# About the strong EULER-GOLDBACH conjecture

Philippe Sainty

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Philippe Sainty. About the strong EULER-GOLDBACH conjecture. Matematicheskie Zametki / Mathematical Notes, In press. hal-03838423

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Distributed under a Creative Commons Attribution 4.0 International License
About the strong EULER-GOLDBACH conjecture.

PHILIPPE SAINTY

University Pierre et Marie Curie Paris, France.

July 2022

Abstract.

In this article, we define a "recursive local" algorithm in order to construct two recurrent numerical sequences of positive prime numbers ($U_{2n}$) and ($V_{2n}$), ($U_{2n}$) function of ($V_{2n}$), such that for any integer $n \ge 2$, their sum is $2n$. To build these, we use a third sequence of prime numbers ($W_{2n}$) defined for any integer $n \ge 3$ by : $W_{2n} = \text{Sup}(p \in \text{IP} : p \le 2n-3)$, where IP is the infinite set of positive prime numbers. The Goldbach conjecture has been verified for all even integers $2n$ between 4 and $4.10^{18}$. In the Table of Goldbach sequence terms given in paragraph § 10, we reach values of the order of $2n= 10^{1000}$. Thus, thanks to this algorithm of "ascent and descent", we can validate the strong Euler-Goldbach conjecture.

Keywords.

Prime numbers, Prime Number Theorem, weak and strong Goldbach's conjectures, Dirichlet's theorem, Bertrand-Tchebychev theorem, gaps between prime numbers.

CONTENTS.

Abstract.

§ 1 Preliminaries and history.

§ 2 Definitions and notations.

§ 3 Introduction.

§ 4 Theorem.

§ 5 Methodology.

§ 6 Proof.

§ 7 Corollary.

§ 8 Remarks.

§ 9 Algorithm and programming with Maxima software.

§ 10 Appendices: Table of Goldbach sequences.

§ 11 Perspectives and generalizations.

§ 12 Conclusion.

References.

§ 1 Preliminaries and history.

Number theory, "the Queen of Mathematics" deals with structures and properties defined over integers (see Euclidean [5], Hardy & Wright [7] and Landau [10]). This theory is built from the notion of prime number. Many problems have been raised and multiple conjectures with statements that are often simple but very difficult to demonstrate have been made. These main components include:

a) Elementary arithmetic.

Determination and properties of prime numbers.
Operations on integers (basic operations, congruence, gcd, lcm, and so on...).

Representations and numbers bases of integers.

Decomposition of integers into product or sum of prime numbers (Fundamental theorem of arithmetic, decomposition of large numbers, cryptography and Goldbach's conjecture).

b) Analytic number theory.

(Riemann Hypothesis). Distribution of prime numbers (Prime Number Theorem, see Hadamard [6], De la Vallée-Poussin [18], Littlewood [11] and Erdos [4]), gaps between prime numbers (see Bombieri [1], Cramer [3], Iwaniec & Pintz [9] and Tchebychev [17]).

c) Algebraic, probabilistic, combinatorics and algorithmic theories of numbers.

d) Modular arithmetic, diophantine approximations, equations and geometry. Arithmetic sequences, functions, and algebraic geometry.

As part of research on the Goldbach conjecture, Chen [2], Hegfollt [8], Ramaré [12], Tchebychev [17], Vinogradov [19], Tao [16] and Zhang [20] have taken essential steps and obtained very promising results for its demonstration. In effect, Helfgott & Platt [8] proved in 2013 Goldbach's weak conjecture. Silva, Herzog & Pardi [15] held the record for calculating the terms of Goldbach sequences after having determined, pairs of positive prime numbers ($p_n; q_n$) verifying : ($p_n + q_n = 2n$) for any integer n such that: ($4 \le 2n \le 4.10^{18}$). In the scientific literature, one does not find explicit constructions of recurrent sequences of positive Goldbach primes ($G_{2n}$) = ($U_{2n}; V_{2n}$), satisfying for any integer $n \ge 2$ the equality: ($U_{2n} + V_{2n} = 2n$). In this article, we define two sequences of prime numbers, using a "local recursive" algorithm simple and efficient allowing to calculate for any integer $n \ge 3$, by successive iterations any term $U_{2n}$ and $V_{2n}$ of a Goldbach sequence. Using the scientific calculation software Maxima on a personal computer, one easily exceeds Silva's record [15] set for $2n = 4.10^{18}$ and we can reach $2n = 10^{500}$ and even $2n = 10^{1000}$.

