \in \mathbb{N}$, then the least possible value of $m - 1$ is when $m = 1$. That is, $m - 1 = 1 - 1 = 0$. The equation then becomes $C_0 = (2n + 1)^2$ or $\sqrt{C_0} = 2n + 1$. This means that the largest possible value of $2n + 1$ is $\sqrt{C_0}$. Hence, $(2n + 1)$ is an odd factor of $C_0$ and $3 \le 2n + 1 \le \sqrt{C_0}$.

**Remark 3.1.** Note that we are only interested on the positive square root of $C_0$ since we are dealing with the set of natural numbers. And whenever we mention $(2n+1)$, as factor of $C_0$ from the given formula, we only refer to those odd factors in the interval $[3, \sqrt{C_0}]$.

**Property 3.3.** $(m - 1)$ is a whole number such that $C_0 + (m - 1)^2$ is a perfect square and $0 \le m - 1 \le \frac{C_0-9}{6}$.

**Proof:** Given that $C_0 = (2n + 1)[(2n + 1) + 2(m - 1)]$,

where,

$C_0$ is any odd-composite number.

$2n + 1$ is an odd factor of $C_0$ and $3 \le 2n + 1 \le \sqrt{C_0}$.

$m - 1$ is a whole number.

$n$ is a natural number.

$m$ is a natural number.

Let $y = 2n + 1$ and $x = m - 1$. The formula then becomes,

$$C_0 = y [y + 2x]$$
or

$$C_0 = y^2 + 2xy$$

If we add $x^2$ to both sides of the equation, the right side turns into a perfect square which means that the value of $C_0 + x^2$ is a perfect square as well,

$$C_0 + x^2 = y^2 + 2xy + x^2$$

$$C_0 + x^2 = (y + x)^2$$

Hence, the sum $C_0 + (m - 1)^2$ is a perfect square.
On the other hand, we know that $m \in \mathbb{N}$ and $(m - 1) \in \mathbb{W}$ in the equation $C_o = (2n + 1)[(2n + 1) + 2(m - 1)]$. This means that the smallest possible value $m - 1$ can have is zero. That is, if $m = 1$, then $m - 1 = 1 - 1 = 0$. This happens when $C_o$ is equal to the square of $(2n + 1)$. Meanwhile, if we manipulate the equation to find the value of $m - 1$, we will have,

$$m - 1 = \frac{C_o - (2n+1)^2}{2(2n+1)} \quad (3.1)$$

Since $n \in \mathbb{N}$, then the smallest possible value of $2n + 1$ is 3. That is $2(1) + 1 = 3$. While 3 is the smallest possible value of $2n + 1$, by Property 3.1, it will be the largest possible value for $m - 1$. So if we substitute this value of $2n + 1$ in equation (3.1) above to find the value of $m - 1$, we will then get,

$$m - 1 = \frac{C_o - [2(1) + 1]^2}{2[2(1) + 1]}$$

$$m - 1 = \frac{C_o - (3)^2}{2(3)}$$

$$m - 1 = \frac{C_o - 9}{6}$$

This means that the largest possible value $(m - 1)$ can have would be $\frac{C_o - 9}{6}$. Therefore, $0 \le m - 1 \le \frac{C_o - 9}{6}$.

## 4. General Formula for Odd-Prime Numbers

For us to derive a general formula for determining odd-prime numbers, let us first go back to Euclid's theorem, the Fundamental Theorem of Arithmetic in (Definition 1.2.1). It states that every natural number greater than 1 can be uniquely expressed as a product of one or more prime numbers. This concept provides us with a possibility of generating prime numbers from composite numbers. In this section, we will explore that possibility. Since we are dealing with odd-composite numbers, our goal is to create a formula that will generate all odd primes. To achieve this, we will first present the corollaries directly resulting from (Theorem 2.1.1).

**Corollary 4.1.** If $x$ is a whole number and $C_o$ is any odd-composite number where $0 \le x \le \frac{C_o-9}{6}$ and $(x^2 + C_o)$ is a perfect square, then every odd factor $y$ of $(C_o)$ such that $(3 \le y \le \sqrt{C_o})$ is equal to the positive square root of the sum $(x^2 + C_o)$ subtracted by $x$.

**Proof:** It has been established that the formula for generating every odd-composite number is given by,

$$C_o = (2n + 1)[(2n + 1) + 2(m - 1)]$$

where,

$C_o$ is any odd-composite number.

$2n + 1$ is an odd factor of $C_o$ and $3 \le 2n + 1 \le \sqrt{C_o}$.

