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 [13], Hadamard [15], Hardy and Wright [16], Landau [22], Tchebychev [35] ). Numerous problems have been raised and conjectures made, the statements of which are often simple but very difficult to prove. These main components include

● Elementary arithmetic .

___ ˽Determination and properties of primes, operations on integers

( Basic operations, congruence, gcd, lcm, ………..).

___ Decomposition of integers into products or sums of primes

( Fundamental theorem of arithmetic, decomposition of large numbers, cryptography and Goldbach's conjecture).

● Analytical number theory .

___ Distribution of primes ( Prime Number Theorem, Hadamard [15], De la Vallée-Poussin [36], Littlewood [25] and Erdos [12], the Riemann hypothesis,.....).

___ Gaps between consecutive primes ( Bombieri,Davenport [3], Cramer [8], Baker,Harmann,Iwaniec, Pintz [4,5,20], Granville [14], Maynard [27], Tao [34], Shanks [30], Tchebychev [35] and Zhang [39] ).

● Algebraic, probabilistic, combinatorial and algorithmic number theories .

___ Modular arithmetic.

___ Diophantine approximations and equations

___ Arithmetic and algebraic functions.

___ Diophantine and number geometry.

2. Definitions Notations and Background

The integers n, k, p, q, r,……..... used in this article are always positive. (2.1)

The symbol ‘‘ / ‘’ means " in relation to". (2.2)

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

( $p_1$ = 2 ; $p_2$ = 3 ; $p_3$= 5 ; $p_4$= 7 ; $p_5$ = 11 ; $p_6$ = 13 ; ......... )

For any integer K ≥ 1 ${\mathcal{P}}_K$ = { p ∈ $\mathcal{P}$: p ≤ 2K } (2.4)

The writing of large numbers (see appendix 12) is simplified using the following constants

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

ln(x) denotes the neperian logarithm of the real x > 0 (2.6)

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

∀ n ∈ ℕ + 3 $W_{2n}$ = Sup( p ∈ $\mathcal{P}$ : p ≤ 2n - 3 ) (2.7)

Any sequence denoted by ($G_{2n}$) = ($U_{2n};V_{2n}$) verifying (2.8) is called a Goldbach sequence.

∀ n ∈ ℕ + 2 $U_{2n}$ ,$V_{2n}$ ∈ $\mathcal{P}$ and $U_{2n}$ + $V_{2n}$ = 2n (2.8)

$U_{2n}$ and $V_{2n}$ are also known as " Goldbach decomponents ".

Iwaniec,Pintz [20] 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 $\exists$ $k_{\epsilon }$ ∈${\mathbb{N}}^{\mathrm{*}}$ / ∀ k ∈ ℕ k ≥ $k_{\epsilon }$ $p_{k+1}$ - $p_k$ < $\epsilon .p_k^{0.525}$ (2.9)

The results obtained on the Cramer-Granville-Maier-Nicely conjecture [1,3,8,14,26,28] 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 ) (2.10)

3. Introduction

Chen [6], Hardy,Littlewood [17], Hegfollt,Platt [18], Ramaré,Saouter [29], Tao [34], Tchebychev [35] and Vinogradov [37] 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 [18] proved the weak Goldbach conjecture in 2013.

Silva,Herzog,Pardi [32] 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)

In previous research work there is no explicit construction of recurrent Goldbach sequences. In this article two sequences of primes are developed using a simple, efficient and « located » algorithm to compute for any integer n $\ge 3$ by successive iterations any term $U_{2n}$ and $V_{2n}$ .

Using Maxima scientific software on a personal computer Silva's record is broken and the values 2n = 10⁵⁰⁰ and even 2n = 10¹⁰⁰⁰ are reached. The binary Goldbach conjecture can be established on the same principle by recurrence by using the weak Chen or Goldbach( - ) conjecture ( Any even integer greater than three is the difference of two primes) demonstrated in Teorem 4.

● Remark.

