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, see Filhoa, Jaimea, de Oliveira Gouveaa, Keller-Füchter, [16]).

● Analytical number theory .

$-$ Distribution of primes : Prime Number Theorem, the Riemann hypothesis, (see 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 ; $p_3\mathrm{ }$= 5 ; $p_{4\mathrm{ }}$= 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 14 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)

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

ln(x) denotes the neperian logarithm of the real x > 0      (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 ; + $\mathrm{\infty }\mathrm{ }$ [ 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 ∈ $\mathcal{P}$   : p ≤ 2n - 3) (2.11) 

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

∀ n ∈ ℕ n ≥ ${\displaystyle \frac{(q+3)}{2}}$   ${Wq}_{2n}$ = Sup (p ∈ $\mathcal{P}$ : p ≤ 2n - q)         (2.12)

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

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

${\mathrm{ }U}_{2n}$ and $V_{2n}$ are also known as "Goldbach partitions or Goldbach 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 $\exists$ $k_{\epsilon }$ ∈${\mathbb{ }\mathbb{N}}^{\mathrm{*}}$ │ ∀ k ∈ ℕ    k ≥ $k_{\epsilon }$     $p_{k+1}$ - $p_k$ < $\epsilon .p_k^{0.525}$ (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

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

and

$p_{k+1}$ - $p_k$ ≤ 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

∀ n ∈ ℕ │ 4 ≤ 2n ≤ 4.10¹⁸    $U_{2n}$ + $V_{2n}$ = 2n    (3.1)

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

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

and

${10}^{10k}$ ≤ 2n ≤ ${10}^{10k}$ + ${10}^9$:     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 ${10}^{10000}$ (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 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)

∀ K ∈ ℕ* $\exists$ p , q ∈ $\mathcal{P}$ │ 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

∀ M ∈ ℕ │ 2 ≤ M ≤ K    then    ${\mathcal{P}}_{Chen}$(M) $⟹$ ${\mathcal{P}}_{Chen}$(K + 1)

We reason through the absurd

Let p , q ∈ ${\mathcal{P}}_K$ │ p ≥ q

∀ P , Q ∈ $\mathcal{P}$ │ P ≥ 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) ; 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

∀ M ∈ ℕ │ M ≤ K $\exists$p, q ∈ $\mathcal{P}$    │    p - q = 2M    (4.5)

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

t$\mathrm{h}\mathrm{i}\mathrm{s}\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{c}\mathrm{t}\mathrm{s}$ (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

   $R_{2K}$ = Inf (p ∈ $\mathcal{P}$: p - 2K ∈ $\mathcal{P})$ and Q_2K = Inf (p ∈ $\mathcal{P}$: 2K + p ∈ $\mathcal{P}$) = $R_{2K}$ - 2K (5.1)

They are defined for any integer K ∈ $\mathbb{N}$*       (5.2)

and satisfy

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

${\forall K\in {\mathbb{N}}^{\mathrm{*}}R}_{2K}$, Q_2K ∈ $\mathcal{P}$    and    $R_{2K}$ - $Q_{2K}$ = 2K       (5.4)

$\forall K \in {\mathbb{N}}^{*}$│ 2 ≤ K ≤ 16     3 ≤ Q_2K ≤ 2K   and   2K + 3 ≤ $R_{2K}$ ≤ 4K     (5.5)

For any integer K large enough

3 ≤ Q_2K ≤ ${\left(2K\right)}^{0.525}$   and   2K + 3 ≤ $R_{2K}$ ≤ 2K + ${\left(2K\right)}^{0.525}$     (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 $R_{2K}$ = +$\mathrm{\infty }$

(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)

$q_{r+1}$ ≤ $q_r$ + $q_r^{0.525}$    (5.6)

It is assumed that M │

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

∀ $T_m$(R), $q_m$ ∈ $\mathcal{P}\exists$h, s ∈ ℕ │ $T_{r+1}$(K) = $T_m$(R) + 2h and $q_{r+1}$ = $q_m$ + 2s (5.7)

then

 $T_m$(R) - $q_m$ ≠ 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) , $q_m$ 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 $n_0$,$n_q$ │

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

2n - ${Wq}_{2s}$ ∈ $\mathcal{P}$     (6.1)

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 ≥ $n_0$      ${Zq}_{2n}$ ≤ ${(2n-q)}^{0.525}$    (6.3)

${Zq}_{2n}$ ≤ o ${\left(2n\right)}^{0.525}$     (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 - 2$M_r$ - ${Wr}_{2k}$ = 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) - ${Wq}_{2(n+1)}$ ∈ $\mathcal{P}$ the proof is validated 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

(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] :

 $\exists m_0$ ∈ ℕ │ $\forall$ n ∈ ℕ : n ≥ $m$ ;

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)

∀ n ∈ ℕ + 2 $V_{2n}$is defined as a function 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 $W_{2n}=+\mathrm{\infty }$     (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

$V_{2n}$ = $W_{2n}$ 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 ] ∈ $\mathcal{P}$      (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

$V_{2n}$ = $V_{2(n-k)}$ and $U_{2n}$ = $U_{2(n-k)}$+ 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

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

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

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 + $U_{2(r+1+M_r-P)}$ ∈ $\mathcal{P}$      (7.12)

then there exists an integer K │

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

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 ".