## § 2 Definitions and notations.

a) The integers $n, k, p, q, r, ...$ and so on... are always positive.

b) We denote by IP the infinite set of positive prime numbers (called simply prime numbers):

$$ \mathrm{IP} = \{ p_k, (k \in \mathrm{IN}^*) : p_k \text{ is the kth-positive prime number}; (p_k < p_{k+1} \text{ and } \lim_{n \to +\infty} p_n = +\infty) \text{ ( } p_1=2; p_2=3; p_3=5; p_4=7; p_5=11; p_6=13; \ldots \text{)}. $$

To simplify the writing of large numbers, we adopt the following notations.

c) $M = 10^9$

d) $R = 4.10^{18}$

e) $G = 10^{100}$

f) $S = 10^{500}$

g) $T = 10^{1000}$

h) $\log(x)$ denotes the decimal logarithm of the strictly positive real $x$ : ($x > 0$).

i) We call Goldbach sequence any numerical sequence denoted by ($G_{2n}$) = ($U_{2n}; V_{2n}$) verifying: For any integer $n \ge 2$, $U_{2n}$ and $V_{2n}$ are prime numbers and $U_{2n} + V_{2n} = 2n$.

## § 3 Introduction.

Let ($W_{2n}$) be the sequence of prime numbers defined by :
For any integer $n \ge 3$,

$$ (1) \qquad W_{2n} = \mathrm{Sup}(p \in \mathrm{IP} : p \le 2n-3) $$

Using case disjunction reasoning, we construct two recurrent sequences of prime numbers ($V_{2n}$) and ($U_{2n}$) as a function of the sequence ($W_{2n}$) by the following process.
$$ (2) \qquad (V_4=2; U_4=2) $$

Let $n$ be an integer : ($n \ge 3$).

1) Either,
($2n - W_{2n}$) is a prime number and $V_{2n}$ and $U_{2n}$ are defined directly in terms of $W_{2n}$.

2) Either,
($2n - W_{2n}$) is a composite number and $V_{2n}$ and $U_{2n}$ are defined from previous terms of the sequence $(G_{2n})$.

## § 4 Theorem.

4.1 There exists a recurrent numerical Goldbach sequence $(G_{2n}) = (U_{2n}, V_{2n})$ such that for any integer $n \ge 2$, we have : $U_{2n}$ and $V_{2n}$ are prime numbers and their sum is $2n$.

$$ (3) \qquad (U_{2n}, V_{2n} \in \text{IP and } U_{2n} + V_{2n} = 2n) $$

4.2 An algorithm makes it possible to explicitly calculate any term $U_{2n}$ and $V_{2n}$.

## § 5 Methodology.

A method similar to that of the sieve is used, (see Selberg [14]). For this purpose, we build a positive sequence of prime numbers ($V_{2n}$) satisfying: $\lim_{n \to \infty} V_{2n} = +\infty$, as a function of $W_{2n} = \text{Sup}(p \in \text{IP} : p \le 2n-3)$, (($W_{2n}$) is an increasing sequence who contains all prime numbers except two, and verifies: $\lim_{n \to \infty} W_{2n} = +\infty$), and a complementary sequence of prime numbers ($U_{2n}$) negligible with respect to ($2n$) for very large integers $n$, i.e for any large enough integer $n$, we have :

$$ (4) \qquad \begin{gathered} \text{Sup}(U_{2k} : (k \in \text{IN} : 3 \le k \le n)) \le \text{K1}(n).\log(2n). \\ (\text{K1}(n)>0; \text{K1}(n) \text{ quasi-constant and hyper slowly increasing function of } n). \end{gathered} $$