1. Chen conjecture̲: 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 [39] demonstrates that the smallest even integer 2K verifying the conjecture is greater than 70 million.

In 2014, James Maynard [27] then Terence Tao [34] lowered this limit to 246.

We validate weak Chen or Goldbach( - ) conjecture by verifying directly in the prime number tables that all even gaps from 2 to 246 are possible between primes.

In addition, the Lagrange-Lemoine-Lévy conjectures [9,19,21,26,28,33,38] and its generalization called ‘’ Bezout-Goldbach « conjecture’ are validated.

Using case disjunction reasoning we construct two recurrent sequences of primes ($V_{2n}$) and ($U_{2n}$) according to the sequence ( $W_{2n}$) by the following process

For any integer n ≥ 2

$U_4$ = 2 and $V_4$= 2 (3.2)

Let n ∈ ℕ + 3

● 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 ($G_{2n}$).

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

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

p - q = 2K 3 ≤ q ≤ 2K and 3 + 2K ≤ p ≤ 4K if K ≥ 2 (4.1)

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 , then 2(K + 1) ( demonstration established by strong recurrence, by the asurd and feedback ).

All pairs of Goldbach( - ) decomponents obtained by this method for K between 2 and 123 are listed in the table in Appendix 13.

Proof . The proof is established by strong recurrence on K . 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 / M ≤ K then ${\mathcal{P}}_{Chen}$ ( M ) $⟹$ ${\mathcal{P}}_C$ ( K + 1 )

We reason through the absurd

∀ p , q ∈ ${\mathcal{P}}_K$ / p ≥ q ∀ h, m ∈ ℕ / p + 2h and q + 2m ∈ $\mathcal{P}$we assume that

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 .

However the strong recurrence hypothesis asserts that

∀ ∈ 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 + 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( - )’s conjecture is validated.

● Remark. Using the same method as in Theorem 4, we can demonstrate the following equivalent property by strong recurrence ::

For any integer n greater than 48

${\mathcal{P}}_{feedback}$ (n) : « There exists an integer K such that 2K + $U_{2(n-k)}\in \mathcal{P}"$(4.7)

To this end, ${\mathcal{P}}_{feedback}$ (49) is true and the heredity of the property ${\mathcal{P}}_{feedback}$ (n ) can be proved by the absurd and returning to the previous terms by noting that there is at least one integer ${\mathrm{M}}_{\mathrm{r}}$ /

2K + $U_{2(n+1-k)}$ = 2( K + $M_r$ ) + $U_{2(r+1-k)}$ = 2P + $U_{2(r+1+M_r-P)}$ (4.8)

by posing P = K + $M_r$ and $r+1+M_r$ ≤ n

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

2P + $U_{2(r+1+M_r-P)}$ ∈ $\mathcal{P}$ (4.9)

then there exists an integer K /

2K + $U_{2(n+1-k)}$ ∈ $\mathcal{P}$ (4.10)

5. Corollary

Let ${(R}_{2K}$) and ${(Q}_{2K}$) 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 ∈ $\mathbb{N}$* and satisfy

lim $R_{2K}$ = +$\mathrm{\infty }$ (5.2)

${\forall K\in {\mathbb{N}}^{\mathrm{*}}R}_{2K}$, $Q_{2K}$̲∈ $\mathcal{P}$and $R_{2K}$ - $Q_{2K}$ = 2K (5.3)

$\forall K\in {\mathbb{N}}^{*}$ / 2 ≤ K ≤ 16 3 ≤ $Q_{2K}$̲≤ 2K and 2K + 3 ≤ $R_{2K}$ ≤ 4K (5.4)

For any integer K ≥ 16

3 ≤ $Q_{2K}$̲≤ 2${\left(2K\right)}^{0.525}$ and 2K + 3 ≤ $R_{2K}$ ≤ 2K + ${\left(2K\right)}^{0.525}$ (5.5)

Proof.

(5.1) : According to the previous theorem, the sequences ( $R_{2K}$) and (Q_2k) are defined by

strong recurrence and finite descent.