(∀ p ∈ ℕ │ 2 ≤ p ≤ n    $U_{2p}$ ,$V_{2p}\in \mathcal{P}$   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

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

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

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

$U_{2(n+1-k)}$, $V_{2(n+1-k)}$ and $U_{2(n+1-k)}$ + 2k ∈ Ƥ     (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 ≤ $U_{2(n-q)}$ ≤ n .

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

$R_{2n}$ = $U_{2(n-k)}$ + 2k ∈ Ƥ (8.5)

following the Bertrand principle and 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) . 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 :

$P_G$(n) $⟹P_G$(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 ≤ $U_{2(n-q)}$ ≤ n .

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

$R_{2n}$ = $U_{2(n-k)}$ + 2k ∈ Ƥ   (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) . 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 return and absurd" directly on the set { $U_{2k}$ : k ≤ n } │

$R_{2n}$ = $U_{2(n-k)}$ + 2k ∈ Ƥ     (8.14)

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

So

$U_{2n}$ = $U_{2\left(n-k_n\right)}$ + 2$k_n$ and $V_{2n}$ = $V_{2\left(n-k_n\right)}$ ∈ Ƥ     (8.15)

(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 -$k_n$) (8.16)

By adding the term $2k_n$ to each member of the equality (8.16) it follows

$U_{2\left(n-k_n\right)}$ + 2$k_n$ + $V_{2\left(n-k_n\right)}$= 2(n - $k_n$) + 2$k_n$      (8.17)

$\iff$ [ $U_{2\left(n-k_n\right)}$ + 2$k_n$] + $V_{2\left(n-k_n\right)}$ = 2n (8.18)

 $\iff$    $U_{2n}$ + $V_{2n}$ = 2n    (8.19)

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 ((2${n)}^{0.525}$)      (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 :

$P_{bhip}$(n) $⟹$ .$P_{bhip}$(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

$V_{2(n+1)}$ = $W_{2(\mathrm{n}+1)}$ and $U_{2(n+1) }$= 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(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 ${.\left[2\right(n+1\left)\right]}^{0.525}$

$⟹$   $U_{2(n+1)}$ ≤ 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 ∈ ${\mathbb{N}}^{\mathrm{*}}$│ $U_{2(n+1)}$ = $U_{2(n+1-p)}$ + 2p (9.5)

According to [4,5,24]

$U_{2(n+1)}$ = 2p + $U_{2(n+1-p)}$= 2p + 2(n + 1 - 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 [8,2].

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 + ${\displaystyle \frac{\mathrm{q}’}{\mathrm{p}’}}$))

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

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

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

.then

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

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

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

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

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

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

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 :

$x_0$ = 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 (${p\mathrm{’}}_s$ , ${q\mathrm{’}}_s$,) of G.D. of 2s such that the product sequence $M_s$ = ${p\mathrm{’}}_s$.${q\mathrm{’}}_s$ = s² - k² is almost increasing (the variations of the geometric mean almost follow those of the arithmetic mean; indeed, if

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

then

${p\mathrm{’}}_{m+1}$.${q\mathrm{’}}_{m+1}$ - ${p\mathrm{’}}_m$.${q\mathrm{’}}_m$ = (${p\mathrm{’}}_{m+1}$- ${p\mathrm{’}}_m$).${q\mathrm{’}}_{m+1}$ + ${p\mathrm{’}}_m$(${q\mathrm{’}}_{m+1}$ - ${q\mathrm{’}}_m$) ≥ 0).

If we choose : ${q\mathrm{’}}_m=$ ${q\mathrm{’}}_{m+1}$ , 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 :

${p\mathrm{’}}_1$ ${\le p\mathrm{’}}_{2,}{p\mathrm{’}}_1$ ${\approx p\mathrm{’}}_2\gg$ ${q\mathrm{’}}_1,{q\mathrm{’}}_2$ and ${q\mathrm{’}}_2$ ≥ ${q\mathrm{’}}_1$,(p’. q’ = n² - k²); we then define admissible bounds for k from a = ${p\mathrm{’}}_1$.${q\mathrm{’}}_1$ and b = . ${p\mathrm{’}}_2$.${q\mathrm{’}}_2$ 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.q’ + r   0 < r < q’   r$\wedge$q’ = 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 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 $p_{k }$ $\gg$ $q_r$ 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 $p_{k }$ $\gg$ $q_r$, $R_k$,

[${\displaystyle \frac{2\mathrm{n}}{\mathrm{q}}}$] = 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 k0 and r0 such that :

  $p_{k0}$ = $D_{r0 }$ and $R_{k0}$ = $q_{r0}$.

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 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

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

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

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

3. 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 $\to$ +$\mathrm{\infty }$)      (13.2)

For any integer n ≥ 5000

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

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 [${\mathrm{l}\mathrm{n}\left(2p\right)]}^{(2.2 - {\displaystyle \frac{1}{p}})}$    (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 [29,31,30] .

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 :

      $U_{2n}$ = $U_{2\left(n-k_n\right)}$ + 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 ${\mathsf{10}}^{\mathsf{4}:}:$

  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 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 ${10}^{1000}$

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 ${p’}_n$ and a successor ${p’}_{n+1}$such that

(E):│${p^{\prime }}_{n+1}$- ${p^{\prime }}_n$│< 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 = ${\mathsf{10}}^{\mathsf{2000}}$

$$

$$

$$

   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 = ${\mathsf{10}}^{\mathsf{3000}}$

  

  

    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 = ${\mathsf{10}}^{\mathsf{5000}}$

  

  

  

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]).