For any integer $n \ge 3$,

If ($2n - W_{2n}$) is a prime number "particular case", then we put:

$$ (5) \qquad V_{2n} = W_{2n} \quad \text{and} \quad U_{2n} = 2n - W_{2n} $$

Else, if ($2n - W_{2n}$) is a composite number "general case",
we return to the sequence $(G_{2n})$ of previous terms, already calculated. Therefore, we are looking for an integer $k$ in order to obtain two terms $U_{2(n-k)}$ and $V_{2(n-k)}$ satisfying the following conditions :

$$ (6) \qquad \begin{gathered} U_{2(n-k)}, V_{2(n-k)} \text{ and } U_{2(n-k)} + 2k \text{ are prime numbers} \\ U_{2(n-k)} + V_{2(n-k)} = 2(n-k) \\ \text{(which is always possible ; see proof in paragraph § 6). Thus, by putting :} \end{gathered} $$

$$ (7) \qquad V_{2n} = V_{2(n-k)} \quad \text{and} \quad U_{2n} = U_{2(n-k)} + 2k $$

we obtain two new prime numbers satisfying :

$$ (8) \qquad U_{2n} + V_{2n} = 2n. $$

We can thus reiterate this process by incrementing $n$ by one unit : ($n \to n+1$).
§ 6 Proof.

For any integer q such that : (1≤ q ≤ n-3), we have : 3 ≤ $U_{2(n-q)}$ ≤ n.

For any integer k such that (2 ≤ 2k ≤ (n-1)/2), there exists two prime numbers p_m and p_r, (m>r)
in the interval [2;n] such that :

$$
(9) \qquad p_m - p_r = 2k
$$

(see Bombieri [1], Cramer [3], Iwaniec & Pintz [9] and Tchebychev [17]).

Then, there exists an integer k, (2 ≤ 2k ≤ n-3) such that :

$$
(10) \qquad R_{2n} = U_{2(n-k)} + 2k \quad \text{is a prime number}
$$

We choose the smallest integer k denoted by $k_n$ such that $R_{2n}$ is a prime number. Then we put :

$$
\begin{gather*}
(11) \qquad U_{2n} = U_{2(n-k_n)} + 2k_n \text{ and } V_{2n} = V_{2(n-k_n)} \\
\qquad \quad \text{(These two terms are prime numbers)}
\end{gather*}
$$

We have constructed in the previous steps two prime numbers, $U_{2(n-k_n)}$ and $V_{2(n-k_n)}$ whose sum is equal to
$2(n-k_n)$ :

$$
(12) \qquad U_{2(n-k_n)} + V_{2(n-k_n)} = 2(n-k_n)
$$

Thus, by adding the term $k_n$ on each member of the equality (12), we obtain :

$$
(13) \qquad U_{2(n-k_n)} + 2k_n + V_{2(n-k_n)} = 2(n-k_n) + 2k_n
$$

$$
(14) \quad \Leftrightarrow \quad \{ U_{2(n-k_n)} + 2k_n \} + V_{2(n-k_n)} = 2n
$$

$$
(15) \quad \Leftrightarrow \quad U_{2n} + V_{2n} = 2n
$$

Finally, this algorithm constructs for any integer n≥3 two sequences of prime numbers (U_{2n}) and (V_{2n})
verifying Goldbach's conjecture.

§ 7 Corollary.

For any integer $n \ge 3$, we easily verify :

7.1 ($W_{2n}$) is a positive increasing sequence of prime numbers.

7.2 $\lim W_{2n} = +\infty$

7.3 $n < W_{2n} \le 2n-3$

7.4 $3 \le 2n - W_{2n} \le U_{2n} \le n$

7.5 $n \le V_{2n} \le W_{2n}$

7.6 $\lim V_{2n} = +\infty$

7.7 $\{W_{2n} : n \in \mathbb{N}^*\} \cup \{2\} = \text{IP}$

§ 8 Remarks.
8.1 There is an infinity of integers $n$ such that $U_{2n} = 3, 5, 7$ or $11$.

8.2 $V_{2n} \sim (2n)$, for $(n \to +\infty)$.

8.3 For any integer $n$ large enough, $U_{2n} \ll V_{2n}$ and $\lim(\frac{U_{2n}}{V_{2n}})=0$.

8.4 $\text{Sup}(U_{2k} : (2 \le k \le n)) \le K_1(n) \cdot \log(2n) : (K_1(n) > 0; K_1(n) \sim cte)$.