(5.2) : $R_{2K}$ ≥ 2K ⇒ lim $R_{2K}$ = +$\mathrm{\infty }$(5.3) : By construction, these sequences thus verify $R_{2K}$ - $Q_{2K}$ = 2K

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

(5.5) : 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 (5.5) more easily.

We examine the following sequences of primes ( PQ( K )).

P3(K ) = 2K + 3

(P3(K ) ; 2K ) → (5;2);(7;4);(11;8);(13;10);(17;14);(19;16);(23;20);(29;26);(29;28);..........

P5(K ) = 2K + 5

(P5(K ) ; 2K ) → (7;2) ; (11;6) ; (13;8) ; (17;12) ; (19;14) ; (23;18) ; (29;24) ; (31;26) ; (37;32)................................

P7(K ) = 2K + 7

(P7(K ) ; 2K ) → (11;4) ; (13;6) ; (17;10) ; (19;12) ; (23;16) ; (29;22) ; (31;24) ; (37;30)...........

P11(K ) = 2K + 11

(P11(K ) ; 2K ) → (13;2) ; (17;6) ; (19;8) ; (23;12) ; (29;18) ; (31;20) ; (37;26) ; (41;30) ; (43;34)......

(P13(K ) ; 2K ) → (17;4) ; (19;6) ; (23;10) ;(29;16) ; (31;18) ; (37;24) ; (41;28) ; (43;30 ; (47;34)......

PQ(K ) = 2K + Q ( K ∈ ${\mathbb{N}}^{*}$: PQ(K ) and Q are primes ) ( see the table in Appendix 14 )

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

Therefore it is generally validated for all K > 15, since we obtain all possible cases of

Chen's weak conjecture starting with P3(K ), then P5(K ), then P7(K ) ...... for 2${\left(2K\right)}^{0.525}$ ≤ Q .

( can be proved by strong recurrence using the same method as inTheorem 4 by “finite descent» ).

Let $c_p$ = ${\displaystyle \frac{40}{21}}$ and Pr(K ) be the following property

« For any integer M < ${\left(0.5Q_K\right)}^{c_p}$, there exists at least a prime Q < $Q_K$ such that

2 M + Q is a prime «

▶ Pr(15) is true ( see Appendix 14 ).

▶ Let’s show : Pr(K) ⇒ Pr(K + 1)

$Q_{K+1}$ ≤ $Q_K$ + $Q_K^{0.525}$ (5.6)

It is assumed thatM /

$P_{K+1}$ - $Q_{K+1}$ ≠ 2M M < ${\left(0.5Q_{K+1}\right)}^{c_p}$ $P_{K+1}$ = p + 2h and $Q_{K+1}$ = q +2s (5.7)

then

p - q ≠ 2(M + s - h) (5.8)

which is impossible according to the hypothesis of strong recurrence since

2(M + s - h) is less than ${\left(0.5Q_K\right)}^{c_p}$ and that all primes p, q satisfy the recurrence hypothesis.

We deduce that Pr(K) ⇒ Pr(K + 1)

Thus the property (5.5) is true.

6. Principle of Proof

To determine pairs of primes that verify Goldbach’s conjecture three sequences of primes

( $W_{2n}),$( $V_{2n}$ ), ( $U_{2n})$ are defined and they verify the following properties

lim $V_{2n}$ = +$\mathrm{\infty }.$ (6.1)

∀ n ∈ ℕ + 2 $V_{2n}$is defined as a function of $W_{2n}$ = Sup( p ∈ Ƥ : p ≤ 2n - 3 ) 6.2)

( $W_{2n}$ ) is an increasing sequence of primes that contains all primes except $p_1$ = 2 (6.3)

lim $W_{2n}=+\mathrm{\infty }$ (6.4)

($U_{2n}$) is a complementary sequence of negligible primes with respect to 2n (6.5)

For any integer n $\ge 3$ (6.6)

