On the Natural Numbers that cannot be Expressed as a Sum of Two Primes

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On the Natural Numbers that cannot be Expressed as a Sum of Two Primes

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Peter Szabó^*

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05 September 2024

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05 September 2024

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Abstract

In this paper, a structural property of prime numbers will be introduced and analyzed. Our analysis is based on the application of the sequence of even integers 2,10,16,22,26,28… which is defined as follows: F(0) = 2, for i > 0, F(i) = F(i − 1)+ l, where l is the smallest even integer that is valid for: F(i − 1)+ 2k − 1 ∈ P, for k = 1, 2, . . . l/2-1, F(i − 1)+ 2k − 1 ∉ P, for k = 2l and P is the set of prime numbers. The sequence F answers the question, which are the natural numbers that cannot be expressed as a sum of two primes. We prove that numbers expressed as j = F(i)+ 1 cannot be written as the sum of two primes, for i = 1, 2, ··· .

Keywords:

Subject: Computer Science and Mathematics - Algebra and Number Theory

1. Introduction

Goldbach’s conjecture is one of the oldest mathematical problems in history. It was discovered by Goldbach in 1742 and remains unsolved up to this day. In 1742, Goldbach and Euler in conversation and in an exchange of letters discussed the representation of numbers as sums of at most three primes. The correspondence led to the formation of the Goldbach conjecture. It reads ‘Every even number is the sum of two primes‘. In fact, there are two conjectures, the binary andthe ternary Goldbach conjecture. These conjectures can be stated as follows:

-: Every even integer greater than two can be written as the sum of two primes (the binary conjecture).
-: Every odd integer greater than five can be written as the sum of three primes (the ternary conjecture).

Over time, there have been many exciting results about conjecture. For a detailed review on this range of problems see the book [1]. Of course, there are many other resources. Interesting stories about primes and unresolved mathematical problems can be found in [2]. Various versions and names of the Goldbach conjecture are also known. The ternary conjecture is often called the Goldbach weak conjecture. In 2013, Harald Helfgott published a proof of Goldbach’s weak conjecture [3]. As of 2018, the proof is widely accepted in the mathematics community, but it has not yet been published in a peer-reviewed journal.

In this work, the discussion will focus on the binary Goldbach conjecture. It is easy to show that the binary conjecture implies the ternary. We examine, what are the numbers which cannot be written as a sum of two primes. We shall show exactly that elements of a uniquely defined sequence

F (i) + 1

, for

i = 1, 2, \dots

(refer to Chapter 4 for definition) cannot be written as a sum of two primes. The question of whether we can write all other integers

j \neq F (i) + 1

greater than three as a sum of two primes remains open.

The paper is organized as follows. Chapter 1 contains a short historical overview about Goldbach’s conjecture and a short description about obtained results. In Chapter 2 basic sets, notations and used technologies are introduced. Prime vectors, matrices and a control matrix function are discussed in Chapter 3. A sequence, the members of which cannot be written as the sum of two prime numbers, is investigated in Chapter 4. Chapter 5 provides a summary of the results achieved.

2. Used Terms and Methods

2.1. Basic Sets and Definitions

The set of all natural numbers (non-negative integers) is denoted by

N = {0, 1, 2, \dots}

. A natural number greater than 1 is prime if its only divisors are 1 and itself. The set of all prime numbers is denoted by

P

. If

n \in N

then

P_{n} = \{p \in P; p \leq n\}

is the set of primes less or equal to n. For any set S the symbol

|S|

will mean the number its elements. The array of numbers consisting of one row and n - elements is called an n-dimensional row vector and denoted by

a = (a_{0}, a_{1}, \dots, a_{n - 1})

a = (a_{j})

. The transpose of the vector

a = (a_{0}, a_{1}, \dots, a_{n - 1})

is an n-dimensional column vector denoted by

a^{T} = {(a_{j})}^{T} = {(a_{0}, a_{1}, \dots, a_{n - 1})}^{T} = (\begin{matrix} a_{0} \\ a_{1} \\ ⋮ \\ a_{n - 1} \end{matrix})

. In this paper the value

a_{j}

- component of the vector

a = (a_{0}, a_{1}, \dots, a_{n - 1})

is a non-negative integer, for

j = 0, 1, \dots, n - 1

2.2. Used Methods and Technologies

The Goldbach conjecture is one of the most intensively studied problem of number theory. There are many interesting results on this problem ([4,5,6,7,8,9,10]). We are not aiming to give a complete account of conjecture related literature here.

