Proof of the Binary Goldbach Conjecture

1. Overview

Number theory, "the queen of mathematics" studies the structures and properties defined on integers and primes (Euclid [15], Hadamard [18], Hardy, Wright [20], Landau [26], Tchebychev [44]). Many problems and conjectures have been formulated simply, but they remain very difficult to prove. These main components include:

● Elementary arithmetic. 

–

Operations on integers, determination and properties of primes. (Basic operations, congruence, gcd, lcm, ………..).

   –Decomposition of integers into products or sums of primes.

    (Fundamental theorem of arithmetic, decomposition of large integers, cryptography and Goldbach's conjecture, Filhoa, Jaimea, de Oliveira Gouveaa, Keller-Füchter, [16]).

● Analytical number theory. 

–

Distribution of primes: Prime Number Theorem, the Riemann hypothesis, (Hadamard [18], De la Vallée-Poussin [45], Littlewood [29] and Erdos [14], ,.....).

–

Gaps between consecutive primes (Bombieri,Davenport [3], Cramer [9], Baker,Harmann,Iwaniec, Pintz [4,5,24], Granville [17], Maynard [31], Tao [43],

   Shanks [40], Tchebychev [44] and Zhang [50]).

● Algebraic, probabilistic, combinatorial and algorithmic number theories. 

–

Modular arithmetic.

    –

Diophantine approximations and equations.

    –

Arithmetic and algebraic functions.

    –

Diophantine and number geometry.

    –

Computational number theory.

2. Definitions Notations and Background

The integers h, m, M, n, N, k, K ,p, q, Q, r,..…used in this article are always positive. (2.1)

The symbol " │ " means: such as or knowing that.. (2.2)

Let $\mathcal{P}$be the infinite set of positive primes $p_k$(called simply primes) (2.3)

($p_1$ = 2; $p_2$ = 3; $p_3$= 5; $p_{4 }$= 7; $p_5$ = 11; $p_6$ = 13;. ........)

For any non-zero integer K ${\mathcal{P}}_K$ = { p ∈ $\mathcal{P}:$p ≤ 2K } (2.4)

Writing the large numbers calculated in Appendix 15 is simplified by defining the following constants:

$$M= {10}^9;R= 4.{10}^8;G={10}^{100};S={10}^{500};T={10}^{1000} (2.5)$$

$$p_k\#={\prod }_1^k{p}_j\mathrm{is} \mathrm{the} \mathrm{primoriale} \mathrm{of}{p}_k. (2.6)$$

ln(x) denotes the neperian logarithm of the strictly positive real number x. (2.7)

exp(x) denotes the exponential of the real number x. (2.8)

Lambert's function is defined as the solution to the complex-valued functional equation in " w(z)":

z = w exp(w) (2.9)

where z is a given complex number and w is the unknown complex-valued function.

Since functional solutions of (2.9) are not injective, the Lambert’s function is multivalued (multibranch).

Main branch: LambertW(0, x) is the inverse function of f defined on [-1; +$\mathrm{\infty }$[ by:

f(x) = x.exp(x) (2.10)

Secondary branch: LambertW(-1, x). This branch is defined for values of  x less than ${\displaystyle \frac{-1}{\mathrm{e}}}$ It corresponds to values of  x that are generally negative and is used where the main branch does not apply.

Remark. Analytical extensions are defined by entire series. 

Let (${ W}_{2n})$ be the sequence of primes defined by

$$\forall n\in \mathbb{N}+3 W_{2n}=\mathrm{Sup} (p\in \mathcal{P}:p\le 2n-3) (2.11)$$

For any odd prime q, let (${ Wq}_{2n})$ be the sequence of primes defined by

$$\forall n\in \mathbb{N} n\ge {\displaystyle \frac{(q+3)}{2}} {Wq}_{2n}=\mathrm{Sup} (p\in \mathcal{P}:p\le 2n-q) (2.12)$$

Any sequence denoted by (${ G}_{2n})$ = (${ U}_{2n };V_{2n }$) verifying (2.11) is called a Goldbach sequence.

$$\forall n\in \mathbb{N}+2 { U}_{2n},V_{2n}\in \mathcal{P}\mathrm{and} U_{2n}+V_{2n}=2n (2.13)$$

${ U}_{2n}$ and $V_{2n}$ are also known as "Goldbach partitions, pairs or decomponents".

Iwaniec,Pintz [24] have shown that for a sufficiently large integer n there is always a prime between n − $n^{23/42 }$ and n. Baker and Harman [4,5] concluded that there is a prime in the interval [ n; n + o ($n^{0.525}\left)\right].$Thus this results provides an increase of the gap between two consecutive primes $p_k$and $p_{k+1 }$of the form

$$\forall \epsilon \text{>}0 \exists k_{\epsilon }\in {\mathbb{ }\mathbb{N}}^{*}│ \forall k\in \mathbb{N} k \ge k_{\epsilon } p_{k+1}-p_k< \epsilon .p_k^{0.525}(2.14)$$

The results obtained on the Cramer-Granville-Maier-Nicely conjecture [1,3,9,17,30,32] imply the following majorization.

For any real c > 2 and for any integer k ≥ 500

$p_{k+1}-p_k\le 0.7 {\mathrm{l}\mathrm{n}}^{\mathrm{c}}(p_k$ (with probability one) (2.15)

and

$p_{k+1}-p_k\le 20.\ln (p_k$) (on average) (2.16)

The following abbreviations have been adopted:

● Lagrange-Lemoine-Levy conjecture (3L) conjecture (2.17)

● Bachet-Bézout-Goldbach conjecture (BBG) conjecture (2.18)

● (Extreme) Goldbach decomponents (E).G.D. (2.19)

3. Introduction

Chen [7], Hardy, Littlewood [21], Hegfollt, Platt [22], Ramaré, Saouter [35], Tao [43],

Tchebychev [44] and Vinogradov [47] have taken important steps and obtained promising results on the Goldbach conjecture (any integer n ≥ 2 is the mean arithmetic of two primes). Indeed, Helfgott, Platt [22] proved the ternary Goldbach conjecture in 2013.

Silva, Herzog, Pardi [41] held the record for calculating the terms of Goldbach sequences after determining pairs of primes ($U_{2n}; V_{2n})$ verifying

$$\forall n \in \mathbb{N} │ 4 \le 2n \le {4.10}^{18} U_{2n}+V_{2n}=2n (3.1)$$

Goldbach's conjecture has also been verified for all even integers 2n satisfying

$${10}^{5k}\le 2n\le {10}^{5k}+{10}^8: k = 3, 4, 5, 6, ........,20$$

and

$${10}^{10k}\le 2n\le {10}^{10k}+{10}^9: k = 20, 21, 22, 23, 24, ........,30$$

by Deshouillers, te Riele, Saouter [11].

In previous research work there is no explicit construction of recurrent Goldbach sequences.

In this article, for any integer n greater than two the E.G.D. $U_{2n}$ and $V_{2n}$ are computed iteratively using a simple and efficient "localised" algorithm.

Using Maxima and Maple scientific computing software on a personal computer Silva's record is broken and many E.G.D. are calculated up to the neighbourhood of

$2n={10}^{500}, {10}^{1000}, {{10}^{50}}^{00} \mathrm{and} \mathrm{G}.\mathrm{D}. \mathrm{around} {10}^{10000}$ (see Sainty [37]

"In Researchgate, Internet Archive, and OEIS, E.G.D. files are supplied: E.G.D. File S around 2n = ${10}^S$ for S = 1, 2, 3,............., 10000").

The binary Goldbach conjecture can be proved globally by strong recurrence on all G.D. using (${Wq}_{2n}$) sequences of primes in the same way via Goldbach(-) conjecture (any even integer greater than one is the difference of two primes) demonstrated in Teorem 4.

Remark.

1. Chen conjecture: For any integer K ≥ 1 there are infinitely many pairs of primes with a difference equal to 2K.

2. De Polignac conjecture: Same as Chen, but with consecutive pairs of primes.

3. What we know: April 2013, Yitang Zhang [50] demonstrates that the smallest even integer 2K verifying the conjecture is greater than 70 million.

In 2014 James Maynard [31] then Terence Tao [43] lowered this limit to 246.

We validate Chen’s weak conjecture by verifying directly in the primes tables that all even gaps from 2 to 246 are possible (see Appendix 16).

In addition, the (3L) conjectures [10,23,25,28,48] and its generalization called

(BBG) conjecture are validated.

Using case disjunction reasoning we construct two recurrent E.G.D. sequences of primes (${ V}_{2n }$) and (${ U}_{2n }$) according to the sequence ($W_{2n})$ by the following process

Firstly,

$U_4=2 \mathrm{and} V_{4 }$ = 2 (3.2)

For any integer n greater than two

●Either

(2n - $W_{2n }$) is a prime

then $V_{2n}$ and $U_{2n}$ are defined directly in terms of $W_{2n}$.

● Either

(2n - $W_{2n})$ is a composite number

then $V_{2n}$ and $U_{2n}$are determined from the previous terms of the sequence (${ G}_{2n})$.

(This process can be reversed by first determining the increasing sequence of primes less than Inf (2n -$W_{2k}$ ∈ $\mathcal{P}$: k ∈ ℕ), which saves a lot of computing time when programming).

4. Theorem (Chen’s Weak or Goldbach(-) Conjecture)

$\forall K\in \mathbb{N}* \exists p q\in \mathcal{P}$ │ p - q = 2K (4.1)

If K ≥ 2 3 ≤ q ≤ 2K and 3 + 2K ≤ p ≤ 4K

Practical method on some examples:

First of all (5 - 3 = 2), then we begin the process at (7 - 3 = 4); we will select the smallest primes for which the difference is precisely 6 (11 - 5 = 6), then 8 (11 - 3 = 8), then 10

(13 - 3 = 10),........., then 2K (demonstration established by strong recurrence, by the asurd and feedback). All pairs of Goldbach(-) partitions obtained by this method for K between 2 and 123 are listed in Appendix 16 to validate it using Tao results.

