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.

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

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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.

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

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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.

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Modular arithmetic.

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Diophantine approximations and equations.

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Arithmetic and algebraic functions.

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Diophantine and number geometry.

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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. .

(2.1)

Let $𝒫$ be the infinite set of positive primes p_k (called simply primes)

(2.3)

(p₁ = 2 ; p₂ = 3 ; p₃ = 5 ; p₄ = 7 ; p₅ = 11 ; p₆ = 13 ; .........)

$$\mathrm{For} \mathrm{any} \mathrm{non}\text{-}\mathrm{zero} \mathrm{integer}\mathrm{ }K \mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }{\mathcal{P}}_K=\{p\in \mathcal{P} :\mathrm{ }p\le 2K\}$$

(2.4)

Writing the large numbers calculated in Appendix 14 is simplified by defining the following constants:

M = 10⁹ ; R = 4.10⁸ ; G = 10¹⁰⁰ ; S = 10⁵⁰⁰ ; T = 10 ¹⁰⁰⁰

(2.5)

$$p_k\#=\prod _1^k{p}_j \mathrm{i}\mathrm{s}\mathrm{ }\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{ }\mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }{p}_k .$$

(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 ; +∞[ 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

∀ n ∈ ℕ + 3     W_2n = Sup ( p ∈ $𝒫$ : p ≤ 2n - 3 )

(2.11)

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

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

(2.12)

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

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

(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

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

(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

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

(2.15)

and

$$p_{k+1}-p_k\le 20.\mathrm{l}\mathrm{n}\left(p_k\right) (on average)$$

(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

$$\forall n \in \mathbb{N}│4\le 2n \le 4.1018 U_{2n}+V_{2n}=2n$$

(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 = 10⁵⁰⁰ , 10^1000, 10⁵⁰⁰⁰ and G.D. around 10¹⁰⁰⁰⁰ (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.

• Chen conjecture : For any integer K ≥ 1 there are infinitely many pairs of primes with a difference equal to 2K.
• De Polignac conjecture : Same as Chen, but with consecutive pairs of primes.
• 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,

U₄ =2  and  V₄ =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 ∈ $𝒫$ : k ∈ ℕ ) , which saves a lot of computing time when programming ).

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

∀ K ∈ ℕ*  ∃ p , q ∈  │  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 $𝒫$_Chen(K) be the following property

" ∀ K ∈ ℕ* ∃ p, q ∈ $𝒫$ │ p - q = 2K  3 ≤ q ≤ 2K and 2K + 3 ≤ p ≤ 4K "

(4.2)

►

$𝒫$_Chen(2) is true : 7 - 3 = 4  q = 3 ≤ 4 and p = 7 ≤ 4 x 2 = 8

►

Let’s show

∀ M ∈ ℕ │  2 ≤ M ≤ K   then   $𝒫$_Chen(M) ⟹ $𝒫$_Chen(K + 1)

We reason through the absurd

Let p , q ∈ $𝒫$_K │ p ≥ q

∀ P , Q ∈ $𝒫$ │ P ≥ Q ∃ 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) ; 2h and 2m are any gaps between primes} contains all even integers between 2 and 2K (according to the recurrence hypothesis on $𝒫$_Chen(K)).

However the strong recurrence hypothesis asserts that

∀ M ∈ ℕ │ M ≤ K ∃ p, q ∈ $𝒫$  │   p - q = 2M

(4.5)

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

this contradicts (4.4).

So

∃ h, m ∈ ℕ │         P - Q = p + 2h - q - 2m = 2(K + 1)

(4.6)

knowing

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

Thus validating the heredity of property $𝒫$_Chen(K).

The property $𝒫$_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 = Inf(p ∈ $𝒫$ : p - 2K ∈ $𝒫$ and Q_2K = Inf (p ∈ $𝒫$ : 2K + p ∈ $𝒫$) = R_2K – 2K

(5.1)

They are defined for any integer K ∈ ℕ*

(5.2)

and satisfy

lim R_2K = +∞

(5.3)

∀ K ∈ ℕ*  R_2K, Q_2K ∈ $𝒫$  and  R_2K - Q_2K = 2K

(5.4)

∀ K ∈ ℕ* | 2 ≤ K ≤ 16   3 ≤ Q_2K ≤ 2K  and  2K + 3 ≤ R_2K ≤ 4K

(5.5)

For any integer K large enough

3 ≤ Q_2K ≤ (2K)^0.525 and 2K + 3 ≤ R_2K ≤ 2K + (2K)^0.525

(5.6)

Proof.

