Analyzing Goldbach Conjecture

1. Introduction

The origin of GC go back to more than two centuries ago: “Goldbach’s original conjecture (sometimes called the ”ternary” Goldbach conjecture), written in a June 7, 1742 letter to Euler, states that every integer is the sum of three primes (Dickson 1957, p. 421).” (extracted from [6]).

There are variants of GC, for instance the ternary Goldbach conjecture is also called the weak Goldbach conjecture and stated as:

Ternary Goldbach Conjecture: Every odd number greater than 5 can be written as the sum of three prime numbers [4].

According to Weisstein [6], the weak version of GC was re-expressed by Euler in the so called “strong” version of GC, as follows:

Goldbach conjecture (GC): asserts that all positive even integers ≥4 can be stated as the sum of two prime numbers.

It is this strong version of GC that proof of is detailed here.

There are some approaches to proof GC ([1,3,5]) based on algebra or Diophantine equations, etc.

The aim of this paper is to offer an account of Goldbach Conjecture (GC), based on the asymmetry of occurrences of prime and composite numbers under any given even integer N≥4. The account of GC presented here is rather simple and is based on analysis of both behavior of sums of integers and the asymmetric occurrence of prime and composite numbers under any even integer N≥4, where this asymmetry ensures the existence of couples of prime numbers that account for GC.

2. Materials and Methods

The main method consists in analyzing the behavior of additions of integers under an even integer N≥4. Then based on observations drawn from this analysis such that positional symmetry between the addends of each sum, see whether is possible to find pairs of prime numbers as addends such that the GC holds.

2.1. Possible Sums Under N

Given an even integer N≥4, according to GC, N = p_i + p_j for some prime numbers p_i and p_j. An addition of integer numbers is involved (subtraction is not considered here), thus to start with, it seems to be convenient to analyze the behavior of all possible sums of two positive integers (k_i, k_j) both less than N such that N = k_i + k_j and determinate whether or not k_i and k_j can be prime numbers.

For instance, consider N= 10, the possible sums under 10 can be obtained by writing together two columns of consecutive integers (c₁, c₂), one in ascending order from 1 to N-1 (9) and the other in descendent order from N-1 to 1 (see Table 1). Thus, any the two integer numbers (k_i, k_j) where i, j ∈{1, … N-1}, in any row in this matrix (N-1 x 2) built by c₁ and c₂, add N. That is, these two columns under any N provide all possible additions of two positive integers (< N) whose result is N.

Following this idea, Figure 1 includes some matrices (from 4 to 30) headed by even integers N and two columns (c₁, c₂) each one, as described above, but this time prime numbers are highlighted (in green and blue respectively) in both columns for each N and a red line indicates the middle location (N/2) of all matrices. See for instance the matrix in Table 1.

2.2. Observations and Definitions

To carry on with this analysis some definitions and observations are drawn by breaking down the parts and the behavior of the sums under N (see Figure 1).

Definition 2.1 (Matrix).

A matrix (A) is a couple of columns (c₁, c₂), such that A = (c₁, c₂) for some even integer N≥4. Column c₁ is a sequence of consecutive integers in ascending order from 1 to N-1 and c₂ is a sequence of consecutive integers in descendent order from N-1 to 1, where k_i + k_j = N being k_i and k_j the two integers in any row of the matrix (Table 1 shows a Matrix for N = 10).

Figure 1 show matrices from N = 4 to N = 30, and a red line indicating the middle location(N/2) of each matrix for N.

Definition 2.2 (Symmetry in a Matrix).

Matrix A for some even integer N is symmetrical in the sense that all of its couples (k_i, k_j) (rows in the Matrix) add N (N = k_i + k_j) and all k_i and k_j are located symmetrically in A, that is to say, k_i and k_j are equidistantly located from location N/2 in the middle of the matrix (Figure 2 illustrates the symmetry of a Matrix for N=24).

For instance, in Figure 2, for matrix N = 24, the pairs (1, 23), (2, 22), (3, 21), …, (11, 13), (12, 12) are examples of couples of values located in symmetric places with respect to N/2 = 24/2 = 12.

2.3. Symmetry as a Function

A Matrix A is generated by an addition function that relates couples of values located in symmetric places. Given and matrix A = (c₁, c₂) defined by an even integer N≥4, A in turn defines a function f: {c₁}→{c₂} as follows:

f(x) = N − x

(1)

Relevant here is that f(x) not only provides the difference between N and any x in column c₁ (or sums N = x + f(x)), but also relates couples of values (x, f(x)) that are located symmetrically with respect to N/2.

It can be observed that this is the case of any row of Matrix A, each row is composed by a couple of values (k₁, k₂), where they not only add N but also are located in symmetric places with respect to N/2. Some examples were provided in Figure 2.

Note that in Figure 1 and Figure 2, there are cases where x and f(x) are prime numbers located symmetrically with respect to N/2 and that add N, that is, in those cases GC holds. In Figure 2, the pairs of primes (5, 19), (7, 17), (11, 13) are examples. From now on all couples of primes that add N in some array A, will be called Goldbach pairs or Goldbach primes (GP).