Proof. An other proof can also be established by strong recurrence on the integer K ≥ 2. Let ${\mathcal{P}}_{Chen}$(K) be the following property

" ∀ K ∈ ℕ* $\exists$ p, q ∈ $\mathcal{P}$│ p - q = 2K 3 ≤ q ≤ 2K and 2K + 3 ≤ p ≤ 4K " (4.2)

►${\mathcal{P}}_{Chen}$(2) is true: 7 - 3 = 4 q = 3 ≤ 4 and p = 7 ≤ 4 x 2 = 8

► Let’s show

$$\forall M\in \mathbb{N} \mathbf{│} 2 \le M \le \mathrm{then} {\mathcal{P}}_{Chen}(M ⟹{\mathcal{P}}_{Chen}(K+1)$$

We reason through the absurd

Let p, q ∈ ${\mathcal{P}}_K$ │ p ≥ q

 $\forall P \in \mathcal{P}│ P \ge Q\exists$ h, m ∈ ℕ │

P = p + 2h and Q = q + 2m

we assume that

P - Q = p + 2h - q - 2m ≠ 2(K + 1) (4.3)

Therefore

p - q ≠ 2(K + 1 - h + m) (4.4)

You can always choose h ≥ m and h - m ≤ K + 1.

The set {2(K + 1 - h + m) > 0; 2h and 2m are any gaps between primes} contains all even integers between 2 and 2K (according to the recurrence hypothesis on ${\mathcal{P}}_{Chen}$(K)).

However the strong recurrence hypothesis asserts that

$\forall M\in \mathbb{N} │ M \le K\exists p, q \in \mathcal{P}$ │ p - q = 2M (4.5)

By choosing: M = K + 1 - h + m

this contradicts (4.4).

So

$\exists$h, m ∈ ℕ │ P - Q = p + 2h - q - 2m = 2(K + 1) (4.6)

knowing

p, p + 2h, q, q + 2m ∈ $\mathcal{P}$ h ≥ m and h - m ≤ K + 1

Thus validating the heredity of property ${\mathcal{P}}_{Chen}$(K).

The property ${\mathcal{P}}_{Chen}$(K) is therefore true. As a result Goldbach(-) conjecture is validated.

5. Corollary

Let${(R}_{2K})$ and ${(Q}_{2K})$ be two sequences of primes determined by

$${ R}_{2K}=\mathrm{Inf} (p\in \mathcal{P}: p-2K\in \mathcal{P}) and \underset{¯}{Q_{2K}}=\mathrm{Inf} (p\in \mathcal{P}: 2K+ \in \mathcal{P})=R_{2K}-2K (5.1)$$

       They are defined for any integer K ∈ N* (5.2)

and satisfy

lim $R_{2K}$ = +$\mathrm{\infty }$ (5.3)

${ \forall K \in {\mathbb{N}}^{*} R}_{2K}$, $\underset{¯}{Q_{2K}}$ ∈ $\mathcal{P}$and $R_{2K}$ - $Q_{2K}$ = 2K (5.4)

$\forall K \in {\mathbb{N}}^{*}$│ 2 ≤ K ≤ 16 3 ≤ $\underset{¯}{Q_{2K}}$ ≤ 2K and 2K + 3 ≤ $R_{2K}$ ≤ 4K (5.5)

For any integer K large enough

$$3 \le \underset{¯}{Q_{2K}}\le {\left(2K\right)}^{0.525}\mathrm{and} 2K+3 \le R_{2K}\le 2K+{\left(2K\right)}^{0.525} \left(5.6\right)$$

Proof.

(5.1); (5.2): According to the previous Theorem 4, the sequences ($R_{2K }$) and $\underset{¯}{{(Q}_{2K)}}$) are defined by strong recurrence (finite descent).

(5.3): $R_{2K}$ ≥ 2K $⟹$ lim $R_{2K}$ = +$\mathrm{\infty }$

(5.4): By construction, these sequences thus verify: $R_{2K}$ - $Q_{2K}$ = 2K

(5.5): The property can be verified directly term-to-term by examining the sequence proposed above.

(5.6): This property is verified up to 2K = 246 by calculations on the previous list.

We prove this result by recurrence

First of all, we order the Goldbach(-) decomponents at a fixed prime q, so as to obtain the estimate (5.6) more easily.

Let $q_r$ be the (r + 1)th prime:

We examine the sequences of primes ($T_r$(K)${)}_{K\in \mathbb{N}}$ satisfying:

$T_1$(K) = 2K + 3

($T_1$(K); 2K) → (5;2); (7 4); (11;8); (13;10); (17;14); (19;16); (23;20); (29;26); (29;28);..

 $T_2$(K) = 2K + 5

($T_2$(K); 2K) → (7;2); (11;6); (13;8); (17;12); (19;14); (23;18); (29;24); (31;26); (37;32).

$T_3$(K) = 2K + 7

($T_3$(K); 2K) → (11;4); (13;6); (17;10); (19;12); (23;16); (29;22); (31;24); (37;30)...........

$T_4$(K) = 2K + 11

(T11(K); 2K) → (13;2); (17;6); (19;8); (23;12); (29;18); (31;20); (37;26); (41;30); (43;34).

(T13(K); 2K) → (17;4); (19;6); (23;10) ;(29;16); (31;18); (37;24); (41;28); (43;30; (47;34)..

. ......

$T_r$(K) = 2K + $q_r$ (K ∈ ${\mathbb{N}}^{*}$: $T_r$(K) and $q_r$ are primes) (see Appendix 16)

For any integer K satisfying ${\left(2K\right)}^{0.525}$ > $q_r$ the property holds for $T_r$(K).

Therefore it is generally validated for all K > $K_{0 }$, since we obtain all possible cases of

Chen's weak conjecture starting with $T_1$(K), then $T_2$(K), then $T_3$(K). ... for ${\left(2K\right)}^{0.525}$ ≤ $q_r$.

(can be proved by strong recurrence using the same method as in Theorem 4 by

"finite descent").

Let a = ${\displaystyle \frac{40}{21}}$ and $P_a$(r) be the following property

"For any integer M │ 2M < ${\left(q_r\right)}^a$ there exists at least a prime q < $q_r$ │ 2 M + q ∈ $\mathcal{P} "$

▶ $P_a$($K_0$) is true (see Appendix 16).

▶ Let’s show: $P_a$(r) $⟹P_a$(r + 1)

$$q_{r+1}\le q_r+q_r^{0.525} \left(5.6\right)$$

It is assumed that M │

$T_{r+1}\left(K\right) - q_{r+1}\text{≠} 2M\mathrm{knowing} 2M < {\left(q_{r+1}\right)}^{c_p}$

$\forall T_m\left(R\right), q_m\in \mathcal{P}\exists h, s \in \mathbb{N} │ T_{r+1}\left(K\right)=T_m\left(R\right)+2h\mathrm{and} <!-- q_{r+1}=q_m$ + 2s (5.7)

then

$$T_m\left(R\right) - q_m \ne 2(M + s - h) \left(5.8\right)$$

which is impossible according to the hypothesis of strong recurrence since

2(M + s - h) is less than Sup ${\left(q_m\right)}^a$and that all primes $T_m$(R) and $q_m$ satisfy the recurrence hypothesis.

We deduce that: ${Pc}_p$(r) $⟹{Pc}_p$(r + 1)

Thus the property (5.6) is true.

6. Lemma (Goldbach’s Fundamental Lemma)

Let q be an odd prime; then

there exists integers $n_0$ and $n_q$ │

For any integer n ≥ $n_q$ there exists an integer s │

$$2n - {Wq}_{2s}\in \mathcal{P} \left(6.1\right)$$

Let (${Zq}_{2n}$) be the sequence of primes defined by

∀ n ∈ ℕ n ≥ $n_q{Zq}_{2n}$ = Inf (2n - ${Wq}_{2k}$ ∈ $\mathcal{P}$: k ∈ ℕ) (6.2)

All G.D. are contains in the set {(2n - ${Zq}_{2n}; {Zq}_{2n}$): n ∈ ℕ + 3}

For any integer n ≥ $n_0{Zq}_{2n}$ ≤ ${\left(2n\right)}^{0.525}$ (6.3)

$${Zq}_{2n}\le o{(n}^{0.525}) (6.4)$$

Proof. The proofs of propositions (6.1), (6.2) and (6.3) are established following the same principle of strong recurrence as in Theorem 4 and Corollary 5 by "return, absurd and finite descent"

(6.1): For any integer n > 3 and for any odd primes r, q │ 3 ≤ r < q,

there exists an integer $M_r$ │

$$2n - {Wq}_{2k}=2n - 2M_r- {Wr}_{2k}=2(n - M_r) - {Wr}_{2k}$$

or

$$2(n+1) - {Wq}_{2k}=2(n+1 - M_r) - {Wr}_{2k}$$

then by recurrence and the absurd the property is validated.

If there were no integer k such that 2(n +1- $M_r$) - ${Wr}_{2k}$ ∈ $\mathcal{P}$, then there would be no integer k such that 2(n +1- $M_r$) - ${Wr}_{2k}$ ∈ $\mathcal{P}$, contradicting the recurrence hypothesis.

(6.2): By strong recurrence:

If

          $2(n+1) - {Wq}_{2(n+1)}\in \mathcal{P}$ then the proof is directly validated

          (see Baker, Harman [4])

else

$$2(n+1) - {Wq}_{2k}=2(n+1 - M_r) - {Wr}_{2k}={Zp}_{2(n+1-M_r)}$$

Then, the property is validated following the recurrence hypothesis hence

$$2(n+1 - M_r) \le 2n$$

and then

$${Zp}_{2(n+1-M_r)}\le {\left(2n\right)}^{0.525}$$

            (Proof to develop).