The analysis of the Goldbach conjecture is performed using Toeplitz and circulant matrices. We applied a similar analysis in [11]. That paper presents an application of Toeplitz matrices and a max-algebraic claim which is equivalent to Goldbach’s conjecture.

In our work, the Goldbach conjecture is examined by methods of combinatorics and classical linear algebra using unique circulant matrices.

3. Matrices and Primes

In this section prime vectors, prime matrices and a control matrix function will be introduced and their main properties will be discussed. Everywhere in this paper n stands for a positive integer greater than 2. The objects defined in the article can be examined using utility programs [12].

Prime Vectors and Matrices

Definition 1.

The column vector

a^{T} = {(a_{0}, a_{1}, \dots, a_{n - 1})}^{T}

= {(a_{i})}^{T}

is called an n - dimensional extended prime vector of size n if for all

i = 0, 1, \dots, n - 1

a_{i} = \{\begin{matrix} 1, i f i i s p r i m e \\ 0, o t h e r w i s e \end{matrix}

The

n \times n

matrix

A = (a_{i j})

is called circulant if

a_{i j} = b_{i - j (m o d n)}

, for some

b_{0}, b_{1}, . . ., b_{n - 1} \in N

(or

R

in general case). This matrix is fully specified by its first column

b^{T} = {(b_{0}, \dots, b_{n - 1})}^{T}

Definition 2.

Let

a^{T} = {(a_{0}, a_{1}, \dots, a_{n - 1})}^{T}

be n - dimensional extended prime vector. The matrix

A_{n} = (a_{i j})

is called an

n \times n

prime matrix , if

a_{i j} = a_{i - j (m o d n)}

(1)

for all

i, j \in \{0, 1, \dots, n - 1\}

Clearly, the prime matrix is a special circulant matrix with entries from the set

\{0, 1\}

. Notice, the circulant matrices can also be defined by a row vector.

Example 1.

The matrix

A_{6}

is a

6 \times 6

prime matrix, and

a^{T} = {(0, 0, 1, 1, 0, 1)}^{T} = {(a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5})}^{T}

is an extended prime vector of size

n = 6

. For the sake of simplicity, we will also refer to

n \times n

prime matrix as A, if this does not cause misunderstanding.

A_{6} = (a_{i j}) = (\begin{matrix} 0 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 & 1 \\ 1 & 0 & 0 & 1 & 0 & 1 \\ 1 & 1 & 0 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1 & 0 & 0 \end{matrix}) = (\begin{matrix} a_{0} & a_{5} & a_{4} & a_{3} & a_{2} & a_{1} \\ a_{1} & a_{0} & a_{5} & a_{4} & a_{3} & a_{2} \\ a_{2} & a_{1} & a_{0} & a_{5} & a_{4} & a_{3} \\ a_{3} & a_{2} & a_{1} & a_{0} & a_{5} & a_{4} \\ a_{4} & a_{3} & a_{2} & a_{1} & a_{0} & a_{5} \\ a_{5} & a_{4} & a_{3} & a_{2} & a_{1} & a_{0} \end{matrix})

Let

A_{n} = (a_{i j})

be an

n \times n

prime matrix. The

i - t h

row of matrix

A_{n}

will be denoted

A_{n / i \cdot}

and the

j - t h

column

A_{n / \cdot j}

for all

i, j = 0, \dots, n - 1

. The vector

A_{n / \cdot n - 1}

is called the prime vector of size n. The

0 - t h

row vector

A_{n / 0 \cdot}

of matrix

A_{n}

is called the reverse prime vector of size n.

Definition 3.

The matrix

T_{n}

T_{n} [i, j]) = A_{n}^{2}

is called a control matrix of order n.

The entry

T_{n} [i, j] = A_{n / i \cdot} A_{n / \cdot j} = \sum_{l = 0}^{n - 1} a_{i l} a_{l j}

is a (dot) product of

i - t h

row and

j - t h

column of prime matrix

A_{n}

, for all

i, j = 0, \dots, n - 1

. The entry

T_{n} [0, 0]

is called the main element of

T_{n}

The set of all control matrices of all orders

n > 2

is denoted by

T

. It is also important to note that the control matrix

T_{n}

as a product of circulant matrices

A_{n} \times A_{n}

is also circulant, see [13].

Lemma 1.

Let

T_{n} \in T .