(5.1) ; (5.2) : According to the previous Theorem 4, the sequences (R_2K) and (Q_2K) are defined by strong recurrence (finite descent).

(5.3) :           R_2K ≥ 2K ⟹ lim R_2K = +∞

(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∈ℕ satisfying:

T1(K) = 2K + 3 (T1(K) ; 2K) → (5;2) ; (7 4) ; (11;8) ; (13;10) ; (17;14) ; (19;16) ; (23;20) ; (29;26) ; (29;28);.. T2(K) = 2K + 5 (T2(K) ; 2K) → (7;2) ; (11;6) ; (13;8) ; (17;12) ; (19;14) ; (23;18) ; (29;24) ; (31;26) ; (37;32)................................ T3(K) = 2K + 7 (T3(K) ; 2K) → (11;4) ; (13;6) ; (17;10) ; (19;12) ; (23;16) ; (29;22) ; (31;24) ; (37;30)........... T4(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).......................... ...................................... Tr(K) = 2K + qr (K ∈ ℕ* : Tr(K) and qr are primes) (see Appendix 16)

For any integer K satisfying (2K)^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 ${(2K )}^{0.525}$ ≤ $q_r$ .

(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 < ${( q_r)}^a$ 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)

$$q_{r+1}\le q_r+q_r^{0.525}$$

(5.6)

It is assumed that M │

$$T_{r+1}\left(K\right)-q_{r+1}\ne 2M \mathrm{k}\mathrm{n}\mathrm{o}\mathrm{w}\mathrm{i}\mathrm{n}\mathrm{g} 2M<{\left(q_{r+1}\right)}^{c_p}$$

$$\forall T_m\left(R\right),q_m\in \mathcal{P} \exists h,s\in \mathbb{N} │ T_{r+1}\left(K\right)=T_m\left(R\right)+2h \mathrm{a}\mathrm{n}\mathrm{d} q_{r+1}=q_m+2s$$

(5.7)

then

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

(5.8)

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 │

$$2n-{Wq}_{2s}\in \mathcal{P}$$

(6.1)

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

$$\forall n\in \mathbb{N}n\ge n_q\hspace{1em}\hspace{1em}\hspace{1em}{Zq}_{2n}=\mathrm{I}\mathrm{n}\mathrm{f} (2n-{Wq}_{2k}\in \mathcal{P}:k\in \mathbb{N})$$

(6.2)

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

$$\mathrm{F}or any integer n\ge n_0\hspace{1em}\hspace{1em}\hspace{1em}{Zq}_{2n}\le {\left(2n\right)}^{0.525}$$

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

( see Baker, Harman [4] )

else

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

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]:

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

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

(6.5)

and

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

(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

$${\mathrm{l}\mathrm{i}\mathrm{m} V}_{2n}=+\mathrm{\infty }.$$

(7.1)

$$\forall n\in \mathbb{N}+2 V_{2n} \mathrm{i}\mathrm{s} \mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d} \mathrm{a}\mathrm{s} \mathrm{a} \mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{o}\mathrm{f} W_{2n}=\mathrm{S}\mathrm{u}\mathrm{p}(p\in \mathcal{P}:p\le 2n–3)$$

(7.2)

$$\left(W_{2n}\right) \mathrm{i}\mathrm{s} \mathrm{a}\mathrm{n} \mathrm{i}\mathrm{n}\mathrm{c}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{s}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e} \mathrm{o}\mathrm{f} \mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{s} \mathrm{a}\mathrm{l}\mathrm{l} \mathrm{o}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{m} \mathrm{e}\mathrm{x}\mathrm{c}\mathrm{e}\mathrm{p}\mathrm{t} p_1=2$$

(7.3)

$$\mathrm{l}\mathrm{i}\mathrm{m} W_{2n}=+\infty$$

(7.4)

(${ U}_{2n }$) 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

$$V_{2n}=W_{2n} \mathrm{a}\mathrm{n}\mathrm{d} U_{2n}=2n-W_{2n}$$

(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

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

(7.7)

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

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

So by setting

$$V_{2n}=V_{2(n-k)} \mathrm{a}\mathrm{n}\mathrm{d} U_{2n}=U_{2(n-k)}+2k$$

(7.8)

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

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

(7.10)

To this end, .