Remark. A better estimate of the following form can be obtained by the same method with probability one or on average using the results of Bombieri [3], Cramer [9], Granville [17],

Nicely [32] and Maier [30]:

$\exists m_0\in \mathbb{N} │ \forall n\in \mathbb{N}\text{:} n \ge m_0$;

For any real c > 2 $U_{2n}$ < 1.7 ${\mathrm{l}\mathrm{n}\left(n\right)}^c$ (with probability one) (6.5)

and

${\exists K\mathrm{’}\ge 3.5│U}_{2n}< K.{\mathrm{l}\mathrm{n}}^{1.3}$ (n) (on average) (6.6)

7. Principle of Proof

To determine the E.G.D. three sequences of primes ($W_{2n}),(V_{2n}$), ($U_{2n})$ are defined and they verify the following properties

lim $V_{2n}$ = +$\mathrm{\infty }.$ (7.1)

 $\forall n\in \mathbb{N}+2 V_{2n} \mathrm{is} \mathrm{defined} \mathrm{as} \mathrm{a} \mathrm{function} \mathrm{of} <!-- W_{2n}$ = Sup (p ∈ Ƥ: p ≤ 2n - 3) (7.2)

($W_{2n}$) is an increasing sequence of primes that contains all of them except $p_1$ = 2 (7.3)

$$\lim W_{2n}=+\infty \left(7.4\right)$$

(${ U}_{2n }$) is a complementary sequence to ($W_{2n}$) of negligible primes with respect to 2n (7.5) For any integer n $\ge 3$

● If (2n - $W_{2n }$) is a prime

then $V_{2n}$ and $U_{2n}$ are defined by

$V_{2n}$ = $W_{2n}$ and $U_{2n}$= 2n - $W_{2n}$ (7.6)

● Otherwise, if (2n - $W_{2n }$) is a composite number

we search for two previous terms of the sequence (${ G}_{2n }$), $U_{2(n-k)}$₎ and $V_{2(n-k)}$ satisfying the following conditions

$$U_{2(n-k)},V_{2(n-k)},[U_{2(n-k)}+2k ] \in \mathcal{P}\left(7.7\right) { U}_{2\left(n-k\right)}+V_{2\left(n-k\right)}=2(n-k)$$

which is always possible (see Theorem 4 and "Goldbach’s fundamental Lemma 6")

So by setting

$$V_{2n}=V_{2(n-k)}\mathrm{and} U_{2n}=U_{2(n-k)} +2k \left(7.8\right)$$

two new primes $V_{2n}$ and $U_{2n}$ satisfying (4.10) are generated │

$U_{2n}+V_{2n}$= 2n (7.9)

This process is then repeated incrementing n by one unit (n ← n + 1).

● Remark. Using the same method as in Theorem 4, we can the following equivalent property by strong recurrence: For any integer n greater than 48

$${\mathcal{P}}_{ret}\left(n\right)\text{:} \text{"} \mathrm{There} \mathrm{exists} \mathrm{an} \mathrm{integer} \mathrm{such} \mathrm{that} 2K + U_{2(n-k)} \in \mathcal{P} " \left(7.10\right)$$

To this end,.

▶ ${\mathcal{P}}_{ret}$(49) is true.

▶ The heredity of the property ${\mathcal{P}}_{ret}$(n): ${\mathcal{P}}_{ret}$ (n) $⟹{ \mathcal{P}}_{ret}$(n + 1)

can be proved by the absurd and returning to the previous terms by noting that

For any integer r: r ≤ n, there is at least one integer ${\mathrm{M}}_{\mathrm{r}}$ │

$$U_{2(n+1-k)}=2{ M}_r+U_{2(r+1-k)}$$

then

$$\begin{array}{cc}2K+U_{2(n+1-k)}\hfill & =2(K+M_r)+U_{2(r+1-k)}\hfill \\ & =2P+U_{2(r+1+{ M}_r-P)} \left(7.11\right)\hfill \end{array}$$

By posing: P = K + ${ M}_r$ and $r + 1 +{ M}_r$ ≤ n

Now, according to the recurrence hypothesis on ${\mathcal{P}}_{ret}$(n) there exists an integer P │

$$2P+U_{2(r+1+{ M}_r-P)}\in \mathcal{P} \left(7.12\right)$$

then there exists an integer K │

$$2K+U_{2(n+1-k)}\in \mathcal{P} \left(7.13\right)$$

In summary, the property ${\mathcal{P}}_{ret}$(n) is hereditary and, as a result, verifiable.

We apply the same type of reasoning using Theorem 4 to the general case with the sequence (${Wq}_{2n}$), showing:

For any integer n > 2 there exists an integer K │

$$2K+q_{2n}\in \mathcal{P}$$

8. Theorem (Goldbach Conjecture)

(i)  There exists at least a recurrent sequence ($G_{2n}$) = ($U_{2n}; V_{2n})$ of primes satisfying the following conditions.

For any integer n ≥ $2$

$U_{2n}, V_{2n} \mathsf{ϵ} \mathcal{P}$ and $U_{2n}$ + $V_{2n}$ = 2n (8.1)

(Any integer n ≥ $2$ is the mean arithmetic of two primes)

(ii)  An algorithm can be used to explicitly compute any E.G.D. ${ U}_{2n}$ and $V_{2n}$ (8.2)

⠀

Proof.

▄ GLOBAL STRONG RECURRENCE:

The proof can be made using the following strong recurrence principle.

Let $P_G$(n) be the property defined for any integer n ≥ 2 by

$P_G$(n): " For any integer p satisfying 2 ≤ p ≤ n there exists two primes$U_{2p}$ and $V_{2p}$ such their sum is

   equal to 2p ".

$$(\forall p\in \mathbb{N} │ 2 \le p \le n U_{2p}, V_{2p } \in \mathcal{P}\mathrm{and} U_{2p}+V_{2p}= 2p)$$

Let's show by strong recurrence that $P_G$(n) is true for any integer n ≥ 2

 ▶ $P_G$(2) is true: it suffices to choose ${ U}_4$ = $V_4$ = 2.

▶ Let's show that the property $P_G$(n) is hereditary: $P_G$(n) $⟹P_G$(n + 1)

Assume property $P_G$(n) is true.

● If (2(n + 1) - $W_{2(n+1) }$) is a prime

then $V_{2(n+1) }$ and $U_{2(n+1) }$ are defined by

$$V_{2(n+1)}=W_{2(n+1)}\mathrm{and} U_{2(n+1)}=2(n+1) - W_{2(n+1)} \left(8.3\right)$$

● Otherwise, if (2(n+1) - $W_{2(n+1) }$) is a composite number

then there exists an integer k to obtain two terms $U_{2(n+1-k)}$₎ and $V_{2(n+1-k)}$ satisfying the following conditions

 $U_{2(n+1-k)},V_{2(n+1-k)}\mathrm{and} U_{2(n+1-k)}+2k\in \text{Ƥ} \left(8.4\right) { U}_{2\left(n+1-k\right)}+V_{2\left(n+1-k\right)}$ = 2(n +1- k)

we use the previous terms of the sequence ($G_{2n }$).

For any integer q │ 1 ≤ q ≤ n - 3 we have

$$3 \le U_{2(n-q) \le n.}$$

Then there exists an integer k 1 ≤ k ≤ n - 3 │

$$R_{2n}=U_{2(n-k)} + 2k \in \mathcal{P} \left(8.5\right)$$

following Theorem 4 since all primes smaller than ${\left(2n\right)}^{0.525 }$ are in the set {${ U}_{2k}$: k ≤ n }

(If there were no such primes, we would have a contradiction with the Goldbach(-) Conjecture (Theorem 4) or with Goldbach’s fundamental Lemma 6). In fact, in an equivalent way (see the previous remark) we can copy the proof of Teorem 4 by performing a similar strong recurrence "finite descent feedback and absurd" directly on the set

{ $U_{2k}$: k ≤ n } │

$R_{2n}$ = $U_{2(n-k)}$ + 2k ∈ Ƥ (8.6)

The smallest integer k │ $R_{2n}$∈ Ƥ is denoted by $k_n$.

So by setting

$U_{2n}$ = $U_{2\left(n-k_n\right)}$ + 2$k_n$ and $V_{2n}$ = $V_{2\left(n-k_n\right)}$ ∈ Ƥ (8.7)

(These two terms are primes)

In the previous steps two primes ${ U}_{2\left(n-k_n\right)}$ and $V_{2\left(n-k_n\right)}$whose sum is equal to 2(n - $k_n$) were

determined.

$U_{2\left(n-k_n\right)}$ + $V_{2\left(n-k_n\right)}$ = 2(n -${ k}_n$) (8.8)

By adding the term $2k_n$ to each member of the equality (8.6) it follows

$U_{2\left(n-k_n\right)}$ + 2$k_n$ + $V_{2\left(n-k_n\right) }$= 2(n - $k_n$) + 2$k_n$ (8.9)

$\iff$ [ $U_{2\left(n-k_n\right)}$ + 2$k_{n }$] + $V_{2\left(n-k_n\right)}$ = 2n (8.10)

$\iff U_{2n}$ + $V_{2n}$ = 2n (8.11)

Two new primes $V_{2(n+1)}$ and $U_{2(n+1)}$ satisfying (${ U}_{2(n+1)}$ + $V_{2(n+1)}$ = 2(n + 1)) are generated.

It follows that $P_G$(n + 1) is true. Then the property $P_G$(n) is hereditary:

$$P_G\left(n\right)⟹P_G(n+1).$$

Therefore for any integer n ≥ 2 the property $P_G$(n) is true.