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

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

(6.7) $V_{2n}$ = $W_{2n}$ and $U_{2n}$= 2n - $W_{2n}$● Otherwise, if ( 2n - ${\mathit{W}}_{2\mathit{n}}$) 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

(6.8) $U_{2(n-k)}$, $V_{2(n-k)}$ and $U_{2(n-k)}$ + 2k are primes $U_{2\left(n-k\right)}$ + $V_{2\left(n-k\right)}$ = 2(n - k )

which is always possible (see Theorem 4)

So by setting

$V_{2n}$ = $V_{2(n-k)}$ and $U_{2n}$ = $U_{2(n-k)}$+ 2k (6.9)

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

$U_{2n}$ + $V_{2n}$ = 2n (6.10)

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

7. Theorem

There exists 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 (7.1)

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

An algorithm can be used to explicitly compute any term$U_{2n}$and$V_{2n}$. (7.2)

Proof .

□ FIRST METHOD :

For any integer n $\ge 3$● If (2n - ${\mathit{W}}_{2\mathit{n}}$) is a prime

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

$V_{2n}$ = $W_{2n}$ and $U_{2n}$= 2n - $W_{2n}$ (7.3)

●Otherwise, if (2n -${\mathit{W}}_{2\mathit{n}}$)is a composite number

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

For any integer q such that 1 ≤ q ≤ n - 3 we have

3 ≤ $U_{2(n-q)}$ ≤ n .

Then, there exists an integer k / 1 ≤ k ≤ n - 3 following the Bertrand principle and Theorem 4 since all primes smaller than 2k are represented by $U_{2(n-j)}$ , ( if there were no such primes, we would have a contradiction with the Theorem 4 , even if it means transforming the indexing of the sequence ($U_{2n}$) . In fact, in an equivalent way ( see the remark following Theorem 4 ) 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 } such that

$R_{2n}$ = $U_{2(n-k)}$ + 2k ∈ Ƥ (7.4)

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 $V_{2n}$ = $V_{2\left(n-k_n\right)}$ (7.5)

( 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 -$k_n$) (7.6)

By adding the term $k_n$ to each member of the equality (5.6), it follows

$U_{2\left(n-k_n\right)}$ + 2$k_n$ + $V_{2\left(n-k_n\right)}$= 2(n - $k_n$) + 2$k_n$ (7.7)

$\iff$ $U_{2\left(n-k_n\right)}$ + 2$k_n$] + $V_{2\left(n-k_n\right)}$ = 2n (7.8)

$\iff$ $U_{2n}$ + $V_{2n}$ = 2n (7.9)

Finally for any integer n ≥ 3 this algorithm determines two sequences of primes ( $U_{2n}$ ) and ( $V_{2n}$ ) verifying Goldbach's conjecture.

□ SECOND METHOD :

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

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

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

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

a) P(2) is true : it suffices to choose $U_4$ = $V_4$ = 2 .

b) Let's show that the property P(n) is hereditary : ∀ k ∈ ℕ + 2 P(n) $\Rightarrow$ $\mathrm{P}$(n+ 1)

Assume property P(n ) is true.

●If (2(n + 1) -${\mathit{W}}_{2(\mathit{n}+1)}$)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) - $W_{2(n+1)}$ (7.10)

●Otherwise, if (2(n+1) -${\mathit{W}}_{2(\mathit{n}+1)}$)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 are primes (7.11) $U_{2\left(n+1-k\right)}$ + $V_{2\left(n+1-k\right)}$ = 2( n +1- k )

( which is always possible : see FIRST METHOD and Theorem 4 ).

Thus by setting

$V_{2(n+1)}$ = $V_{2(n+1-k)}$ and $U_{2(n+1)}$ = $U_{2(n+1-k)}$+ 2k (7.12)

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(n + 1) is true. Then the property P(n ) is hereditary : P(n) => P(n + 1).