It is worth mentioning that since {c₁} = {c₂}, then f: {c₁}→{c₂} is equivalent to f: {c₁}→{c₂}.

The account of GC demands distinguishing between two sets.

Definition 3.1 (Sets of composed and prime numbers under N).

Given any even integer N≥4, the following sets can be defined:

• Let P be the set of prime numbers less than N.

P = {p_k / p_k is a prime number, 0 < p_k < N}

b.

Let C be the set of composed odd numbers less than N.

C = {c_k / c_k is a composed odd number, 0 < c_k <N}

The set C considers odd composite numbers because excepting number 2, all components of symmetrical couples (p_i, d_i) of a Matrix A, where a prime number (p_i) is considered, the other number (d_i) is an odd number.

Definition 3.2 (Symmetry of a set in a Matrix A).

For some property X of some subset S of couples (rows) of integers (a_i, b_i) taken from a Matrix A for some even N integer greater than 4, if the property X holds for both integers in all couples (a_i, b_i) in S, then S is a symmetrical set with respect to property X.

It would be useful to know whether these two sets (P and C) are symmetrical or asymmetrical. Symmetry of a set in a Matrix for some N even, refers to symmetrical occurrences in both cases. That is, for every prime number p_i under an even integer N, is there another prime number p_j such that f(p_i) = p_j? or for every composed number c_i under an even integer N, is there another composed number c_j such that f(c_i) = c_j?

Now, the interest is to find out whether the occurrence of prime and composite numbers under some even N are symmetrical by introducing two lemmas, then a proof of Goldbach Conjecture (GC) can be outlined. According to Definition 3.2 here X would be the property of being prime and in the other case X is the property of being composite.

3. Results

3.1. Proof of GC

Based on Lemma 4.1 and Lemma 4.2, proof of GC is outlined below.

Proof. 

In order to offer an account for GC, it is necessary to take into account two sets: the set of prime numbers (P) under some even integer N≥4 and the set (Δ) of their differences with respect to N, namely between each prime in P and N, defined in terms of f(x) as in equation (1) as follows:

Δ = { d_i / d_i = f(p_i) = N– p_i, where p_i is a prime number such that p_i <N}

Since Δ is the set of values generated by f(p_i), where p_i is a prime number, then any pair (p_i, d_i) includes values located symmetrically. Furthermore N = p_i + d_i (4)

Figure 3. Function f: P→ Δ, where P = {p_k / p_k is a prime number, 0 < p_k < N} and Δ = {d_i / d_i = f(p_i) = N– p_i}.

Figure 3. Function f: P→ Δ, where P = {p_k / p_k is a prime number, 0 < p_k < N} and Δ = {d_i / d_i = f(p_i) = N– p_i}.

Since the set P includes only prime numbers under some even N, now, the main interest is to see whether there exists at least one prime number (d_i) in Δ mapped by f(p_i) for any even N≥4, since this would proof GC.

But now this is clear, given that P is asymmetrical (by Lemma 1), it means that any d_i in Δ (f(p_i) = d_i), both not necessarily is a prime number and not necessarily composed number.

That is, Δ is not a set of only prime numbers nor a set of only composite numbers. This means that, {Δ ∩ C}≠ ϕ and {Δ ∩ P}≠ϕ, where ϕ denotes the empty set.

Thus, for any even integer N≥4, Δ must integrate both composed and prime numbers, that is, Δ ⊂ {P ∪ C} for any N. Then, there must exists some prime number in Δ, that is, f(x) must generate at least a couple of prime numbers (p_i, p_j=f(p_i)) such that p_i = N - p_j and p_j = N - p_i and then N = p_i + p_j. Therefore, GC holds for any even integer N≥4. This proves Goldbach Conjecture. ■

Observe that, in case that some d_i is a composed number, of Δ, there must exists some p_i in P, such that f(d_i) = p_i, as pointed out by Lemma 2.

This proof is not touched by the gap theorem for prime numbers, which claims that there are arbitrary sequences of composite numbers (m! + 2, m! + 3, m! + 4, …, m!+ m). An arbitrary sequence of m-2 composites begins in the integer m!+2. The proof of GC above is based on matrices of integers under any even N≥4. Thus, when N = m! + 2 then N>(m-2) (the length of the sequence of composite numbers), and also N/2>(m-2), in this way, any gap of (m-2) composite numbers would be absorbed by a Matrix for N= m! + 2.

4. Conclusions

A simple proof of GC is outlined above. It is based on visualization of behavior of sums of primes and a definition of symmetry (Definition 2.2), which is described by the function f: {c₁}→{c₂} (f(x) = N-x), that defines all sums of N and all differences among x and N. Then Δ is introduced to focus the attention on both the values of f(x) where x takes values of prime numbers, and the differences described by N-x. Therefore, asymmetry of composed and prime numbers ensures that the set of differences Δ includes both prime and composite numbers (always for any even integer N greater or equal than 4) and therefore GC holds.

As stated, and shown by Lamb, once GC is proven, you get the ternary or weak Goldbach Conjecture for free [4].