It follows

$\forall$ n ∈ ℕ + 2 there are two primes $U_{2n}$ and $V_{2n}$ and such their sum is 2n: $U_{2n}$ + $V_{2n}$ = 2n

⠀

▄ ALGORITHM:

 For any integer n $\ge 3$

● If (2n - $W_{2n }$) is a prime

then $V_{2n }$ and $U_{2n}$ are defined by

 $V_{2n}$ = $W_{2n}$ and $U_{2n}$= 2n - $W_{2n}$ (8.12)

● Otherwise, if (2n - $W_{2n})$ is a composite number

we use the previous terms of the sequence ($G_{2n }$).

For any integer q │ 1 ≤ q ≤ n - 3 we have

$$3 \le U_{2(n-q)}\le n$$

Then there exists an integer k 1 ≤ k ≤ n - 3 │

$$R_{2n}=U_{2(n-k)}+2k \in \mathcal{P} \left(8.13\right)$$

following Theorem 4 since all primes smaller than ${\left(2n\right)}^{0.525 }$ are in the set {${ U}_{2k}$: k ≤ n }

(If there were no such primes, we would have a contradiction with the Theorem 4 or with Goldbach’s fundamental Lemma 6), see previous GLOBACH STRONG RECURRENCE

Finally, for any integer n ≥ 3 this algorithm determines two sequences of primes ($U_{2n}$)

and ($V_{2n}$) verifying Goldbach's conjecture.

9. Lemma

The sequence (${ U}_{2n})$ verifies the following majorization

For any integer n ≥ 65

$U_{2n}$ ≤ (2${n)}^{0.525}$ (9.1)

and

$U_{2n}$ = o ($n^{0.525}$) (9.2)

Proof. According to the programm 12.2 and Appendix 14 the majorization (9.1) is verified

for any integer n │ 65 ≤ n ≤ 2000.

For any integer n > 2000 the proof is established by recurrence. For this purpose let $P_{bhip}$(n) be the following property

${ P}_{bhip}$(n): " $U_{2n}$ ≤ ${\left(2n\right)}^{0.525}$ ". (9.3)

⠀

▶ $P_{bhip}$(2000) is true according to program 13.2 and the table in appendix 14.

▶ For any integer n ≥ 2000 let’s show that $P_{bhip}$(n) is hereditary:

$$P_{bhip}\left(n\right)⟹.P_{bhip}(n+1)$$

Assume that $P_{bhip}$(n) is true: then

● If (2(n + 1) - $W_{2(n+1)})$ is a prime

then $V_{2(n+1)}$ and $U_{2(n+1)}$ are defined by

$$V_{2(n+1)}=W_{2(\mathrm{n}+1)}\mathrm{and} U_{2(n+1) }=2(n+1) - W_{2(n+1)} \left(9.4\right)$$

According to the results in [4,5,24] (see Lemma 9) there is a constant K > 0 such that

$2(n+1) - K. { \left[2\right(n+1\left)\right]}^{0.525}< W_{2(n+1)}<2(n+1)$

$$⟹U_{2(n+1)}=2(n+1) W_{2(n+1)}< K {.\left[2\right(n+1\left)\right]}^{0.525}$$

  $⟹U_{2(n+1)}\le K {.\left[2\right(n+1\left)\right]}^{0.525}$

● Otherwise, if (2(n + 1) - $W_{2(n+1)}$) is a composite number

$\exists p\in {\mathbb{N}}^{*} │ U_{2(n+1)}=U_{2(n+1-p)}$ + 2p (9.5)

According to [4,5,24]

$U_{2(n+1)}=2p U_{2(n+1-p)}=2p (n+1-p)-W_{2(n+1-p)}=2(n+1) - W_{2(n+1-p)}$ (9.6)

Via " Goldbach's fundamental Lemma 6 " it follows that$U_{2(n+1)}<K.[{2(n+1)]}^{0.525}$ (9.7)

$P_{bhip}$(n + 1) is true then $P_{bhip}$(n) is hereditary.

So for any integer n ≥ 2000 the property $P_{bhip}$(n) is true.

Finally $U_{2(n+1)}$ ≤ ${\left[2\right(n+1\left)\right]}^{0.525}$

● Remark. A more precise estimate can be obtained using the Cipolla or Axler frames [2,8].

10. Propositions

 A) Link between Goldbach conjecture and the fundamental theorem of arithmetic.

A log-exp correspondence is established by linking the sum and product of primes via Goldbach's conjecture and the fundamental theorem of arithmetic, since if G.D. of 2n are p' and q', and if 2n decomposes into factors P " and Q " │

(p’, q’ ∈ Ƥ │ p’ $\gg$ q’ and P " $\gg$ Q "); then,

2n = P ".Q " = p’ + q’ and p’ - q’ = 2K

ln(P ".Q ") = ln(P ") + ln(Q ")

 = ln(p’ + q’) = ln(p’(1 +${\displaystyle \frac{q\text{'}}{p\text{'}}}$))

 ≈ ln(p’) +${\displaystyle \frac{q\text{'}}{p\text{'}}}$

By choosing p’ = next or prevprime(P ") (P " = p’ +/- a) we obtain a q' localization of the form 

$$q\text{'}\approx [p\text{'}.ln(Q\text{"}\left)\right] \approx [ p\text{'}.ln({\displaystyle \frac{2n}{P \text{"}}})] \approx [ p\text{'}.ln\left({\displaystyle \frac{2n}{p\text{'}}}\right)]$$

.then

2n ≈ p’(1 + ln(${\displaystyle \frac{2n}{p\text{'}}}$))

2n ≈ p’(1 + ln(${\displaystyle \frac{2n}{P \text{"}}}$)) ≈ p’(1 + ln(${\displaystyle \frac{2n}{p\text{’} +/- a}}$))

2n ≈ p’(1 + ln(2n / (p’(1 +/-${\displaystyle \frac{ a}{p\text{'}}}$)))

2n ≈ p’(1 + ln((${\displaystyle \frac{2n}{p\text{'}}}$).(1 / (1 +/-${\displaystyle \frac{ a}{p\text{'}}}$))))

2n ≈ p’(1 + ln(${\displaystyle \frac{2n}{p\text{'}}}$) + ln(1/(1 +/-${\displaystyle \frac{ a}{p\text{'}}}$)))

2n ≈ p’(1 + ln(${\displaystyle \frac{2n}{p\text{'}}}$)) -/+ a

You can solve equations like these using the scientific software Maple via the command,

$$solve(2n+/- a=x.(1+ln{\displaystyle \left(\frac{2n}{x}\right)), x)}$$

to locate p' and proceed by successive next or prevprime to determine two G.D. of 2n,

(programming possible in Algorithm 14). This procedure appears to generalise Pocklington's theorem, and we observe that the G.D. and their number G(E) are related to the number of prime factors in the decomposition of 2n.

Examples:

● evalf(solve([90 = x*(1 + ln(96/x)), x < 96], x)); {x = 64.12418697}; p’ = 67 q’ = 29

● evalf(solve([1000 = x*(1 + ln(1100/x)), x < 1100], x)); { x = 665.6361412}

prevprime(665); 661

isprime(1100 - 661); true; p’ = 661 q’ = 439

● evalf(solve([9700 = x*(1 + ln(10000/x)), x < 10000], x)); { x = 7652.697929}

prevprime(7652); 7649

isprime(10000 - 7649); true; p’ = 7649 q’ = 4351

● evalf(solve([99950 = x*(1 + ln(100000/x)), x < 100000], x)); { x = 96854.43333}

a: = prevprime(96799); a: = 96797# obtained after 3 or 4 iterations of the command prevprime()

isprime(100000 - a); true; p’ = 96799 q’ = 3201

Solutions are:

$x_0$= Re(- (2n +/- a) / LambertW(-1,- (2n +/- a) / (2n.e)))

and

$x_1$= Re(- (2n +/- a) / LambertW(- (2n +/- a) / (2n.e)))

Remarks. For any composite number n greater than three,

● gcd(n, p’)= gcd(n ,2n - p’) = gcd(n ,q’) = gcd(n ,K)= gcd(n,p’.q’) = gcd(n,n² - K²) = 1

● gcd(K,p’) = gcd(K,q’) = gcd(n.K, p’) = gcd(n.K, q’) = 1

● The smallest E.G.D. of 2n is less than the square of its greatest prime factor.

● For any non-zero integer R, the smallest of G.D.’s of R.$p_k$# is greater than $p_k$.

B)  Method of locating G.D. products (Difference in squares: N² - K² or decentered dichotomy by geometric mean (see code RSA, [37]).

Locally (around 2n), there exists a sub-sequence (${p\text{'}}_s, {q\text{'}}_s$,) of G.D. of 2s such that the product sequence${ M}_s$=${p\text{'}}_s$.${q\text{'}}_s$= s² - k² is almost increasing (the variations of the geometric mean almost follow those of the arithmetic mean; indeed, if

${p\text{'}}_{m+1}{\ge p\text{'}}_{m }{, p\text{'}}_{m+1}{ \approx p\text{'}}_m\gg$ ${q\text{'}}_m, {q\text{'}}_{m+1}$ and ${q\text{'}}_{m+1}$ ≥ ${q\text{'}}_{m }$

then 

$${p\text{'}}_{m+1}.{q\text{'}}_{m+1}-{p\text{'}}_m.{q\text{'}}_m=({p\text{'}}_{m+1}-{p\text{'}}_{m }){q\text{'}}_{m+1}+{p\text{'}}_m.({q\text{'}}_{m+1}-{q\text{'}}_m)\ge 0).$$

If we choose:${q\text{'}}_m =$ ${q\text{'}}_{m+1}$, we minimize and better controls the deviation ${ M}_{s+1}$ - ${ M}_s$

Thus, it is possible to determine Goldbach decomponents of 2n by the following algorithm, choosing a neighborhood of 2n of amplitude c.ln²(n) in agreement with the estimates made on the G(E) distribution function associated with the Goldbach comet.