$m - 1$ is a whole number such that $(C_o + (m - 1)^2)$ is a perfect square and $0 \le m - 1 \le \frac{C_o-9}{6}$.

$n$ is a natural number.

$m$ is a natural number.

Again, we let $y = 2n + 1$ and $x = m - 1$. The formula then becomes,
$$ 
\begin{aligned}
C_o &= y[y+2x] \\
    &\text{or} \\
C_o &= y^2 + 2xy
\end{aligned} 
\qquad (4.1) $$

Equation (4.1) is quadratic in two variables and we will solve for the value of y in terms of x. First, we need to rearrange the equation such that all terms are in the left side. The equation is now,

$$ y^2 + 2xy - C_o = 0 $$

Using the quadratic formula, we have the values obtained from $y^2 + 2xy - C_o = 0$ as follows,

$$ a = 1, \quad b = 2x, \quad \text{and} \quad c = -C_o $$

To find the value of y, which represents $(2n + 1)$, an odd factor of $C_o$ and $[3 \le \sqrt{C_o} \le]$ we write the following solution:

$$ 
\begin{aligned}
y &= \frac{-b + \sqrt{b^2 - 4ac}}{2a} \\
y &= \frac{-2x + \sqrt{(2x)^2 - 4(1)(-C_o)}}{2(1)} \\
y &= \frac{-2x + \sqrt{4x^2 + 4C_o}}{2} \\
y &= \frac{-2x + 2\sqrt{x^2 + C_o}}{2} \\
y &= -x + \sqrt{C_o + x^2}
\end{aligned} 
$$

where,

y is an odd factor of $C_o$ and $3 \le y \le \sqrt{C_o}$.

x is a whole number such that $(C_o + x^2)$ is a perfect square and $0 \le x \le \frac{C_o-9}{6}$.

$C_o$ is any odd-composite number.

**Remark 4.1.** Note that we are only interested on the positive square root of $(C_o + x^2)$ since we are dealing with the set of natural numbers.

**Corollary 4.2.** If x is a whole number and $C_o$ is an odd-composite number where $0 \le x \le \frac{C_o-9}{6}$ and $(x^2 + C_o)$ is a perfect square, then every odd factor $(y + 2x)$ of $(C_o)$ such that $C_o = y(y+2x)$, is equal to the positive square root of the sum $(x^2 + C_o)$ added by x.

**Proof:** Since we already know that every y factor of $C_o$, where $3 \le y \le \sqrt{C_o}$ can be expressed as $y = -x + \sqrt{C_o + x^2}$, then we can express $(y + 2x)$, the other factor of $C_o$ as follows,

$$ y + 2x = -x + \sqrt{C_o + x^2} + 2x $$

giving us with,

$$ y + 2x = x + \sqrt{C_o + x^2} $$
where,

y is an odd factor of $C_O$ and $3 \le y \le \sqrt{C_O}$.

x is a whole number such that $(C_O + x^2)$ is a perfect square and $0 \le x \le \frac{C_O-9}{6}$.

$C_O$ is an odd composite number.

And if we replace both values we obtained for the pair of factors of an odd-composite number $C_O$, we have,

$$C_O = y(y+2x)$$

$$C_O = (-x + \sqrt{C_O + x^2})(x + \sqrt{C_O + x^2})$$

$$C_O = -x^2 + C_O + x^2$$

$$C_O = C_O$$

Therefore, $y + 2x = x + \sqrt{C_O + x^2}$.

After presenting the two corollaries of Theorem 2.1.1, let's now proceed with one of the most significant result of our study: the formula for odd prime numbers. The theorem is presented as follows:

**Theorem 4.1.** For every odd composite number ($C_O$), with $k$ number of $(y=2n+1)$ factors, where $3 \le (y=2n+1) \le \sqrt{C_O}$, there exists a greatest whole number ($x_k$) in the interval $[0, \frac{C_O-9}{6}]$ such that $(C_O + x_k^2)$ is a perfect square and $-x_k + \sqrt{C_O + x_k^2}$ is the smallest prime factor of ($C_O$).