8.5 The first integer $n$ such that: $U_{2n} \ne 2n - W_{2n}$ is obtained for $n=49$ and $G_{98} = (79;19)$.
(This type of term “densifies” in the Goldbach sequence ($G_{2n}$) when $n$ increases, in the sense
Of Schnirelmann [13] and there is an infinity of them; we can calculate their proportion
by interval).

8.6 Let $q \ge 5$ be an odd integer, then we could generalize this process with sequences ($W'_{2n}$) = $\text{Sup}(p \in \text{IP} : p \le 2n-q)$ and we obtain other sequences ($G'_{2n}$) of Goldbach sequences independent of ($G_{2n}$).

8.7 The sequence ($G_{2n}$) is extremal in the sense that for any given integer $n \ge 2$, $V_{2n}$ and $U_{2n}$ are the greatest and the smallest possible prime numbers such that: $U_{2n} + V_{2n} = 2n$.

## § 9 Algorithm.

9.1 Algorithm written in natural language.

Input three variables integers: N, $n$ and $P$.

Input : $p_1=2, p_2=3, p_3=5, p_4 = 7, ...,..., p_N$, the first $N$ prime numbers
: $n=3$.
: $P=M, R, G, S$, or $T$ given in paragraph § 2.

Start:

Algorithm body :

A) Calculate : $W_{2n} = \text{Sup}(p \in \text{IP} : p \le 2n-3)$

If $T_{2n} = (2n - W_{2n})$ is a prime number, then we put:

$$ (16) \qquad U_{2n} = T_{2n} \quad \text{and} \quad V_{2n} = W_{2n} $$

else,

B) If $T_{2n}$ is a composite number, then let $k$ be an integer; we put: $k=1$.

B.1) while : $U_{2(n-k)} + 2k$ is a composite number,
assign to $k$ the value $k+1$
return to B1)

end while.

We put : $k_n = k$ and

$$ (17) \qquad U_{2n} = U_{2(n-k_n)} + 2k_n \quad \text{and} \quad V_{2n} = V_{2(n-k_n)} $$

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

End

Output for integers less than $10^4$: Print ($2n = ...$ ; $2n-3 = ...$ ; $W_{2n} = ...$ ; $T_{2n} = ...$ ; $V_{2n} = ...$ ; $U_{2n} = ...$).

Output for large integers: Print ($2n-P = ...$ ; $2n-3-P = ...$ ; $W_{2n}-P = ...$ ; $T_{2n} = ...$ ; $V_{2n}-P = ...$ ; $U_{2n} = ...$).

9.2 Algorithm written with the language of the Maxima scientific computing software.

r:0 ; n1:10**500 ; for n:5*10**499+10000 thru 5*10**499+10010 do
(k:1, a:2*n, c:a-3, test:0, b:prev_prime(a-1),
if primep(a-b)
```pascal
then print(a-n1,c-n1,b-n1,a-b,b-n1,a-b)
else ( r:r+1,
      while test=0 do
        ( if ( primep(c) and primep(a-c) )
        then ( test:1, print(a-n1,a-n1-3,b-n1,a-b,c-n1,a-c, "Ret ",r))
        else ( test:0, c:c-2*k )));
```

## § 10 Appendices.

Application of the algorithm (paragraph § 9) : Table of Goldbach's sequences ($U_{2n}$) and ($V_{2n}$) calculated by the Maxima program written in paragraph § 9.2.

The abbreviation Ret) listed in the Table below indicates the results given by the algorithm in the case B) of return to the previous terms of the sequence ($G_{2n}$). WATCH OUT ! For large integers n, ($2n>10^9$, for instance), in order to simplify the writing of big numbers, we write the results in the form: 2n-P, (2n-3)-P, $W_{2n}$-P, $T_{2n}$, $V_{2n}$-P and $U_{2n}$ with P=M, R, G, S, or T given in paragraph § 2.