Another possible method.

By off-center dichotomy using geometric means, similar to that used to crack RSA codes (see Sainty [37]). 

>

n2:= 1000;

# To determine two G.D.s of 2n = 1000, we choose two decomponents of a lower integer, m2 and two decomponents of a higher integer, r2 to 2n; we easily calculate

m2 < 2n = n2 < r2 and their differences km2 and kr2; then we examine their products which are assumed to preserve order, (if the initial decomponents are well chosen:

${p\text{'}}_1$ ${ \le p\text{'}}_{2, }{p\text{'}}_1$ ${ \approx p\text{'}}_2\gg$ ${q\text{'}}_1, {q\text{'}}_2$ and ${q\text{'}}_2$ ≥ ${q\text{'}}_{1 }$,(p'. q' = n² - k²); we then define admissible bounds for k from a = ${p\text{'}}_1$.${q\text{'}}_1$ and b =. ${p\text{'}}_2$.${q\text{'}}_2$

 min2 = trunc(evalf(sqrt(n² - b),Digits)) and max2 = trunc(evalf(sqrt(n² - a),Digits)); decomponents of 2n are deduced by iterating the nextprime() command from n + min2, (choose a gap of the order of c.ln²(n) between m2 and r2.

pinf: = prevprime(735); pinf: = 733

qinf: = nextprime(17); qinf: = 19

psup: = nextprime(1050); psup: = 1051

qsup: = nextprime(29); qsup: = 31

m2: = pinf + qinf; m2: = 752

r2: = psup + qsup; r2: = 1082

km2: = pinf - qinf; km2: = 714

kr2: = psup - qsup; kr2: = 1020

a: = m2*m2 - km2*km2; a: = 55708# a: = pinf.qinf

b: = r2*r2 - kr2*kr2; b: = 130324 # b: = psup.qsup

min2: = trunc(evalf(sqrt(0.25*n2*n2 - b),digits); min2: = 466

max2: = trunc(evalf(sqrt(0.25*n2*n2 - a),digits); max2: = 485

n:= trunc(0.5*n2);

em: = nextprime(n + min2 - 1);em: = 967

nextprime(em); 971

em2: = 0.5*n2 + max2; em2: = 985.0

q: = n2 - 971; q: = 29

isprime(q); true

C)  Euclidean divisions of 2n by its presumed Goldbach decomponents

To determine two Goldbach decomponents of 2n, the following parameters can be used:

If p’ + q’ = 2n, (p’, q’ ∈ Ƥ │ p’ $\gg$ q’) then we perform the Euclidean division of p' by q' under the following conditions: 

$$p\text{'}=m.q\text{’}+r0<r<q\text{'} r \wedge q\text{'}=1 r \wedge m=1$$

We deduce that q’ =${\displaystyle \frac{(2n - r)}{(m+1)}}$or 2n = (m + 1).q’ + r (dual view point).

which leads to the algorithm.

        (To develop) 

Implementation:

We perform the Euclidean division of 2n by odd primes in ascending order.

3,5,7,11,.....

20 = 3 x 6 + 2 = (3 x 5 + 2) + 3 = 17 + 3

22 = 3 x 7 + 1 = (3 x 6 + 1) + 3 = 19 + 3

24 = 3 x 8 = 5 x 4 + 4 = (5 x 3 + 4) + 5 = 19 + 5

26 = 3 x 8 + 2 = (3 x 7 + 2) + 3 = 23 + 3

28 = 3 x 9 + 1 = (3 x 8 + 1) + 3 = 5 x 5 + 3 = (5 x 4 + 3) + 5 = 23 + 5

30 = 3 x 10 = 5 x 6 = 7 x 4 + 2 = (7 x 3 + 2) + 7 = 23 + 7

32 = 3 x 10 + 2 = (3 x 9 + 2) + 3 = 29 + 3

34 = 3 x 10 + 4 = (3 x 9 + 4) + 3 = 31 + 3

36 = 3 x 12 = 5 x 7 + 1 = (5 x 6 + 1) + 5 = 31 + 5

38 = 3 x 12 + 2 = (3 x 11 + 2) + 3 = 35 + 3 = 5 x 7 + 3 = (5 x 6 + 3) + 5 = 33 + 5

= 7 x 5 + 3 = (7 x 4 + 3) + 7 = 31 + 7

. ..

500 = 3 x 166 + 2 = (3 x 165 + 2) + 3 = 497 + 3 = 5 x 100 = 7 x 71 + 3 = (7 x 70 + 3) + 7 = 493 + 7 = 11 x 45 + 5 = (11 x 44 + 5) + 11 = 489 + 11 = 13 x 38 + 6 = (13 x 37 + 6) + 13 = 487 + 13

For large integers, we will begin Euclidean division with a prime divisor of the order of nextprime(trunc(c.ln(n)).

Remark. This point of view allows us to give another proof of the Binary Goldbach Conjecture equivalent but more explicit byidentifying uniqueness, coincidence and consistency using euclidean division of

2n by $p_k$ ∈ Ƥ $p_k$ > n: 2n = $p_k$ + $R_k$ and division of 2n by $q_r$ ∈ Ƥ $p_{k }$ $\gg$ $q_r$ which gives 

$$2n=m.q_r+t_r$$

hence 2n = ((m - 1).$q_r$+$t_r$) +$q_r$=$D_{r }$+$q_r$;

$p_{k }$,$q_r$are increasing sequences,$R_k$and$D_{r }$= (m - 1).$q_r$+$t_r$are decreasing sequences. By uniqueness of Euclidean division and since$D_{r }$ $\gg$ $q_r$ and $p_{k }$ $\gg$ $q_r, R_k$, 

$$[ {\displaystyle \frac{2n}{q}}]=m=1+[ {\displaystyle \frac{p}{q}}]$$

($D_{r }$is the result of the euclidean division of 2n by $q_r$)

We deduce that there exists integers $i_n$ and $j_n$ such that:

$$p_{i_n}=D_{ j_n } and R_{i_n}=q_{j_n.}$$

11. Theorem

For any integer n$\ge$ 3 it is easy to check

($W_{2n}$) is a positive increasing sequence of primes (11.1)

{ $W_{2n}$: n $\in$ IN + 3 } $\cup \{ 2 \}$ = $\mathcal{P}$ (11.2)

lim $W_{2n}$ = +$\mathrm{\infty }$ (11.3)

($U_{2n}$) and (${ V}_{2n }$) are sequences of primes and the set { $U_{2k}$: k ≤ n } (11.4)

contains all primes less than ln(n)

n$\le$ $V_{2n}$ ≤ $W_{2n}$ (11.5)

3$\le {2n-W_{2n}\le U}_{2n}\le$n (11.6)

lim $V_{2n}$ = +oo (11.7)

Proof.

(11.1): For any integer n ≥ 2 ${ \mathcal{P}}_n \subset {\mathcal{P}}_{n+1}$. Therefore, $W_{2n}$ ≤ $W_{2(n+1)}$. So the sequence (${ W}_{2n }$) is increasing.

(11.2): Any prime except $p_1$ = 2 is odd, hence the result.

(11.3): lim $W_{2n}$ = lim $p_k$ = +oo

(11.4): By definition $V_{2n}$ = $W_{2n}$ or there exits an integer k ≤ n - 2 │ $V_{2n}$ = $V_{2(n-k)}$.

So the terms of the sequence ($V_{2n)}$) are primes.

(11.5): According to Lemma 9, for any integer n ≥ 65

$U_{2n}$ < ${\left(2n\right)}^{0.525}$

therefore

$${ U}_{2n}< {\left(2n\right)}^{0.55}<n$$

and

$V_{2n}=2n-U_{2n}$> 2n - n > n

For any integer n │ 3 ≤ n ≤ 65 verification is carried out according to the computer program in paragraph 14.3 and the table in appendix 15.

We can also see that by construction $V_{2n}$ ≥ $U_{2n}$ because if we assume the opposite then $V_{2n}$ is not the largest prime number verifying

${\displaystyle \frac{1}{2}}$ ($U_{2n}$ + $V_{2n}$) = n.

So

$V_{2n}$ ≥ n

 According to (11.5) n $\le$ $V_{2n} ⟹U_{2n}$ = 2n - $V_{2n}$ ≤ 2n - n ≤ n (11.6)

 $V_{2n}$≤$W_{2n} ⟹$ 2n - $W_{2n}$ ≤ 2n - $V_{2n}$ = $U_{2n}$ (11.7)

 By (11.5) for any integer n ≥ 2: n $\le$ $V_{2n}$

lim $V_{2n}$ = +oo.

12. Lemma

We dissociate the following cases mod 6 for any even integer 2n: n ≥ 3 │ p + q = 2n p, q ∈ Ƥ

If 2n = 6m then (p; q) = (6r + 5; 6(m - r - 1) + 1) or (6r+1; 6(m - 1 - r) + 5)

If 2n = 6m + 2 then (p; q) = (6r + 1; 6(m - r) + 1)

If 2n = 6m + 4 then (p; q) = (6r + 5; 6(m- 1 - r) + 5)

Table 1. Sum of integers 1, 5 mod 6 (in $\mathbb{Z}$/6$\mathbb{Z}$).

p + q mod 6	1	5
1	2	0
5	0	4

(To adapt with 2n = 30m + k).

Table 1. Sum of integers 1, 5 mod 6 (in $\mathbb{Z}$/6$\mathbb{Z}$).

p + q mod 6	1	5
1	2	0
5	0	4

(To adapt with 2n = 30m + k).

Table 2. Sum of integers 1, 7, 11, 13, 17, 19, 23, 29 mod 30 (in $\mathbb{Z}$/30$\mathbb{Z}$).