**Proof:** It has been shown from Property 3.1 that given an odd composite number $C_O$, with $k$ number of odd factors ($y = 2n + 1$) in the interval $[3, \sqrt{C_O}]$, the value of ($x = m - 1$) increases as the value of ($y = 2n + 1$) decreases and vice versa. Therefore, we can also show that if we replace $x$ by its largest value ($x_k$) from the formula given by Corollary 4.1, we will obtain the smallest odd factor ($y_1$) of ($C_O$). That is,

$$y_1 = -x_k + \sqrt{C_O + x_k^2}.$$

Since $y_1$ represents the smallest odd factor, $(2n + 1)$ of ($C_O$), we can say that $y_1$ is a prime number. It is because the Fundamental Theorem of Arithmetic states that every composite number can be expressed uniquely as a product of primes. That is why, the smallest odd factor of any odd-composite number beside 1 would always be a prime number. And since we are dealing with odd composites, their prime factors must all be odd numbers as well. Therefore, $y_1$ is the smallest prime factor of the composite odd number ($C_O$) and we can write it in the equation,

$$p_{CO} = -x_k + \sqrt{C_O + x_k^2}$$

where,

$P_{CO}$ is the smallest prime factor of $C_O$.

$C_O$ is an odd-composite number.

$k \in \mathbb{N}$, is the number of $2n + 1$ factors of $C_O$ and $3 \le 2n + 1 \le \sqrt{C_O}$.

$x_k$ is the $k$th or biggest whole number in the interval $[0, \frac{C_O-9}{6}]$, such that $(C_O + x_k^2)$ is a perfect square. QED!
Theorem 4.1 basically is a formula for getting the smallest prime factor of any odd composite number $C_O$. It can be traced from a function whose range is the interval $[3, \sqrt{C_O}]$ and whose domain is the interval $[0, \frac{C_O-9}{6}]$. Every odd factor ($y = 2n + 1$) of $C_O$ lies within the range, while every whole number ($x = m - 1$) such that the sum ($C_O + x^2$) is a perfect square lies within the specified domain. Those other whole numbers $x$ from the domain of the function which do not make the sum ($x^2 + C_O$) a perfect square will result in irrational numbers for both $\sqrt{x^2 + C_O}$ and $-x + \sqrt{x^2 + C_O}$. This is because the square root of a number that is not a perfect square is an irrational number. Furthermore, any irrational number added to or subtracted by a whole number remains an irrational number.

As stated earlier, there are two cases for the pairs of factors of any odd composite number. The first case is when it is a square of an odd number, and the second case is when it is a product of distinct odd factors. This means that if we want to create an odd composite number, we can simply pick up any odd number $\ge 3$, then we square or multiply it by any bigger odd number. Let's try and pick the odd number 7. To make 7 an odd-composite number, we can square it to have $7^2 = 49$ or multiply it by any bigger odd number. Let's say 15. The product, $7 \times 15 = 105$ is an odd composite number. These two cases were actually generalized in Theorem 2.1.1. That is, the formula for generating all odd composite numbers given by $C_O = (2n + 1)[2n + 1 + 2(m - 1)]$. Notice that when $m = 1$, $C_O = (2n + 1)^2$. This is in fact the first case of the pair with factors that are equal. When $m > 1$, the pair of factors falls under the second case which means that an odd composite number has distinct factors. So, in order to produce an odd composite number, we either use the formula $C_O = (2n + 1)[2n + 1 + 2(m - 1)]$ or utilize any of the two cases. The bottom line is for us to be able to produce odd-composite numbers that we can plug in the formula, $P_{C_O} = -x_k + \sqrt{x_k^2 + C_O}$, $0 \le x_k \le \frac{C_O-9}{6}$ to generate odd prime numbers. Let's take $C_O = 105$ and see the table of values of the function in its specified domain.

<table><caption>**Table 1**
Table of values for $f(x) = y = -x + \sqrt{105 + x^2}$, $0 \le x \le \frac{C_O-9}{6}$</caption><thead><tr><th>x</th><th>f(x) = y = -x + √x<sup>2</sup> + C<sub>o</sub></th></tr></thead><tbody><tr><td>0</td><td>10.24695076596...</td></tr><tr><td>1</td><td>9.29563014099...</td></tr><tr><td>2</td><td>8.44030650891...</td></tr><tr><td>3</td><td>7.67707825203...</td></tr><tr><td>4</td><td>7</td></tr><tr><td>5</td><td>6.40175425099...</td></tr><tr><td>6</td><td>5.87434208704...</td></tr><tr><td>7</td><td>5.40967364599...</td></tr><tr><td>8</td><td>5</td></tr><tr><td>9</td><td>4.63818169699...</td></tr><tr><td>10</td><td>4.31782106328...</td></tr><tr><td>11</td><td>4.03329637837...</td></tr><tr><td>12</td><td>3.77973383806...</td></tr><tr><td>13</td><td>3.55294535725...</td></tr><tr><td>14</td><td>3.34935157290...</td></tr><tr><td>15</td><td>3.16590212458...</td></tr><tr><td>16</td><td>3</td></tr></tbody></table>
We see on table 1 that x-values are from 0 to 16. This represents the specified domain where the ordered pair $(x_k, P_{CO})$ can be found. In fact, the very first ordered pair from the bottom of the table corresponding to a pair of whole number coordinates represents exactly the ordered pair $(x_k, P_{CO})$. In the case of $C_O = 105$, $x_k = 16$, If we substitute these values in the formula, we will then have,