Therefore for any integer n ≥ 2 the property P(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

8. Lemma

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

For any integer n ≥ 65

$U_{2n}$ ≤ (2${n)}^{0.55}$ (8.1)

Proof . According to the programm 11.2 and appendix 12 the majorization (8.1) is verified

For any integer n such that 65 ≤ n ≤ 2000 . For any integer n > 2000 the proof is established by recurrence. For this purpose let P1(n ) be the following property

P1(n ) : “ There exists a strictly increasing sequence of positive numbers ($C_n$) such that

$U_{2n}$ ≤ $C_n{\left(2n\right)}^{0.525}$ “ . (8.2)

▶ P1(2000) is true according to program 11.2 and the table in appendix 12.

▶ For any integer n ≥ 2000 let’s show that P1(n ) is hereditary : P1(n ) ⇒ P1(n + 1).

Assume that P1(n ) is true : then

●If (2(n + 1) -${\mathit{W}}_{2(\mathit{n}+1)}$)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) - $W_{2(n+1)}$ (8.3)

According to the results in [4,5,20] there is a constant K > 0 such that

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

⇒ $U_{2(n+1)}$ < K ${.\left[2\right(n+1\left)\right]}^{0.525}$⇒ $U_{2(n+1)}$ ≤ $C_{n+1}{.\left[2\right(n+1\left)\right]}^{0.525}$● Otherwise, if (2(n + 1) - ${\mathit{W}}_{2(\mathit{n}+1)}$) is a composite number

$\exists$ p ∈ ${\mathbb{N}}^{\mathrm{*}}$/ $U_{2(n+1)}$ = $U_{2(n+1-p)}$ + 2p (8.4)

According to [4,5,18] the smallest integer p defined in (6.4) verifies

2p < K.[ ${U_{2(n+1-p)}]}^{0.525}$ and $U_{2(n+1-p)}$ < $C_{n+1-p}{.\left[2\right(n+1-p\left)\right]}^{0.525}$ (8.5)

It follows

$U_{2(n+1)}$ < K .$C_{n+1-p}^{0.525}{.\left[2\right(n+1-p\left)\right]}^{0.275625}$ + $C_{n+1-p}{.\left[2\right(n+1-p\left)\right]}^{0.525}$ (8.6)

Then

$U_{2(n+1)}$ < $C_{n+1}.[{2(n+1)]}^{0.525}$ (8.7)

and by setting $C_n$ = ${\left(2n\right)}^{0.025}$It follows

$U_{2(n+1)}$ < ${\left[2\right(n+1\left)\right]}^{0.55}$ (8.8)

P1(n + 1) is true then P1(n ) is hereditary.

So for any integer n ≥ 2000 the property P1(n ) is true.

(The inequality (6.7) is verified with the aid of the software Maple studying the functions of the type f : x $\to$ a ${.x}^{0.275625}$ + b . $x^{0.525}$ increased by g : x $\to$ $x^{0.55}$ a and b being two strictly positive real parameters).

● Remark. A more precise estimate can be obtained using the Cipolla or Axler frames [7,2].

9. Theorem

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

($W_{2n}$) is a positive increasing sequence of primes. (9.1)

{ $W_{2n}$ : n $\in$ IN + 3 } $\cup \left\{2\right\}$ = $\mathcal{P}$ (9.2)

lim $W_{2n}$ = +oo (9.3)

( $U_{2n}$ ) and ($V_{2n}$) are sequences of primes and the set { $U_{2k}$ : k ≤ n} contains all primes less than ln( n ) (9.4)

(9.5) n $\le$ $V_{2n}$ ≤ $W_{2n}$(9.6) 3$\le {2n-W_{2n}\le U}_{2n}\le$n

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

Proof .

(9.1) 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.

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

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

(9.4) By definition $V_{2n}$ = $W_{2n}$ or there exits an integer k ≤ n - 2 such that $V_{2n}$ = $V_{2(n-k)}$ ;

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

(9.5) According to Lemma 6, for any integer n ≥ 65

$U_{2n}$ < ${\left(2n\right)}^{0.55}$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 11.2 and the table in appendix 12.