+ mod 30	1	7	11	13	17	19	23	29
1	2	8	12	14	18	20	24	0
7	8	14	18	20	24	26	0	6
11	12	18	22	24	28	0	4	10
13	14	20	24	26	0	2	6	12
17	18	24	28	0	4	6	10	16
19	20	26	0	2	6	8	12	18
23	24	0	4	6	10	12	16	22
29	0	6	10	12	16	18	22	28

Proof.

Table 2. Sum of integers 1, 7, 11, 13, 17, 19, 23, 29 mod 30 (in $\mathbb{Z}$/30$\mathbb{Z}$).

+ mod 30	1	7	11	13	17	19	23	29
1	2	8	12	14	18	20	24	0
7	8	14	18	20	24	26	0	6
11	12	18	22	24	28	0	4	10
13	14	20	24	26	0	2	6	12
17	18	24	28	0	4	6	10	16
19	20	26	0	2	6	8	12	18
23	24	0	4	6	10	12	16	22
29	0	6	10	12	16	18	22	28

Proof.

13. Properties

For any integer k ≥ 2 there are infinitely many integers n │ $U_{2n}$ = $p_k$ (13.1) $V_{2n}$ ~ 2n (n $\to$ +$\mathrm{\infty }$) (13.2)

For any integer n ≥ 5000

$${ U}_{2n}\ll V_{2n} and \lim \left({\displaystyle \frac{U_{2n}}{V_{2n}}}\right)=0 \left(13.3\right)$$

(Cesaro sums of E.G.D. ${ U}_{2n}$ and $V_{2n}$ have interesting properties)

The smallest integer n │${ U}_{2n}\ne$2n - $W_{2n}$is obtained for n = 49 and $G_{98 }$= (79; 19) (13.4)

(This type of terms increases in the Goldbach sequence ($G_{2n})$ as n increases in the sense of the Schnirelmann density and there are an infinite number of them; their proportion per interval can be computed using the results given in [39]).

The sequence ($G_{2n}$) is "extremal " in the sense that for any integer n $\ge 2$ (13.5)

$V_{2n}$ and $U_{2n}$ are the largest and smallest possible primes │ $U_{2n}$ + $V_{2n}$ = 2n.

The Cramer-Granville-Maier-Nicely conjecture [9],[ 17],[30,32] is verified with probability one. It leads to the following majorization

For any integer p ≥ 500

$U_{2p}$ ≤ 0.7 [${\mathrm{l}\mathrm{n}(2p\left)\right]}^{(2.2 - {\displaystyle \frac{1}{p}})}$ (with probability one) (13.6)

The proof is similar to that of Lemma 9 and is validated by the studying functions of the type

f: x$\to$a. g (x) + b${\left[\mathrm{l}\mathrm{n}\right(g\left(x\right)\left)\right]}^c$ (a,b > 0; c > 2) with

g: x$\to$ 0.7 [l${\mathrm{n}\left(x\right)]}^{(c - {\displaystyle \frac{1}{x}})}$ and h: x $\to$ 0.7${\left[\mathrm{l}\mathrm{n}\right(x\left)\right]}^{(2.2 - {\displaystyle \frac{1}{x}})}$ by using Maple software.

A better estimate can be obtained via [29,30,31].

According to Bombieri [3] and using the same method as in the proof of Lemma 9,

we obtain the following estimate of${ U}_{2n}$

$\forall$ $\epsilon$ > 0 $U_{2n}$ = O (${\mathrm{l}\mathrm{n}}^{1.3+\epsilon }(n$)) (on average) (13.7)

14. Algorithm

14.1. Algorithm Written in Natural Language

Inputs:

Input four integer variables: k, N, n, P

Input: $p_{1 }$= 2, $p_{2 }$= 3, $p_3$ = 5, $p_4=$ 7,. ................, $p_N$ the first N primes.

: n ← 3

: P = M, R, G, S or T as indicated in paragraph 2

Algorithm body:

A) Compute: $W_{2n}$ = Sup(p $\in \mathcal{P}$: p ≤ 2n - 3)

⠀

If ${\mathit{T}}_{2\mathit{n}}$= (2n -${\mathit{W}}_{2\mathit{n}}$) is a prime

$U_{2n}$ ← $T_{2n}$ and $V_{2n }$ ← $W_{2n}$(14.1.1)

otherwise

B) If ${\mathit{T}}_{2\mathit{n}}$ is a composite number

Let: k = 1

B.1) While $U_{2\left(n-k\right)}$ + 2k is a composite number

 assign to k the value k + 1 (k $⟵$ k + 1).

 return to B1)

End while

Assign to k the value $k_n$ ($k_n⟵k$)

Let:

$$U_{2n}= U_{2\left(n-k_n\right)}+ 2k_n\mathrm{and} V_{2n}= V_{2\left(n-k_n\right)} \left(\mathrm{14.1.2}\right)$$

Assign to n the value n + 1 (n$⟵ n+1$and return to A)

End:

Outputs for large integers less than 10⁴:

Print (2n = ●; 2n - 3 = ●; $W_{2n}$ = ●; $T_{2n}$= ●; $V_{2n}$ = ●; $U_{2n}$ = ●)

Outputs for large integers:

Print (2n - P = ●; 2n - 3 - P = ●; $W_{2n}$ - P = ●; $T_{2n}$ = ●; ${V_{2n}-P=\mathrm{●} ;U}_{2n}$ = ●)

14.2. Program Written with Maxima Software for 2n around ${10}^{1000}$_

c: 10**1000; forn: c +40000step 2 thruc +40100do

(b:2,test: 0, b: next_prime(b) ,e: n - b ,

if primep(e)

then print(n -c, b, e - c)

else whiletest= 0do(e: n -b, if primep(e)

 then test:1, print(n -c, b, e - c)

else test: 0 ,b:next-prime(b)) ;

14.3. Program Written with Maplesoft Maple for 2n Around ${10}^{1000}$

G: = 10^1000:

V: = [1, 11, 13, 17, 19, 23, 29]:

A: = G + 500000:

B: = A + 59:

b:=2:

st: = time():

for q from A by 6 to B do # Program modulo 30. using the results of Lemma 11

Possibility of inverting the two loops or defining three similar structures with s: = 0, 1, 2.

for s from 0 to 2 do

n: = q + s + s:

b: = trunc(0.59b - 20); # Improving computation time: the idea is to recognise that for any integer n large

      enough there exists a Goldbach decomponent ${p\text{'}}_n$ and a successor ${p\text{'}}_{n+1}$such that

         E:│${p\text{'}}_{n+1}$- ${p\text{'}}_n$│< k.${\mathrm{l}\mathrm{n}}^2$(n); this reduces the number of

         " nextprime(●) " operations which take up the most computing time.

(If G = ${10}^{500}$: Computingtime is around 10 sec for thirty terms);The algorithm can be refined by exploiting

      frame (E). Cesàro averages can also be used to determine the initial condition for b.

t:= 0:

 R: = [[1, 5], [1,5]]: Q: = [[1, 7, 11, 13, 17, 19, 23, 29], [1, 13, 19], [11, 17, 23], [7, 13, 17, 19, 23, 29], [1, 7, 19], [11, 17, 23, 29], [1, 11, 13, 19, 23, 29], [1, 7, 13], [17, 23, 29], [1, 7, 11, 17, 19, 29], [1, 19, 7, 13], [11, 23, 29], [1, 23, 7, 17, 11, 13], [7, 19, 13], [11, 17, 29]]:

  while t = 0 do.

b: = nextprime(b + 100); # Additional test possible by improving Lemma 11. (with V mod 30).

 # Possibility of replacing nextprime with a faster procedure (see Sainty [37]).(the computation time is greatly reduced by replacing with b:=nextprime(b + k(b,G)), k(b,G) constant around 150 for G=${10}^{1000}$, k(b,G) chosen randomly with the rand procedure or very slowly increasing as a function of b and G), but in general we don't obtain the E.G.D. but any Goldbach decomponents.

  e: = n - b;

 K: = e mod 6;

   if K in R[s+1] then

     if isprime(e)

     Then t: = 1;

      print(n - G, b, e - G);

   end if;

  end if;

  end do:

  end do:

end do:

Computingtime:= time() - st;

Comments: Possible test with igcd(n, b) = 1 and igcd(n, 2n - b) = 1

(or igcd(n,b.(n-b)) = 1) then isprime(b) and isprime(2n - b) may be faster than the nextprime() command, if we can improve the igcd algorithm.