$$ 
\begin{aligned}
p_{CO} &= -x_k + \sqrt{C_o + x_k^2} \\
P_{CO} &= -16 + \sqrt{105 + 16^2} \\
P_{CO} &= -16 + \sqrt{361} \\
P_{CO} &= -16 + 19 \\
P_{CO} &= 3
\end{aligned}
 $$

Notice also from the table that all whole number values for $y = f(x)$ are odd numbers. These are $(y_1 \text{ or } P_{CO} = 3)$ if $(x_3 \text{ or } x_k = 16)$, $(y_2 = 5)$ if $(x_{k-1} \text{ or } x_2 = 8)$ and $(y_k \text{ or } y_3 = 7)$ if $(x_1 = 4)$. These values are the odd factors of 105 in the interval $[3,\sqrt{105}]$. Since 3 is the factor of 105 which corresponds to the greatest value of $x$ in the domain such that $(C_O + x_k^2 = 105 + 16^2 = 361)$ is a perfect square, we say that it is also the smallest prime factor of 105. While the formula $P_{CO} = -x_k + \sqrt{C_o + x_k^2}$, $0 \le x_k \le \frac{C_o-9}{6}$ guarantees to provide the smallest prime factor of any odd composite number $C_O$, one might wonder, “Does this formula generate all odd prime numbers?” Let us now take it as a proposition.

**Proposition 4.1.** The formula for finding the smallest prime factor of any odd composite numbers generates all odd primes.

**Proof:** By Fundamental Theorem of Arithmetic, every odd-composite number can be expressed as a product of odd-primes. Theorem 2.1.1 addresses the identification of all odd-composite numbers, whereas Theorem 4.1 focuses on determining the smallest prime factor of any given odd-composite number. If we let $P_n$ be any odd prime number where $n \ge 2$, then $P_n$ is a prime number $\ge 3$ and so all products in the set $\{P_n^2, P_n^3, (P_n)(P_{n+1}), (P_n)(P_{n+1})(P_{n+2}), \dots\}$ are odd composite numbers with $P_n$ as a smallest prime factor. And since every odd-prime is the smallest prime factor of that set of odd-composite numbers, we therefore say the formula $p_{CO} = -x_k + \sqrt{C_o + x_k^2}$, $0 \le x_k \le \frac{C_o-9}{6}$ generates all odd primes. Hence, proposition 4.1 is confirmed.

Another important observation that we may think of is, “Are there infinitely many odd-composite numbers which have any odd-prime as the smallest prime factor?” Again, let us take it as a proposition.

**Proposition 4.2.** If $P_n$ is any odd-prime number where $n \ge 2$, there exists infinitely many odd-composite numbers whose smallest prime factor is $P_n$.

**Proof:** Euclid’s Theorem states that there exist infinitely many prime numbers. It is also a known fact that all of the primes are odd except for 2. If we raise any odd prime $P_n$, where $n \ge 2$, to the power of $m$, where $m \ge 2$, we obtain infinitely many odd-composites as shown in the set $\{P_n^2, P_n^3, P_n^4, \dots, P_n^m\}$. On the other hand, if $P_n$ or its powers of $m \in \mathbb{N}$ is multiplied to any bigger primes or their products raised to the power of $k \in \mathbb{N}$, we obtain odd-composites such
as these numbers shown in the set $\{(P_n \cdot P_{n+1}), (P_n \cdot P_{n+2}), (P_n^2 \cdot P_{n+1}), (P_n^2 \cdot P_{n+2}), (P_n^2 \cdot P_{n+3}), \dots\}$. Both sets of these products consist of infinitely many odd-composite numbers with $P_n$ as the smallest prime factor providing evidence that proposition 4.2 is true.