►

$𝒫$_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

$$2K+U_{2(n+1-k)}=2(K+M_r)+U_{2(r+1-k)}$$

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

(7.11)

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 │

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

(7.12)

then there exists an integer K │

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

(7.13)

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)

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

For any integer n ≥ $2$

$$U_{2n}, V_{2n} \mathsf{ϵ} \mathcal{P} \mathrm{a}\mathrm{n}\mathrm{d} U_{2n}+V_{2n}=2n$$

(8.1)

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

$$\left(\mathrm{ii}\right) \mathrm{A}n \mathit{\text{algorithm can be used to explicitly compute any E.G.D.}} { U}_{2n} \mathrm{and} V_{2n}$$

(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 ".

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

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

$$V_{2(n+1)}=W_{2(n+1)} \mathrm{a}\mathrm{n}\mathrm{d} U_{2(n+1)}=2(n+1)-W_{2(n+1)}$$

(8.3)

• 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

$$U_{2(n+1-k)}, V_{2(n+1-k)} \mathrm{a}\mathrm{n}\mathrm{d} U_{2(n+1-k)}+2k\in \mathcal{P}$$

(8.4)

$$U_{2\left(n+1-k\right)}+V_{2\left(n+1-k\right)}=2(n+1-k)$$

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 │

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

(8.5)

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} │

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

(8.6)

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

So by setting

$$U_{2n}=U_{2\left(n-k_n\right)}+2k_n \mathrm{a}\mathrm{n}\mathrm{d} V_{2n}=V_{2\left(n-k_n\right)}\in \mathcal{P}$$

(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

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

(8.9)

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

(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

$$\forall n\in \mathbb{N}+2 \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} \mathrm{a}\mathrm{r}\mathrm{e} \mathrm{t}\mathrm{w}\mathrm{o} \mathrm{p}\mathrm{r}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{s} U_{2n} \mathrm{a}\mathrm{n}\mathrm{d} V_{2n} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h} \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{r} \mathrm{s}\mathrm{u}\mathrm{m} \mathrm{i}\mathrm{s} 2n : U_{2n}+V_{2n}=2n$$

▄ ALGORITHM:

For any integer n $\ge 3$

• If ( 2n - $W_{2n }$) is a prime

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

$$V_{2n}=W_{2n} \mathrm{a}\mathrm{n}\mathrm{d} U_{2n}=2n-W_{2n}$$

(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

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

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

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

(8.13)

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

$$U_{2n}\le (2{n)}^{0.525}$$

(9.1)

and

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

(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

$$P_{bhip}\left(n\right): { \mathrm{U}}_{2\mathrm{n}}\text{≤}{\left(2\mathrm{n}\right)}^{0.525} .$$

(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

$$V_{2(n+1)}=W_{2(\mathrm{n}+1)} \mathrm{a}\mathrm{n}\mathrm{d} U_{2(n+1) }=2(n+1)-W_{2(n+1)}$$

(9.4)

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

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

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

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

• 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]

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

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

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

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

.then

$$\begin{array}{l}2n\approx p’( 1+ln({\displaystyle \frac{2\mathrm{n}}{p^{\text{'}}}}))\\ 2n\approx p’(1+ln ({\displaystyle \frac{2\mathrm{n}}{\mathrm{P} \text{"}}}))\approx p’(1+ln({\displaystyle \frac{2\mathrm{n}}{\mathrm{p}\text{’}+/- \mathrm{a}}}))\\ 2n\approx p’(1+ln(2n/((p’(1+/-{\displaystyle \frac{ \mathrm{a}}{\mathrm{p}\text{'}}})))\\ 2n\approx p’(1+ln(({\displaystyle \frac{2\mathrm{n}}{p^{\text{'}}}}).(1/1+/-{\displaystyle \frac{ \mathrm{a}}{\mathrm{p}\text{'}}}))))\\ 2n\approx p’(1+ln({\displaystyle \frac{2\mathrm{n}}{p\text{'}}})+ln(1/(1+/-{\displaystyle \frac{ \mathrm{a}}{\mathrm{p}\text{'}}})))\\ 2n\approx p’(1+ln({\displaystyle \frac{2\mathrm{n}}{p\text{'}}}))-/+a\end{array}$$