The value

T_{n} [0, 0] = a_{0} a_{0} + \sum_{i = 1}^{n - 1} a_{i} a_{n - i}

(2)

is the number of all prime pairs

p, q \in P_{n}

such that

p + q = n

Proof.

Suppose that

p, q \in P_{n}

is an arbitrary pair of primes and

p + q = n

. Let

T_{n} [0, 0] = A_{n / 0 \cdot} A_{n / \cdot 0}

and

A_{n / \cdot 0} = a^{T} = {(a_{0}, a_{1}, \dots, a_{n - 1})}^{T}

is the extended prime vector,

A_{n / 0 \cdot} = (a_{0}, a_{n - 1}, a_{n - 2}, \dots, a_{1})

is the reverse prime vector . Clearly

q = n - p \in P_{n}

, therefore

a_{p} a_{q} = a_{p} a_{n - p} = 1

. If

i \notin P_{n}

n - i \notin P_{n}

then

a_{i} = 0

a_{n - i} = 0,

thus

a_{i} a_{n - i} = 0

. □

We will examine when the equation

T_{n} [0, 0] = 0

holds.

Theorem 1.

An arbitrary natural number n can not be written as a sum of two primes if only if

T_{n} [0, 0] = 0

is fulfilled.

Proof.

T_{n} [0, 0] = 0

then there is no pair of primes

p, q \in P_{n}

for which

p + q = n

. If n can not be written as a sum of two primes then for all

p, q \in P_{n}

p + q \neq n

, therefore

T_{n} [0, 0] = 0

. □

Theorem 2.

T_{n}, T_{n + 1} \in T

then

T_{n + 1} [0, 0] = 0

if only if

T_{n} [0, n - 1] = 0

Proof.

⇐ Suppose that

T_{n} [0, n - 1] = 0

. Therefore,

T_{n} [0, n - 1] = A_{n / 0 \cdot} A_{n / \cdot n - 1} = (a_{0}, a_{n - 1}, a_{n - 2}, \dots, a_{1}) (\begin{matrix} a_{1} \\ a_{2} \\ ⋮ \\ a_{n - 1} \\ a_{0} \end{matrix}) = 2 a_{0} a_{1} + \sum_{i = 1}^{n - 1} a_{n - i} a_{i + 1} = 0

(3)

It follows that

\sum_{i = 1}^{n - 1} a_{n - i} a_{i + 1} = 0

. We know that

a_{0} = a_{1} = 0

, thus,

T_{n + 1} [0, 0] = A_{n + 1 / 0 \cdot} A_{n + 1 / \cdot 0} = (a_{0}, a_{n}, a_{n - 1}, \dots, a_{1}) (\begin{matrix} a_{0} \\ a_{1} \\ ⋮ \\ a_{n - 1} \\ a_{n} \end{matrix}) = a_{0} a_{0} + a_{n} a_{1} + \sum_{i = 1}^{n - 1} a_{n - i} a_{i + 1} = 0

(4)

⇒ Equation 4 implies 3. □

4. Not Two Primes Sum Partition Sequence

4.1. Definition of an Auxiliary Sequence

Definition 4.

Let us define a sequence:

F (0) = 2, f o r i > 0, F (i) = F (i - 1) + l

(5)

where l is the smallest even integer with the properties:

\begin{matrix} F (i - 1) + 2 k - 1 \in P, & f o r & k = 1, 2, \dots, \frac{l}{2} - 1 \\ F (i - 1) + 2 k - 1 \notin P, & f o r & k = \frac{l}{2} \end{matrix}

{\{F (i)\}}_{i = 0}^{\infty}

will be called the auxiliary sequence. The value

l = F (i) - F (i - 1)

is termed the distance between adjacent elements

F (i)

and

F (i - 1)

It is not difficult to compute some first elements of sequence

: 2, 10, 16, 22, 26, 28, \dots

F (i)

is the

i - t h

element of the sequence, for

i = 0, 1, 2, \dots

Note that in the rest of the paper, notation

n \in F

will mean, that there is an index

i \in N

, such that

F (i) = n

and

n \notin F

will mean, that there is no such index

i \in N

. We denote

F = \{F (i) : i \in N\}

4.2. Basic Properties of the Auxiliary Sequence

Lemma 2.

All elements of the sequence

{\{F (i)\}}_{i = 0}^{\infty}

are even numbers.

Proof.

The statement yields from Definition 5. The first item

F (0) = 2

is even and, the distance

l = F (i) - F (i - 1)

is even for each i, therefore

F (i)

is even for each i. □

Lemma 3.