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 (9.5) n $\le$ $V_{2n}$ ⇒ $U_{2n}$ = 2n - $V_{2n}$ ≤ 2n - n ≤ n (9.6)

therefore

$V_{2n}$ ≤ $W_{2n}$ ⇒ 2n - $W_{2n}$ ≤ 2n - $V_{2n}$ = $U_{2n}$ (9.7) By (9.5) for any integer n ≥ 2 : n $\le$ $V_{2n}$So

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

10. Remarks

10.1 For any integer k ≥ 2 there are infinitely many integers n such that $U_{2n}$ = $p_k$.

10.2 $V_{2n}$ ~ 2n for (n $\to$ +oo) .

10.3 For any sufficiently large integer n / n ≥ 5000

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

10.4 The smallest integer n such that

$U_{2n}\ne$2n - $W_{2n}$ is obtained for n = 49 and $G_{98}$= ( 79 ; 19 ).

( 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 [29] ).

10.5 If q is an odd integer greater than four we could generalize this algorithm with sequences (${W\mathrm{\text{'}}}_{2n}$)

defined by

(10.5.1) $\forall$ n ∈ $\mathbb{N}$/ n ≥ ${\displaystyle \frac{(q+3)}{2}}$ ${W\mathrm{\text{'}}}_{2n}$ = Sup( p ∈ $\mathcal{P}$: p ≤ 2n - q )

Other Goldbach’s sequences$\left({G\text{'}}_{2n}\right)$independent of ($G_{2n}$) are thus generated.

10.6 The sequence ( $G_{2n}$ ) is ‘‘extremal’’ in the sense that for any integer n$\ge 2V_{2n}$ and $U_{2n}$ are the largest and smallest possible primes such that $U_{2n}$ + $V_{2n}$ = 2n.

10.7 The Cramer-Granville-Maier-Nicely conjecture [8,14,19,21,23,24,26,28,33]

is verified with probability one. It leads to the following majorization

For any integer p ≥ 500

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

The proof is similar to that of lemma 8 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${\left[\mathrm{l}\mathrm{n}\right(x\left)\right]}^{(2.2-{\displaystyle \frac{1}{x}})}$ using Maple software.

● Remark. A better estimate can be obtained via [26,28,30] .

10.8 According to Bombieri [3] and using the same method as in the proof of Lemma 8, on average, we obtain the following estimate of $U_{2n}$(10.8.1) $\forall$ $\epsilon$ > 0 $U_{2n}$ = O ( ${\mathrm{l}\mathrm{n}}^{1.3+\epsilon }(2n$)) ( on average )

11. Algorithm

11.1. 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 :

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

Let :

(11.1.1) $U_{2n}$ = $T_{2n}$ and $V_{2n}$ = $W_{2n}$otherwise

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 $\to$ k + 1 ).

return to B1)

End while

Assign to k the value $k_n$ ( k $\to k_n$ )

(11.1.2) Let :

$U_{2n}$ = $U_{2\left(n-k_n\right)}$ + 2$k_n$ and $V_{2n}$ = $V_{2\left(n-k_n\right)}$Assign to n the value n + 1 ( n$\to n+1$ and return to A)

End :

Outputs for integers less than${10}^{4:}:$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 = ● ; $T_{2n}$ = ● ; ${V_{2n}-P=\mathrm{●};U}_{2n}$ = ● )

11.2. Program Written with Maxima Software for 2n = ${10}^{500}$

n1 : 10**500 ; for n :5*10**499 + 10000 thru 5*10**499 + 10010 do

( a :2*n , c :a - 3 , test : 0 , b : prev_prime(a - 1) ,d :a - b ,

if primep(d )

then print(a - n1 , c - n1 , b - n1 , d , b - n1 , d )

else ( while test = 0 do (e :a - c , if ( primep(c ) and primep(e ) )

then ( test:1 , print(a - n1 , b - n1 , d , c - n1 , e ," ** " ))

else ( test : 0 , c : c - 2 ))) ;