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

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

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:

x₀ = Re(- (2n +/- a) / LambertW(-1,- (2n +/- a) / (2n.e)))

and

x₁ = 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

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

then

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

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:

$$p^{’}=m\text{.}q\text{’}+r \mathit{0}\text{<}r\text{<}{p}^{’} r\wedge p^{’}=\mathit{1} r\wedge m=\mathit{1}$$

We deduce that q’ =${\displaystyle \frac{(2\mathrm{n} - \mathrm{r} )}{(\mathrm{m}+1)}}$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 by identifying uniqueness, coincidence and consistency using euclidean division of

2n by p_k ∈ Ƥ p_k > n : 2n = p_k + R_k and

2n by q_r ∈ Ƥ p_k ≫ 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 ≫ q_r and p_k ≫ q_r, R_k,

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

(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 ≥ 3 it is easy to check

(W_2n) is a positive increasing sequence of primes

(11.1)

{W_2n : n ∈ IN + 3} ∪ {2} = $𝒫$

(11.2)

lim W_2n = +∞

(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 ≤ V_2n ≤ W_2n

(11.5)

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

(11.6)

lim V_2n = +∞

(11.7)

Proof.

(11.1) : For any integer n ≥ 2 $𝒫$_n ⊂ $𝒫$_n+1. Therefore, W_2n ≤ W_2(n+1). So the sequence (W_2n) is increasing.

(11.2) : Any prime except p₁ = 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 13.2 and the table in appendix 14.

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

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

So

$$V_{2n}\ge n$$

$$\mathrm{A}\mathrm{c}\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{d}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{t}\mathrm{o}\left(11.5\right)\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{n}\le V_{2n}\hspace{1em}\hspace{1em}\hspace{1em}⟹\hspace{1em}\hspace{1em}{U}_{2n}=2\mathrm{n}-V_{2n}\le 2\mathrm{n}-\mathrm{n}\le \mathrm{n}$$

(11.6)

$$V_{2n}\le W_{2n}⟹2n-W_{2n}\le 2n-V_{2n}=U_{2n}$$

(11.7)

By (11.5) for any integer n ≥ 2 :    n ≤ V_2n

$$\mathrm{l}\mathrm{i}\mathrm{m}{V}_{2n}=+\mathrm{\infty }.$$

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: 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: 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 → +∞)

 (13.2)

For any integer n ≥ 5000

$${ U}_{2n}\ll V_{2n}\hspace{1em}\hspace{1em}\hspace{1em}and\hspace{1em}\hspace{1em}\hspace{1em}lim \left({\displaystyle \frac{U_{2n}}{V_{2n}}}\right)=0$$

 (13.3)

(Cesaro sums of E.G.D. U_2n and V_2nhave interesting properties)

The smallest integer n │ U_2n ≠ 2n - W_2n is obtained for n = 49 and G₉₈ = (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 ≥ 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

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

 (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${\left[ln\right(g\left(x\right)\left)\right]}^c$   ( a,b > 0 ; c > 2) with

g : x $\to$ 0.7 [l${n\left(x\right)]}^{(c-{\displaystyle \frac{1}{x}} )}$  and  h : x $\to$ 0.7${ \left[ln\right(x\left)\right]}^{(2.2-{\displaystyle \frac{1}{x}} )}$ 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

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

(13.7)

(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 :
$U_{2n}$ = $U_{2\left(n-k_n\right)}$ + 2$k_n$ and $V_{2n}$ = $V_{2\left(n-k_n\right)}$	(14.1.2)
Assign to n the value n + 1 (n$⟵ n+1$and return to A)   End :   Outputs for 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 = ● ; $T_{2n}$ = ● ; ${V_{2n}-P= ● ; U}_{2n}$ = ● )

14.2. Program Written with Maxima Software for 2n Around 10¹⁰⁰⁰

c : 10**1000 ; for n :c + 40000 step 2 thru c + 40100 do

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

if primep(e )

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

else while test = 0 do ( 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 10¹⁰⁰⁰

	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 nextprime( ), if we can improve the gcd algorithm. RESULTS :           G = ${10}^{1000}$
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 = ${10}^{2000}$       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 = ${10}^{3000}$       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 = ${10}^{5000}$ 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, 81739     100076, 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 (${G\text{'}}_{2n}$) independent of $\left(G_{2n}\right)$ may be studied using the increasing sequences of primes ( ${W\text{'}}_{2n} )$ defined by

For any integer n $\ge 3$

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

(18.1.1)