<table><thead><tr><th>2n</th><th>I</th><th>2n-3</th><th>W<sub>2n</sub></th><th>T<sub>2n</sub>=2n-W<sub>2n</sub></th><th>V<sub>2n</sub></th><th>U<sub>2n</sub></th></tr></thead><tbody><tr><td>4</td><td></td><td>1</td><td></td><td></td><td>2</td><td>2</td></tr><tr><td>6</td><td></td><td>3</td><td>3</td><td>3</td><td>3</td><td>3</td></tr><tr><td>8</td><td></td><td>5</td><td>5</td><td>3</td><td>5</td><td>3</td></tr><tr><td>10</td><td></td><td>7</td><td>7</td><td>3</td><td>7</td><td>3</td></tr><tr><td>12</td><td></td><td>9</td><td>7</td><td>5</td><td>7</td><td>5</td></tr><tr><td>14</td><td></td><td>11</td><td>11</td><td>3</td><td>11</td><td>3</td></tr><tr><td>16</td><td></td><td>13</td><td>13</td><td>3</td><td>13</td><td>3</td></tr><tr><td>18</td><td></td><td>15</td><td>13</td><td>5</td><td>13</td><td>5</td></tr><tr><td>20</td><td></td><td>17</td><td>17</td><td>3</td><td>17</td><td>3</td></tr><tr><td>22</td><td></td><td>19</td><td>19</td><td>3</td><td>19</td><td>3</td></tr><tr><td>24</td><td></td><td>21</td><td>19</td><td>5</td><td>19</td><td>5</td></tr><tr><td>26</td><td></td><td>23</td><td>23</td><td>3</td><td>23</td><td>3</td></tr><tr><td>28</td><td></td><td>25</td><td>23</td><td>5</td><td>23</td><td>5</td></tr><tr><td>30</td><td></td><td>27</td><td>23</td><td>7</td><td>23</td><td>7</td></tr><tr><td>32</td><td></td><td>29</td><td>29</td><td>3</td><td>29</td><td>3</td></tr><tr><td>34</td><td></td><td>31</td><td>31</td><td>3</td><td>31</td><td>3</td></tr><tr><td>36</td><td></td><td>33</td><td>31</td><td>5</td><td>31</td><td>5</td></tr><tr><td>38</td><td></td><td>35</td><td>31</td><td>7</td><td>31</td><td>7</td></tr><tr><td>40</td><td></td><td>37</td><td>37</td><td>3</td><td>37</td><td>3</td></tr><tr><td>And so on</td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>
<table><tr><td>70</td><td>67</td><td>67</td><td>3</td><td>67</td><td>3</td></tr><tr><td>72</td><td>69</td><td>67</td><td>5</td><td>67</td><td>5</td></tr><tr><td>74</td><td>71</td><td>71</td><td>3</td><td>71</td><td>3</td></tr><tr><td>76</td><td>73</td><td>73</td><td>3</td><td>73</td><td>3</td></tr><tr><td>78</td><td>75</td><td>73</td><td>5</td><td>73</td><td>5</td></tr><tr><td>80</td><td>77</td><td>73</td><td>7</td><td>73</td><td>7</td></tr><tr><td>82</td><td>79</td><td>79</td><td>3</td><td>79</td><td>3</td></tr><tr><td>84</td><td>81</td><td>79</td><td>5</td><td>79</td><td>5</td></tr><tr><td>86</td><td>83</td><td>83</td><td>3</td><td>83</td><td>3</td></tr><tr><td>88</td><td>85</td><td>83</td><td>5</td><td>83</td><td>5</td></tr><tr><td>90</td><td>87</td><td>83</td><td>7</td><td>83</td><td>7</td></tr><tr><td>92</td><td>89</td><td>89</td><td>3</td><td>89</td><td>3</td></tr><tr><td>94</td><td>91</td><td>89</td><td>5</td><td>89</td><td>5</td></tr><tr><td>96</td><td>93</td><td>89</td><td>7</td><td>89</td><td>7</td></tr><tr><td>R1)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>98</td><td>95</td><td>89</td><td>9</td><td>79</td><td>19</td></tr><tr><td>100</td><td>97</td><td>97</td><td>3</td><td>97</td><td>3</td></tr><tr><td>And so on........</td><td></td><td></td><td></td><td></td><td></td></tr></table>