RESULTS:

G = 10¹⁰⁰⁰ 

b: b:= nextprime(b+rand(100..150)) b:= nextprime(b+100) b:= nextprime(b+150) 

n - G b n - G - b 500000, 54133, 445867 500002, 40693, 459309 500004, 422393, 77611 500006, 49157, 450849 500008, 222991, 277017 500010, 259451, 240559 500012, 521981, -21969 500014, 622561, -22547 500016, 342929, 157087 500018, 25097, 474921 500020, 95083, 404937 500022, 201821, 298201 500024, 226337, 273687 500026, 255859, 244167 500028, 8147, 491881 500030, 83833, 416197 500032, 43261, 456771 500034, 162251, 337783 500036, 179203, 320833 500038, 12601, 487437 500040, 608471,-108431 500042, 157103, 342939 500044, 145531, 354513 500046, 440303, 59743 500048, 162577, 337471 500050, 258637, 241413 500052, 111791, 388261 500054, 139661, 360393 500056, 126397, 373659 500058, 40739, 459319 500060, 106121, 393939 ComComputtime:= 179.343 sec

500000, 139387, 360613 500002, 40693, 459309 500004, 731447, -231443 500006, 54139, 445867 500008, 205651, 294357 500010, 100109, 399901 500012, 40693, 459319 500014, 261823, 238191 500016, 82913, 417103 500018, 300889, 199129 500020, 12583, 487437 500022, 233591, 266431 500024, 159871, 340153 500026, 106087, 393939 500028, 608459, -108431 500030, 30347, 469683 500032, 43261, 456771 500034, 201833, 298201 500036, 186859, 313177 500038, 95101, 404937 500040, 121763, 378277 500042, 9029, 491013 500044, 148663, 351381 500046, 304847, 195199 500048, 157109, 342939 500050, 40459, 459591 500052, 8171, 491881 500054, 223037, 277017 500056, 49207, 450849 500058, 301349, 198709 Computtime:= 188.250 sec

500000, 361069, 138931 500002, 40693, 459309 500004, 535637, -35633 500006, 277789, 222217 500008, 205651, 294357 500010, 138959, 361051 500012, 40693, 459319 500014, 145501, 354513 500016, 198659, 301357 500018, 26309, 473709 500020, 77347, 422673 500022, 160709, 339313 500024, 162553, 337471 500026, 106087, 393939 500028, 263009, 237019 500030, 151813, 348217 500032, 24049, 475983 500034, 400031, 100003 500036, 145037, 354999 500038, 854257, -354219 500040, 121763, 378277 500042, 8161, 491881 500044, 145987, 354057 500046, 304847, 195199 500048, 12611, 487437 500050, 163729, 336321 500052, 100151, 399901 500054, 155291, 344763 500056, 126397, 373659 500058, 208277, 291781 500060, 67547, 432513  Computime:= 163.828 sec

b:= nextprime(b+rand(150..175)) b:= nextprime(b+rand(140..160))

n-G b n-b-G	n-G b n-b-G
500000, 139387, 360613 500002, 90481, 409521 500004, 422393, 77611 500006, 145007, 354999 500008, 604339, -104331 500010, 138959, 361051 500012, 221021, 278991 500014, 334843, 165171 500016, 297779, 202237 500018, 167267, 332751 500020, 54577, 445443 500022, 139409, 360613 500024, 336491, 163533 500026, 12589, 487437 500028, 263009, 237019 500030, 145517, 354513 500032, 334861, 165171 500034, 163697, 336337 500036, 318979, 181057 500038, 221047, 278991 500040, 761591, -261551 500042, 178691, 321351 500044, 54601, 445443 500046, 174989, 325057 500048, 84229, 415819 500050, 163729, 336321 500052, 159899, 340153 500054, 155291, 344763 500056, 166183, 333873 500058, 151841, 348217 Computtime:= 174.438 sec	500000, 112429,-387571 500002, 40693, 459309 500004, 277787, 222217 500006, 82903, 417103 500008, 148627, 351381 500010, 139397, 360613 500012, 40693, 459319 500014, 145501, 354513 500016, 388313, 111703 500018, 258329, 241689 500020, 77347, 422673 500022, 453683, 46339 500024, 67511, 432513 500026, 221197, 278829 500028, 263009, 237019 500030, 112459, 387571 500032, 178681, 321351 500034, 208253, 291781 500036, 274019, 226017 500038, 14071, 485967 500040, 162257, 337783 500042, 361111, 138931 500044, 52903, 447141 500046, 582299, -82253 500048, 8167, 491881 500050, 67537, 432513 500052, 111791, 388261 500054, 126641, 373413 500056, 126397, 373659 500058, 40739, 459319 Computtime:= 138.578 sec	Record: 116 sec; see in researchgate files PDFGOLDBACHTEST4,10 (For n from G+5000000 to 5000058 by 2), [37].

500000, 9473, 490527

500002, 24019, 475983

500004, 8123, 491881

500006, 9479, 490527

500008, 25087, 474921

500010, 57917, 442093

500012, 8999, 491013

500014, 9001, 491013

500016, 40697, 459319

500018, 9491, 490527

500020, 9007, 491013

500022, 139409, 360613

500024, 9011, 491013

500026, 9013, 491013

500028, 8147, 491881

500030, 26321, 473709

500032, 24049, 475983

500034, 54167, 445867

500036, 57943, 442093

500038, 9511, 490527

500040, 57947, 442093

500042, 8161, 491881

500044, 24061, 475983

500046, 162263, 337783

500048, 8167, 491881

500050, 12613, 487437

500052, 8171, 491881

500054, 9041, 491013

500056, 9043, 491013

500058, 40739, 459319

Computingtime: 343.453 sec

    G = 10²⁰⁰⁰

n - G n - b - G b n - G b n - b - G

40000, 39957, 43 40050, 86117, -46067

40002, 39091, 911 40052, 503, 39549

40004, 39957, 47 40054, 97, 39957

40006, 39549, 457 40056, 89393, -49337

40008, 25369, 14639 40058, 101, 39957

40010, 39957, 53 40060, 103, 39957

40012, 39549, 463 40062, 971, 39091

40014, 17737, 22277 40064, 107, 39957

40016, 39957, 59 40066, 109, 39957

40018, 39957, 61 40068, 977, 39091

40020, 39091, 929 40070, 113, 39957

40022, 39141, 881 40072, 523, 39549

40024, 39957, 67 40074, 983, 39091

40026, 35443, 4583 40076, 16937, 23139

40028, 39957, 71 40078, 937, 39141

40030, 39957, 73 40080, 4637, 35443

40032, 39091, 941 40082, 941, 39141

40034, 35443, 4591 40084, 127, 39957

40036, 39957, 79 40086, 4643, 35443

40038, 39091, 947 40088, 131, 39957

40040, 39957, 83 40090, 541, 39549

40042, 23139, 16903 40092, 4649, 35443

40044, 39091, 953 40094, 137, 39957

40046, 39957, 89 40096, 139, 39957

40048, 39549, 499 40098, 31991, 8107

40100, 1009, 39091

G = 10³⁰⁰⁰

n - G b n - b - G

100000, 36529, 63471

100002, 77069, 22933

100004, 22717, 77287

100006, 181873, -81867

100008, 12239, 87769

100010, 4547, 95463

100012, 4549, 95463

100014, 22727, 77287

100016, 59497, 40519

100018, 24847, 75171

100020, 12251, 87769

100022, 12253, 87769

100024, 4561, 95463

100026, 22739, 77287

100028, 22741, 77287

100030, 4567, 95463

100032, 12263, 87769

100034, 36563, 63471

100036, 42649, 57387

100038, 12269, 87769

100040, 23143, 76897

100042, 36571, 63471

100044, 43973, 56071

100046, 4583, 95463

100048, 24877, 75171

100050, 12281, 87769

G = 10⁵⁰⁰⁰

  n- G b n - b - G n - G b n - b - G n - G b n - b - G

100000, 31147, 68853 100050, 12611, 87439 100100, 31247, 68853

100002, 309371, -209369 100052, 12613, 87439 100102, 31249, 68853

100104, 105071, -4967

100106, 13649, 86457

  100004, 31151, 68853 100054, 13597, 86457 100108, 640669, -540561

100006, 31153, 68853 100056, 105023, -4967 100110, 12671, 87439

100008, 12569, 87439 100058, 12619, 87439 100112, 31259, 68853

100114, 87991, 12123

100116, 122033, -21917

100118, 18379, 81739

  100010, 13553, 86457 100060, 54151, 45909

100012, 31159, 68853 100062, 108971, -8909

100014, 108923, -8909 100064, 103091, -3027

100016, 12577, 87439 100066, 87943, 12123

100018, 592237, -492219 100068, 18329, 81739

100020, 104987, -4967 100070, 13613, 86457

100022, 12583, 87439 100072, 31219, 68853

100024, 13567, 86457 100074, 264881, -

100026, 18287, 81739100076, 12637, 87439

100028, 12589, 87439 100078, 107971, -7893

100030, 31177, 68853 100080, 12641, 87439

100032, 61871, 38161 100082, 76913, 23169

100034, 13577, 86457 100084, 13627, 86457

 100036, 31183, 68853 100086, 12647, 87439 100038, 108947, -8909 10038, 108947, -8909 100088, 61927,

    38161

 100040, 12601, 87439 100090, 13633, 86457

100042, 31189, 68853 100092, 12653, 87439

100044, 457091, -357047 100094, 61933, 38161

100046, 18307, 81739 100096, 87973, 12123

100048, 13591, 86457 100098, 12659, 87439

100120, 31267, 68853

100122, 61961, 38161

100124, 31271, 68853

100126, 13669, 86457

100128, 12689, 87439

100130, 31277, 68853

100132, 76963, 23169

100134, 122051, -21917

100136, 12697, 87439

100138, 13681, 86457

100140, 18401, 81739

100142, 12703, 87439

100144, 13687, 86457

100146, 152993, -52847

100148, 13691, 86457

100150, 13693, 86457

   1000000, 35509, 964491

   1000002, 113, 999889

   1000004, 69193, 930811

   1000006, 95233, 904773

   1000008, 69197, 930811

   1000010, 31873, 968137

   1000012, 35521, 964491

   1000014, 69203, 930811

   1000016, 127, 999889

   1000018, 35527, 964491

   1000020, 131, 999889

 Maple program corrected and improved, (see Sainty [37]).

18. Perspectives and Generalizations

18.1.

Other Goldbach sequences (${\mathrm{G}\mathrm{\text{'}}}_{2\mathrm{n}}$) independent of $\left({\mathrm{G}}_{2\mathrm{n}}\right)$ may be studied using the increasing sequences of primes (${\mathrm{W}\mathrm{\text{'}}}_{2\mathrm{n}})$ defined by

For any integer n $\ge 3$

$${ W\text{'}}_{2n}=\mathrm{Sup} (p \mathsf{ϵ}\mathcal{P}: p\le f(n\left)\right) \left(\mathrm{18.1.1}\right)$$

f is a function defined on the interval J = [3; +$\mathrm{\infty }$[ and satisfying the following conditions

● f is strictly increasing on the interval J

● f (3) = 3 and $\underset{x\to +\infty }{\lim }f\left(x\right)$ = +$\mathrm{\infty }$

$$\text{●} \forall x \mathsf{ϵ} I f\left(x\right)\le 2x-3$$

For example, one of the following functions defined on J can be selected.