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 $\underset{x\to +\infty }{\lim }f\left(x\right)$ = +$\mathrm{\infty }$
• $\forall x \mathsf{ϵ} I f\left(x\right)\le 2x-3$

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 ( $U_{2n }$) on variable intervals and the Caesaro sums of $U_{2n}$ 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$l_{2n}$ ), (${ Vl}_{2n} ) \mathrm{a}\mathrm{n}\mathrm{d}$(${Ul}_{2n} ) \mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{d} \mathrm{b}\mathrm{y}$

$$\mathrm{F}\mathrm{o}\mathrm{r} \mathrm{a}\mathrm{n}\mathrm{y} \mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{e}\mathrm{r} n\ge 3 {Wl}_{2n}=\mathrm{S}\mathrm{u}\mathrm{p}(\mathrm{p}\mathsf{ϵ}\mathcal{P}:p\le n-1)$$

(18.5.1)

• If Tl_2n = (2n + 1 - 2 Wl_2n) is a prime

then let

$${ Vl}_{2n}={Wl}_{2n} \mathrm{a}\mathrm{n}\mathrm{d} { Ul}_{2n}={Tl}_{2n}$$

(18.5.2)

• If Tl_2n is a composite number

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

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

(18.5.3)

then let

$${ Vl}_{2n}={Vl}_{2\left(n-k\right)} \mathrm{a}\mathrm{n}\mathrm{d} {Ul}_{2n}={ Ul}_{2\left(n-k\right)}+2k$$

(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

$${K . Ub}_{2n}+Q. {Vb}_{2n}=2n$$

(18.5.5)

• 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 (-) :

$$R_{2K}=\mathrm{I}\mathrm{n}\mathrm{f}\left(p\in \mathcal{P}:p-2K\in \mathcal{P} \right) \mathrm{a}\mathrm{n}\mathrm{d} \underset{\_}{Q_{2K}}=\mathrm{I}\mathrm{n}\mathrm{f}(p \in \mathcal{P} :2K+p \in \mathcal{P} )=R_{2K}-2K$$

• GOLDBACH (+) :

$$V_{2K}=\mathrm{S}\mathrm{u}\mathrm{p}(p\in \mathcal{P} :2K-p\in \mathcal{P} ) \mathrm{a}\mathrm{n}\mathrm{d} \underset{\_}{U_{2K}}=\mathrm{I}\mathrm{n}\mathrm{f}(p\in \mathcal{P} :2K-p\in \mathcal{P})=2K-V_{2K}$$

$$(\mathrm{I}\mathrm{t} \mathrm{i}\mathrm{s} \mathrm{p}\mathrm{o}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{e} \mathrm{t}\mathrm{o} \mathrm{e}\mathrm{n}\mathrm{v}\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{g}\mathrm{e} \mathrm{s}\mathrm{y}\mathrm{m}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{s} \mathrm{i}\mathrm{n} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{G}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{b}\mathrm{a}\mathrm{c}\mathrm{h} \mathrm{t}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}).$$

• For any integer n greater than one, there exists two integers K_Min and K_Max such that the G.D. f 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 :

$$0.57.{\displaystyle \frac{E}{{\mathrm{l}\mathrm{n}}^2(E)}}<G(E)<3.62.{\displaystyle \frac{E}{{\mathrm{l}\mathrm{n}}^2(E)}} ,(\mathrm{V}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{a}-\mathrm{C}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{l}\mathrm{a} [46],\mathrm{W}\mathrm{o}\mathrm{o}\mathrm{n} [49]).$$

$$\mathrm{A}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e} \mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e} \mathrm{o}\mathrm{f} G(E)\approx 1.62.{\displaystyle \frac{E}{{\mathrm{l}\mathrm{n}}^2(E)}}$$

19. Conclusions

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

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

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. $U_{2n}$ and $V_{2n }$ 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:

$$U_{2k}\le 5.{\mathrm{l}\mathrm{n}}^{1.3}\left(2n\right) \mathrm{a}\mathrm{n}\mathrm{d} 2n -5.{\mathrm{l}\mathrm{n}}^{1.3}\left(2n\right)\le V_{2k }\le 2n$$

(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.

Framing and mean value of the Goldbach comet by functions of the type f : x -> a.x /${\mathbf{l}\mathbf{n}}^2$(x) , ( via AI CLAUDE : to be specified ).