## 5. Proof of the Goldbach's Conjecture

Goldbach's Conjecture in Number Theory is an assertion that every even counting number greater than or equal to 4 is expressible as the sum of two primes. It was proposed in 1742 by the Russian mathematician Christian Goldbach in a letter to the Swiss mathematician Leonhard Euler. This conjecture is one of oldest and popular unsolved problems in Number Theory. It has been tested and verified to hold true for very large even numbers but despite all that, a proof has yet to be discovered. Now that we already have a general formula for odd primes, we shall utilize it to show whether this conjecture holds for all even counting numbers $\ge 4$. We know for a fact that 4 is the only even counting number expressible as a sum of two even primes, that is, $4 = 2 + 2$. If we set this unique case aside and demonstrate that the rest of the even counting numbers greater than 4 can be expressed as a sum of two odd primes, then the Goldbach Conjecture is proved. The sequence of even numbers $\ge 6$ can be written in the set $\{6, 8, 10, 12, 14, \dots\}$. To find the $g$th term of this arithmetic sequence, we can use the formula,

$$a_g = a_1 + (g - 1)d$$

where,

$a_g$ = $g$th term

$g$ = the number of terms

$a_1$ = first term

$d$ = common difference

Finding for the $g$th term of the sequence we get,

$$a_g = 6 + (g - 1)(2)$$

$$a_g = 6 + 2g - 2$$

$$a_g = 4 + 2g$$

Hence, the $g$th term of the sequence of even numbers $\ge 6$ is $(4 + 2g)$, $g \in \mathbb{N}$.

**Theorem 5.1:** Every even counting number $\ge 6$ is a sum of two odd primes.

**Proof:** Let $P_1$ and $P_2$ be a pair of odd primes. Since both are odd-primes, then we can utilize the odd-prime formula defined in theorem 4.1 to write each as $P_1 = (P_{CO})_1 = -(x_k)_1 + \sqrt{(C_o)_1 + (x_k^2)_1}$ and $P_2 = (P_{CO})_2 = -(x_k)_2 + \sqrt{(C_o)_2 + (x_k^2)_2}$. Notice that each component term in the formula was assigned a subscript for proper distinction between the two odd-primes. After defining every even number $\ge 6$ as $(4 + 2g)$, $g \in \mathbb{N}$, we can now write theorem 5.1 into the following equation,

$$4 + 2g = (P_{CO})_1 + (P_{CO})_2 \qquad (5.1)$$

where,

$4+2g$ is an even counting number $\ge 6$, $g \in \mathbb{N}$.

$(P_{CO})_1$ is the first odd prime addend.

$(P_{CO})_2$ is the second odd prime addend.
Since $4+2g, (P_{CO})_1$, and $(P_{CO})_2$ are all counting numbers, then it follows that $4+2g > (P_{CO})_1$, $(P_{CO})_2$. And because we also know that $(P_{CO})_1$ and $(P_{CO})_2$ are both odd primes, then $4+2g - (P_{CO})_1 \ge 3$ and $4+2g - (P_{CO})_2 \ge 3$. It is based on the fact that 3 is the smallest odd prime number. Theorem 5.1 is logically equivalent with equation (5.1) because they both state that every even number greater than or equal to 6 is a sum of two odd-primes. Proving the latter is true shall also confirm that theorem 5.1 is also true. If we manipulate equation (5.1), then by addition property of equality we have,

$$4 + 2g - (P_{CO})_1 = (P_{CO})_2.$$

If we replace $(P_{CO})_1$ and $(P_{CO})_2$ by their respective values based on the formula of odd-primes, we will then have,

$$4 + 2g - [-(x_k)_1 + \sqrt{(C_o)_1 + (x_k^2)_1}] = -(x_k)_2 + \sqrt{(C_o)_2 + (x_k^2)_2}$$

If we further manipulate the equation until we have $(C_o)_2$ (odd composite number), the only term on the right side, we would be able to show that the left side of the equation is going to be a product of two odd counting numbers. The equation then becomes,

$$\{[4 + 2g + (x_k)_1 - \sqrt{(C_o)_1 + (x_k^2)_1}] + (x_k)_2\}^2 - (x_k^2)_2 = (C_o)_2$$