The sequence

{\{F (i)\}}_{i = 0}^{\infty}

is unbounded.

Proof.

Let

F (i)

be an arbitrary element of sequence F. From Lemma 2 follows that

F (i) = 2 k

is an even number. Let

2 k = 2^{j} n

, where

j \geq 1

and n is the part of

2 k

prime factor, which contains primes greater than 2 or

n = 1

. Therefore, n is an odd number. Now we consider the number

3^{j} n

. This is not a prime, odd number greater than

2 k .

Let

l - 1

be the smallest, non prime, odd number greater than

F (i) = 2 k

. Then l is the smallest integer from Definition 5, therefore

F (i + 1) = F (i) + l

, i.e. F is unbounded. The value

F (i + 1)

is the next member of sequence F, which is greater than

F (i)

. □

Lemma 4.

The distance

l = F (i + 1) - F (i)

is less than or equal to six, for

i > 1

Proof.

k > 2

is even and

k + 1

and

k + 3

are primes then

k + 1

cannot be

0 (m o d 3)

and also cannot be

1 (m o d 3)

(because then

k + 3

would be

0 (m o d 3)

). So

k + 1

2 (m o d 3)

and hence

k + 5

0 (m o d 3)

. So if k is in the sequence and neither

k + 2

, nor

k + 4

are then

k + 6

is in the sequence. □

Lemma 5.

l_{0} = F (0),

and

l_{j} = F (j) - F (j - 1)

for all

j = 1, 2, \dots, i

then

F (i) = \sum_{j = 0}^{i} l_{j}

Proof.

We shall prove it by induction on k. If

k = 0

then

F (0) = l_{0} .

For

k = i - 1

suppose, that

F (i - 1) = \sum_{j = 0}^{i - 1} l_{j}

and

l_{j} = F (j) - F (j - 1)

for

j = 1, 2, \dots, i - 1

. Now we consider the case

k = i .

Let denote

l_{i} = F (i) - F (i - 1)

, then

F (i) = F (i - 1) + l_{i} = \sum_{j = 0}^{i - 1} l_{j} + l_{i} = \sum_{j = 0}^{i} l_{j}

and

l_{j} = F (j) - F (j - 1)

for

j = 1, 2, \dots, i .

□

4.3. Not Two Primes Sum Partition Sequence

Based on the Theorem 2 the sequence

{\{F (i) + 1\}}_{i = 1}^{\infty}

(6)

describes those numbers that cannot be written as the sum of two prime numbers or shortly we can call the squence as the not two primes sum partition sequence.

We can define a set

M = \{j \in N; (\forall i \in N) j \neq F (i) + 1\}

(7)

by the sequence (6).

If it is proved that this set is the set of those numbers which can be written as the sum of two primes then this statement implies the binary Goldbach conjecture. In this case, the set M (it is also a sequence) can be called as the two primes sum partition sequence.

5. Conclusion

In summary, these results show that the sequence

{\{F (i) + 1\}}_{i = 1}^{\infty}

determines the set of natural numbers, which cannot be written as a sum of two primes.

The proof of the statement that

M = \{j \in N; (\forall i \in N) j \neq F (i) + 1\}

is the set of numbers that can be written as the sum of two primes, would mean solving the binary Goldbach conjecture. However, this remains an open question. This work is at the same time an introduction, a description of the basic concepts, for proving the statement formulated above.

Funding

This research received no external funding.

Data Availability Statement

Data sharing is not applicable.

Acknowledgments

This article has been greatly improved by the comments and advice of various people. I should like to thank Professor Peter Butkovič for invaluable assistance and providing language help. I am incredibly grateful to Professor Ján Plavka for constructive discussions and proof reading the article.

Conflicts of Interest

The author declare no conflicts of interest.

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On the Natural Numbers that cannot be Expressed as a Sum of Two Primes

Abstract

1. Introduction

2. Used Terms and Methods

2.1. Basic Sets and Definitions

2.2. Used Methods and Technologies

3. Matrices and Primes

Prime Vectors and Matrices

4. Not Two Primes Sum Partition Sequence

4.1. Definition of an Auxiliary Sequence

4.2. Basic Properties of the Auxiliary Sequence

4.3. Not Two Primes Sum Partition Sequence

5. Conclusion

Funding

Data Availability Statement

Acknowledgments

Conflicts of Interest

References

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