15. Perspectives and Generalizations

15.1 Other Goldbach sequences (${G\text{'}}_{2n}$) and (${G"}_{2n}$) independent of $\left(G_{2n}\right)$ may be studied using the increasing sequences of primes ( ${W\text{'}}_{2n}),\left(\mathrm{s}\mathrm{e}\mathrm{e}10.5\right)$ and (${W"}_{2n}$) defined by

For any integer n $\ge 3$${W"}_{2n}$ = Sup( p$\mathsf{ϵ}\mathcal{P}:p\le$ f (n) )

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

● f is strictly increasing on the interval I

● f (3) = 3 and $\underset{x\to +\infty }{\lim }f\left(x\right)$ = +$\mathrm{\infty }$● $\forall x\mathsf{ϵ}If\left(x\right)\le 2x-3$For example, one of the following functions defined on I 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 number x )

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

15.2 Using this method it would be interesting to study the Schnirelmann density [31] of primes

3 , 5 , 7, 11 ,........ ... in the sequence ( $U_{2n}$) on variable intervals .

15.3 It is possible to exceed the values shown in the table of 2n = ${10}^{1000}$ by perfecting this algorithm starting from n , exploiting the fact that one of Goldbach's decomponents can be chosen

equal to 12p + 1,

( the set of Goldbach decomponents consists of primes of the form 6p +/- 1 ) using Cipolla-Axler-Dusart type functions [2,7,10,11] to better identify the terms of ( $G_{2n}),\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{s}$a$\mathrm{n}\mathrm{d}\mathrm{m}\mathrm{o}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}$ $\mathrm{s}\mathrm{o}\mathrm{f}\mathrm{t}\mathrm{w}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{a}\mathrm{s}$ Maple.

15.4 Diophantine equations and conjectures of the same nature ( Lagrange-Lemoine-Levy conjecture [9,19,21,23,24,33] ) can be processed using similar reasoning and algorithms.

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

then let

${Vl}_{2n}$= ${Wl}_{2n}$ and ${Ul}_{2n}$= ${Tl}_{2n}$● If ${Tl}_{2n}$ is a composite number

then there exists an integer k / 1 $\le k\le n-3$such hat

${Ul}_{2\left(n-k\right)}$ + 2k is a prime

then let

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

Using the same type of reasoning a generalization called «Bezout-Goldbach 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

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

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

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

15.5. Remark

GOLDBACH( - ) :

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

GOLDBACH( + ) :

$V_{2K}$ = Sup ( p ∈ $\mathcal{P}$: 2K - p ∈ $\mathcal{P}$) and $U_{2K}$̲= Inf ( p ∈ $\mathcal{P}$ : 2K - p ∈ $\mathcal{P}$) = 2K - $V_{2K}$( Is it possible to envisage a symmetry in the Goldbach triangle parametrized by arithmetic sequences between the representations of primes and even integers ? )

16. Conclusion

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

$\forall$n ∈ $\mathbb{N}+2$ $U_{2n}$ and $V_{2n}$ are primes $\mathrm{a}\mathrm{n}\mathrm{d}{U}_{2n}$ +$V_{2n}$= 2n

has been developed using an simple and efficient "located" algorithm.

16.2 The record of Silva [29] is beaten on a personal computer and ten Goldbach decomponents

$U_{2n}$ and $V_{2n}$ are obtained for values of the order 2n =${10}^{1000}$ for a computation time of less than

three hours.

16.3 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 just need to know the primes $p_l$ and $V_{2r}$such that

(16.3.1) $p_l$ ≤ 7.${\mathrm{l}\mathrm{n}}^{1.3}\left(2n\right)$ and 2n - 7.${\mathrm{l}\mathrm{n}}^{1.3}$(2n) ≤ $V_{2r}$≤ 2n (on average )

This property allows quick computing of $U_{2n}$ and $V_{2n}$ .

16.4 Therefore the Lagrange-Lemoine-Levy 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.