We see that the left side of the equation assembled to be a difference of two squares which means that it can be written as a product of the sum and difference of two numbers. If we let,

$$A = 4 + 2g + (x_k)_1 - \sqrt{(C_o)_1 + (x_k^2)_1} \quad \text{and} \quad B = (x_k)_2$$

we can then write it as,

$$(A + B)^2 - B^2 = (C_o)_2$$

which implies $(A + B - B)(A + B + B) = (C_o)_2$

or $(A)(A + 2B) = (C_o)_2 \qquad (5.2)$

Since $A = 4 + 2g + (x_k)_1 - \sqrt{(C_o)_1 + (x_k^2)_1}$ is an odd counting number $\ge 3$ as it is a difference between an even number $4+2g$ and an odd-prime $-(x_k)_1 + \sqrt{(C_o)_1 + (x_k^2)_1}$, then $A+2B$ is also an odd number. This is because the sum of an odd number A and an even number 2B is also an odd number. Hence, the product of A and $A+2B$ is an odd-composite number. In equation (5.2), $(A)(A + 2B) = (C_o)_2$ is an odd-composite number, so by theorem 4.1, we know that there exists a greatest whole number $B = (x_k)_2$ such that $(A)(A + 2B) + B^2$ is a perfect square and $-B + \sqrt{(A)(A + 2B) + B^2}$ is the smallest prime factor $(P_{CO})_2$ of $(A)(A + 2B) = (C_o)_2$. Thus we have,

$$(P_{CO})_2 = -B + \sqrt{(A)(A + 2B) + B^2}$$

$$(P_{CO})_2 = -B + \sqrt{A^2 + 2AB + B^2}$$

$$(P_{CO})_2 = -B + (A + B)$$

$$(P_{CO})_2 = A \qquad (5.3)$$

Since $A = 4 + 2g + (x_k)_1 - \sqrt{(C_o)_1 + (x_k^2)_1}$ and $(P_{CO})_1 = -(x_k)_1 + \sqrt{(C_o)_1 + (x_k^2)_1}$, then we can write,
$$4 + 2g - (P_{co})_1 = A \qquad (5.4)$$

Plugging in (5.4) in (5.3) we get,

$$4 + 2g - (P_{co})_1 = (P_{co})_2$$

Therefore we have,

$$4 + 2g = (P_{co})_2 + (P_{co})_1$$

This means that whatever even number we choose, as long as it is greater than or equal to 6, there always exists a pair of odd primes whose sum equals the given even number. Therefore, every positive even integer greater than or equal to 6 can be expressed as the sum of two odd-primes. And with even number 4 shown to be a sum of two primes as well, that is $4 = 2 + 2$, then all positive even numbers greater than or equal to 4 can be expressed as a sum of two primes. That's exactly what the Goldbach Conjecture states. Hence, it is true. Q.E.D.

# 6. The Twin Prime and Polignac's Conjectures

The Twin Prime Conjecture in Number Theory is an assertion that there are infinitely many pairs of consecutive primes that differ only by 2. Examples are 3 and 5, 5 and 7, 11 and 13, 17 and 19 etc. No one really knows how old this conjecture is. Some say it had to go back all the way from the time of the ancient Greeks in 300 BC when Euclid, first proved the existence of infinitely many prime numbers. Back then, it was only logical to ask that if there exist infinitely many primes, then there may also be infinitely many twin primes. It is the reason why this conjecture is sometimes called Euclid's Twin Prime Conjecture.

The first formal statement about the twin prime conjecture was given in 1846 by a French mathematician Alphonse de Polignac when he wrote that any positive even number can be expressed in infinite ways as the difference between two consecutive primes. This became the reason why the twin prime conjecture nowadays is also known as the Polignac's conjecture because it means that proving the Polignac Conjecture would also validate the twin Prime Conjecture.

## 6.1. Proof of the Polignac's Conjecture:

Let $P_n$ and $P_{n+1}$ where $n \in \mathbb{N}$, $n > 1$, be a pair of two consecutive odd-prime numbers. Since both of them are odd-primes, we will use the odd-prime formula defined in theorem 4.1 to denote them as,

$$P_{n+1} = (P_{co})_2 = -(x_k)_2 + \sqrt{(C_o)_2 + (x_k^2)_2} \quad \text{and} \quad P_n = (P_{co})_1 = -(x_k)_1 + \sqrt{(C_o)_1 + (x_k^2)_1}$$

**Remark 6.1.1.** To distinguish between the two consecutive primes, subscripts are used on every component term of the odd-prime formula. A subscript 2 for the component terms of $P_{n+1}$ and a subscript 1 for the component terms of $P_n$.