■ f: x $\to$a x + 3 - 3a (a$\in \mathbb{R}$: 0 < a$\le 2)$

■ g: x$\to$[ 4$\sqrt{3x}-$9 ] ([x] is the integer part of the real x)

■ h: x $\to 6 \mathrm{l}\mathrm{n}\left( {\displaystyle \frac{x}{3}} \right)$ + 3

18.2.

Using this method it would be interesting to study the Schnirelmann density [39] of primes 3, 5, 7, 11 ,......... .. in the sequence (${\mathrm{U}}_{2\mathrm{n}}$) on variable intervals and the Caesaro sums of ${\mathrm{U}}_{2\mathrm{n}}$ E.D.G.’s with a view to more efficient programming for their calculation.

18.3.

It is possible to exceed the values shown in the table of 2n = ${10}^{1000}$ (many E.G.D have been calculated for values of 2n in the order of ${10}^{2000},{10}^{5000}$ and G.D. in the order of ${10}^{10000}$ Sainty [37]), by perfecting this algorithm, exploiting the fact that one of Goldbach's decomponents can be chosen equal to 4p + 3, (G.D. are primes of the form

6m + 1 or 6m + 5 and can be expressed more precisely using primes of the form 30m + r:

r ∈ [1,7,11,13,17,19,23,29], see Table mod 30, Lemma 11), by using De Pocklington Theorem [6,34,36], Primality tests [37], Cipolla-Axler-Dusart type functions and improvment of primes frames [2,8,12,13,37] via a new Prime number Theorem to better identify the terms of (${ G}_{2n }),\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{e}$ $\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{w}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}$ C++, or Assembleur compilation.

18.4 G.D.’s of order 2n = ${10}^{10000}$ can be determined more quickly by replacing the instruction b:= 2 by b:= trunc(c.b + d) and b: = nextprime(b) with

b: = nextprime(b + k(b, G)), where k(b, G) is a constant of around 150 for G = 10¹⁰⁰⁰ and is chosen randomly using the rand procedure or increases very slowly as a function of b and G. An increasing sequence of primes, $b_k$, can also be determined in stages by replacing the initial value b:= 2 by b:= trunc($k_0.$b - $k_1$.${\mathrm{l}\mathrm{n}}^{\mathrm{s}}$(n) - $k_2$) and by setting c: = trunc(a.${\mathrm{l}\mathrm{n}}^d$(b)),

1 ≤ d, s ≤ 2 and b: = b + c for each stage, followed by b: = nextprime(b) until the next stage, (see Sainty [37]); Note that for any even integer 2n large enough there exists G.D.

${p\text{'}}_n,{p\text{'}}_{n+1},{q\text{'}}_n,{q\text{'}}_{n+1}$│ ${ p\text{'}}_n$ + ${q\text{'}}_n$ = 2n and ${p\text{'}}_{n+1}$ + ${q\text{'}}_{n+1}$ = 2(n + 1) with

${p\text{'}}_{n+1 }$- ${p\text{'}}_n$ and ${q\text{'}}_{n+1}$ - ${q\text{'}}_n$ < k.${\mathrm{l}\mathrm{n}}^2$(n). It is therefore advisable to develop adaptive algorithms based on this model using A.I., as a function of the program's G parameter.

18.5 Diophantine equations and conjectures of the same nature ((3L) conjecture [9,21,23,26,27,44]) can be processed using similar reasoning and algorithms.

▄ To validate the (3L) conjecture we study the following sequences of primes (W$l_{2n}$), (${ Vl}_{2n}) \mathrm{a}\mathrm{n}\mathrm{d}({Ul}_{2n}) \mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d}\mathrm{b}\mathrm{y}$$\mathrm{F}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{y}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{e}\mathrm{r}n\ge 3 { Wl}_{2n}$= Sup (p $\mathsf{ϵ}\mathcal{P}: p\le n-1)$ (18.5.1)

● If ${Tl}_{2n }$= (2n + 1 - 2 ${Wl}_{2n})$ is a prime

then let

$${ Vl}_{2n}={Wl}_{2n}\mathrm{and} { Ul}_{2n}={Tl}_{2n} \left(\mathrm{18.5.2}\right)$$

● If ${Tl}_{2n}$ is a composite number

then there exists an integer k 1 $\le k\le n-3 │$

$${Ul}_{2\left(n-k\right)}+2k \in \mathcal{P} \left(\mathrm{18.5.3}\right)$$

then let

${ Vl}_{2n}$= ${Vl}_{2\left(n-k\right)}$ and ${Ul}_{2n}$=${ Ul}_{2\left(n-k\right)}$+ 2k (18.5.4)

▄ Using the same type of reasoning a generalization, the (BBG) conjecture of the following form can be validated

● Let K and Q be two odd integers prime to each other:

For any integer n │ 2n$\ge 3$(K + Q) there exist two primes ${Ub}_{2n}$ and ${Vb}_{2n}$ verifying

$$K. {Ub}_{2n}+Q.{Vb}_{2n}=2n \left(\mathrm{18.5.5}\right)$$

● Let K and Q be two integers of different parity prime to each other:

For any integer n │ 2n $\ge 3$(K + Q) there are two primes ${Ub}_{2n}$ and ${ Vb}_{2n}$ verifying

K${.Ub}_{2n}$ + Q. ${Vb}_{2n}$ = 2n + 1 (18.5.6)

18.6. Remark

● GOLDBACH (-):

$R_{2K}$ = Inf (p ∈ $\mathcal{P}$: p - 2K ∈ $\mathcal{P})$ and $Q_{2K}$= Inf (p ∈ $\mathcal{P}:$2K + p ∈ $\mathcal{P})$ = $R_{2K}$ - 2K

● GOLDBACH (+):

$V_{2K}$ = Sup (p ∈ $\mathcal{P}:$2K - p ∈ $\mathcal{P})$ and $U_{2K}$= Inf (p ∈ $\mathcal{P}$: 2K - p ∈ $\mathcal{P}$) = 2K - $V_{2K}$

    (It is possible to envisage symmetries in the Goldbach triangle).

● For any integer n greater than one there exists two integers $K_{Min}$ and $K_{Max}$ such that the G.D. of 2n are n - K and n + K │ $K_{Min}$ ≤ K ≤ $K_{Max}$.

18.7.

The sequences (${W\mathrm{q}}_{2n}$) generate all the G.D. and may enable us to better estimate the values of distribution function G of the Goldbach's Comet, probably of type:

      0.57 ${\displaystyle \frac{E}{{\mathrm{l}\mathrm{n}}^2\left(E\right)}}$ < G(E) < 3.62 ${\displaystyle \frac{E}{{\mathrm{l}\mathrm{n}}^2\left(E\right)}}$, (Vella-Chemla [46], Woon [49]).

        Average value of G(E) ≈ 1.62 ${\displaystyle \frac{E}{{\mathrm{l}\mathrm{n}}^2\left(E\right)}}$

19. Conclusions

19.1.

A recurrent and explicit Goldbach sequence ($G_{2n}$) = (${ U}_{2n }$; $V_{2n})$ verifying

$\forall n\in \mathbb{N}+2 U_{2n}, V_{2n}\in \mathcal{P} \mathrm{a}\mathrm{n}\mathrm{d}{ U}_{2n}+{ V}_{2n}$

has been developed using an simple and efficient "localised" algorithm. The Goldbach conjecture has been proved by strong recurrence (absurd and finite descent), and a reversible Goldbach tree uniquely associated with each even integer 2n: 2n ≥ 8 allows a better understanding of this conjecture. A relation (Proposition 10) is established between the fundamental theorem of arithmetic and the Goldbach conjecture (sum and product of primes), allowing fast computation of G.D. of very large even integers via a "localisation" of G.D.’s using a generalized Pocklington-type algorithm and further proof of Goldbach's binary conjecture via Euclidean divisions of 2n by primes and consistent increasing and decreasing sequences.

19.2.

The records of Silva [41] and Deshouillers, te Riele, Saouter [11] are beaten on a personal computer. Hundreds E.G.D. ${\mathrm{U}}_{2\mathrm{n}}$ and ${\mathrm{V}}_{2\mathrm{n}}$ are obtained for values around

2n =${ 10}^{1000 }$, twenty-six around 2n =${ 10}^{2000 },$seventy-five around 2n =${ 10}^{5000 }$ and ten G.D. around 2n = ${ 10}^{10000 }$ for a computation time of less than three hours (see Sainty [37]).

19.3.

For a given integer n ≥ 49 the evaluation of the terms $U_{2n}$ and $V_{2n}$ does not require the computing of all previous terms $U_{2k}$ and $V_{2k}$ │ 1 $\le$k < n - 1; we will only consider those that verify:

$U_{2k}\le 5.{\mathrm{l}\mathrm{n}}^{1.3}\left(2n\right)\mathrm{and} 2n- 5.{\mathrm{l}\mathrm{n}}^{1.3}\left(2n\right) \le V_{2k }$ ≤ 2n (on average) (19.3.1)

This property allows any E.G.D $U_{2n}$ and $V_{2n}$ to be calculated quite quickly, the upper limit being defined by the scientific software and the computer's ability to determine the largest prime preceding 2n - 2 (next or prevprime(2n - 2) function).

19.4.

Therefore the (BBG), the (3L) and the binary Goldbach(- /+) conjectures "Any even integer greater than three is the sum and difference of two primes" are true.

In fact these two conjectures are intertwined.