Proposition 4.2 states that there are infinitely many odd-composite numbers which has any odd-prime $P$ as a smallest prime factor. We can then pick up one of those odd-composites and let,

$$(C_o)_2 = (P_{n+1})(P_{n+2}) \quad \text{and} \quad (C_o)_1 = (P_n)(P_{n+1})$$

Here we see that the smallest prime factors of $(C_o)_2$ and $(C_o)_1$ are $P_{n+1}$ and $P_n$ respectively.
The expression $P_{n+1} - P_n$ which represents a difference between consecutive two odd primes can now be written as,

$$ -(x_k)_2 + \sqrt{(C_O)_2 + (x_k^2)_2} - [-(x_k)_1 + \sqrt{(C_O)_1 + (x_k^2)_1}] \quad (6.1) $$

The Polignac's conjecture predicts that there are infinite ways of expressing every positive even number as a difference between two consecutive odd primes and the first logical way to verify this is to equate expression (6.1) to the general form of positive even numbers. To do this, we'll use the expression $2z, z \in \mathbb{N}$. Thus, we get,

$$ -(x_k)_2 + \sqrt{(C_O)_2 + (x_k^2)_2} - [-(x_k)_1 + \sqrt{(C_O)_1 + (x_k^2)_1}] = 2z \quad (6.2) $$

If we manipulate the equation until we have $(C_O)_2$ the only term on the left side, we then get,

$$ -(x_k)_2 + \sqrt{(C_O)_2 + (x_k^2)_2} = 2z + \sqrt{(C_O)_1 + (x_k^2)_1} - (x_k)_1 \\ (C_O)_2 = [2z + \sqrt{(C_O)_1 + (x_k^2)_1} - (x_k)_1 + (x_k)_2]^2 - (x_k^2)_2 $$

Notice that the expression from the right side is a difference of two squares and if we let,

$$ A = 2z + \sqrt{(C_O)_1 + (x_k^2)_1} - (x_k)_1 \quad \text{and} \quad B = (x_k)_2 $$

we can write the equation into,

$$ \begin{aligned} (C_O)_2 &= (A + B)^2 - B^2 \\ (C_O)_2 &= (A + B - B)(A + B + B) \\ (C_O)_2 &= (A)(A + 2B) \end{aligned} $$

Since $A = 2z + \sqrt{(C_O)_1 + (x_k^2)_1} - (x_k)_1$ is an odd counting number because it is a sum of an even number $2z$ and an odd-prime $[-(x_k)_1 + \sqrt{(C_O)_1 + (x_k^2)_1}]$, then the sum $A+2B$ would also be odd. This is because when an odd number $A$ is added to an even number $2B$, the sum is an odd number. Hence, the product of $A$ and $(A + 2B)$ is an odd-composite number which is exactly what $(C_O)_2$ represents. Since $(C_O)_2 = (A)(A + 2B)$ is an odd-composite number, by theorem 4.1 we know that there exists a greatest whole number $B = (x_k)_2$ such that $(A)(A + 2B) + B^2$ is a perfect square and $-B + \sqrt{(A)(A + 2B) + B^2}$ is the smallest prime factor $(P_{CO})_2$ of $(A)(A + 2B) = (C_O)_2$. Thus we have,

$$ \begin{aligned} (P_{CO})_2 &= -B + \sqrt{(A)(A + 2B) + B^2} \\ (P_{CO})_2 &= -B + \sqrt{A^2 + 2AB + B^2} \\ (P_{CO})_2 &= -B + (A + B) \\ (P_{CO})_2 &= A \end{aligned} \quad (6.3) $$

With $P_{n+1} = (P_{CO})_2$, we can write (6.3) as,

$$ P_{n+1} = A \quad (6.4) $$

And with $A = 2z + \sqrt{(C_O)_1 + (x_k^2)_1} - (x_k)_1$ and $P_n = (P_{CO})_1 = -(x_k)_1 + \sqrt{(C_O)_1 + (x_k^2)_1}$, we can write (6.4) as,
$$P_{n+1} = 2z - P_n$$

Thus we have,

$$P_{n+1} - P_n = 2z$$

This means that every positive even number can be expressed as the difference between two consecutive odd primes. However, it is insufficient to conclusively prove Polignac's conjecture. For the purpose of advancing our argument, we will now assume, for the sake of contradiction, that Polignac's conjecture is false and that there are only finitely many ways to express any positive even number as the difference between two consecutive odd primes.

Let's say $P_n$ and $P_{n+1}$ are the last pair of primes that differ by any positive even number $2z$, $z \in \mathbb{N}$. By Euclid's theorem, we know that there exist infinitely many pairs of consecutive primes. Again, with the use the odd-prime formula in Theorem 4.1, we shall create an expression that is a difference between two consecutive odd primes. To do so, we let $P_{d+n}$, $P_{d+n+1}$, and $P_{d+n+2}$ where $d \in \mathbb{N}$, be three consecutive odd primes that are sufficiently larger than $P_{n+1}$.

By proposition 4.2, there are infinitely many odd-composite numbers which has any odd-prime as a smallest prime factor. And so if we pick up one of those and let,

$$(C_o)_2 = (P_{d+n+2})(P_{d+n+1}) \quad \text{and} \quad (C_o)_1 = (P_{d+n+1})(P_{d+n})$$

Then by the theorem 4.1, the expression below is a difference between two consecutive odd primes.

$$-(x_k)_2 + \sqrt{(C_o)_2 + (x_k^2)_2} - [-(x_k)_1 + \sqrt{(C_o)_1 + (x_k^2)_1}] \quad (6.5)$$

This is because $P_{d+n+1} = -(x_k)_2 + \sqrt{(C_o)_2 + (x_k^2)_2}$ and $P_{d+n} = -(x_k)_1 + \sqrt{(C_o)_1 + (x_k^2)_1}$

**Remark 6.1.2.** To distinguish between the two consecutive primes, subscripts are used on every component term of the odd-prime formula. A subscript 2 for the component terms of the bigger prime of the pair and a subscript 1 for the component terms of the smaller one.

We now equate expression (6.5) to $2z$, $z \in \mathbb{N}$ to see if the equation is true.

$$-(x_k)_2 + \sqrt{(C_o)_2 + (x_k^2)_2} - [-(x_k)_1 + \sqrt{(C_o)_1 + (x_k^2)_1}] = 2z \quad (6.6)$$

By manipulating equation (6.4) until $(C_o)_2$ is the only term on the left side, we can see that the expression from the right side is a difference of two squares.

$$-(x_k)_2 + \sqrt{(C_o)_2 + (x_k^2)_2} = 2z + \sqrt{(C_o)_1 + (x_k^2)_1} - (x_k)_1$$

$$(C_o)_2 = [2z + \sqrt{(C_o)_1 + (x_k^2)_1} - (x_k)_1 + (x_k)_2]^2 - (x_k^2)_2$$

If we let,

$$A = 2z + \sqrt{(C_o)_1 + (x_k^2)_1} - (x_k)_1 \quad \text{and} \quad B = (x_k)_2$$

We will then have,

$$(C_o)_2 = (A + B)^2 - B^2$$

$$(C_o)_2 = (A + B - B)(A + B + B)$$
$$ (C_o)_2 = (A)(A + 2B) \qquad (6.7) $$

With the equation in (6.7), we get,

$$ A = P_{d+n+1}. $$

This is because $(C_o)_2 = (P_{d+n+1})(P_{d+n+2})$ is a product of two consecutive odd-primes and primes are indivisible. And since $A = 2z + \sqrt{(C_o)_1 + (x_k^2)_1} - (x_k)_1$ and $P_{d+n} = -(x_k)_1 + \sqrt{(C_o)_1 + (x_k^2)_1}$, we then have,

$$ P_{d+n+1} = 2z + P_{d+n}. $$

Thus,

$$ P_{d+n+1} - P_{d+n} = 2z. $$

This means that any positive even integer $2z$ will continue to exist infinitely as a difference between two consecutive odd-primes after it appeared as a difference between two smaller consecutive ones. This is a contradiction to what we have earlier assumed that $P_n$ and $P_{n+1}$ are the last pair of consecutive primes that differ by any positive even number because $P_{d+n}$ and $P_{d+n+1}$ are both consecutive odd primes that are sufficiently larger than $P_{n+1}$ and represent a pair of consecutive odd-primes that differ by $2z$. This provides us with evidence to support the validity of Polignac's Conjecture, and by extension, the Twin Prime Conjecture. Thus, both conjectures are true. Q.E.D.

## 7. References

Encyclopædia Britannica. "Twin Prime Conjecture." Accessed January 29, 2025. https://www.britannica.com/science/twin-prime-conjecture.

Goldbach, Christian. *Letter to Leonhard Euler, June 7, 1742*. In *Correspondence of Leonhard Euler*, edited by Franz Lemmermeyer and Martin Mattmüller. Basel: Birkhäuser, 2015.

Weisstein, Eric W. "Goldbach Conjecture." *MathWorld—A Wolfram Web Resource*. Accessed January 29, 2025. https://mathworld.wolfram.com/GoldbachConjecture.html.

Weisstein, Eric W. "Twin Prime Conjecture." *MathWorld—A Wolfram Web Resource*. Accessed January 29, 2025. https://mathworld.wolfram.com/TwinPrimeConjecture.html.