<table><tr><td>120</td><td>117</td><td>113</td><td>7</td><td>113</td><td>7</td></tr><tr><td>R2)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>122</td><td>119</td><td>113</td><td>9</td><td>109</td><td>13</td></tr><tr><td>124</td><td>121</td><td>113</td><td>11</td><td>113</td><td>11</td></tr><tr><td>126</td><td>123</td><td>113</td><td>13</td><td>113</td><td>13</td></tr><tr><td>R3)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>128</td><td>125</td><td>113</td><td>15</td><td>109</td><td>19</td></tr><tr><td>130</td><td>127</td><td>127</td><td>3</td><td>127</td><td>3</td></tr><tr><td>132</td><td>129</td><td>127</td><td>5</td><td>127</td><td>5</td></tr><tr><td>134</td><td>131</td><td>131</td><td>3</td><td>131</td><td>3</td></tr><tr><td>136</td><td>133</td><td>131</td><td>5</td><td>131</td><td>5</td></tr><tr><td>138</td><td>135</td><td>131</td><td>7</td><td>131</td><td>7</td></tr><tr><td>140</td><td>137</td><td>137</td><td>3</td><td>137</td><td>3</td></tr></table>
<table><tbody><tr><td>142</td><td>139</td><td>139</td><td>3</td><td>139</td><td>3</td></tr><tr><td>144</td><td>141</td><td>139</td><td>5</td><td>139</td><td>5</td></tr><tr><td>146</td><td>143</td><td>139</td><td>7</td><td>139</td><td>7</td></tr><tr><td>R4)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>148</td><td>145</td><td>139</td><td>9</td><td>137</td><td>11</td></tr><tr><td>150</td><td>147</td><td>139</td><td>11</td><td>139</td><td>11</td></tr><tr><td>And so on..................</td><td>....................................</td><td>....................................</td><td>....................................</td><td>....................................</td><td>....................................</td></tr><tr><td>180</td><td>177</td><td>173</td><td>7</td><td>173</td><td>7</td></tr><tr><td>182</td><td>179</td><td>179</td><td>3</td><td>179</td><td>3</td></tr><tr><td>184</td><td>181</td><td>181</td><td>3</td><td>181</td><td>3</td></tr><tr><td>186</td><td>181</td><td>181</td><td>5</td><td>181</td><td>5</td></tr><tr><td>188</td><td>181</td><td>181</td><td>7</td><td>181</td><td>7</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>190</td><td>187</td><td>181</td><td>9</td><td>179</td><td>11</td></tr><tr><td>192</td><td>189</td><td>181</td><td>11</td><td>181</td><td>11</td></tr><tr><td>194</td><td>191</td><td>191</td><td>3</td><td>191</td><td>3</td></tr><tr><td>196</td><td>193</td><td>193</td><td>3</td><td>193</td><td>3</td></tr><tr><td>198</td><td>195</td><td>193</td><td>5</td><td>193</td><td>5</td></tr><tr><td>200</td><td>197</td><td>197</td><td>3</td><td>197</td><td>3</td></tr><tr><td>And so on..................</td><td>....................................</td><td>....................................</td><td>....................................</td><td>....................................</td><td>....................................</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>500</td><td>497</td><td>491</td><td>9</td><td>487</td><td>13</td></tr><tr><td>502</td><td>499</td><td>499</td><td>3</td><td>499</td><td>3</td></tr><tr><td>504</td><td>501</td><td>499</td><td>5</td><td>499</td><td>5</td></tr><tr><td>506</td><td>503</td><td>503</td><td>3</td><td>503</td><td>3</td></tr><tr><td>508</td><td>505</td><td>503</td><td>5</td><td>503</td><td>5</td></tr><tr><td>510</td><td>507</td><td>503</td><td>7</td><td>503</td><td>7</td></tr><tr><td>And so on..................</td><td>....................................</td><td>....................................</td><td>....................................</td><td>....................................</td><td>....................................</td></tr><tr><td>1000</td><td>997</td><td>997</td><td>3</td><td>997</td><td>3</td></tr><tr><td>1002</td><td>999</td><td>997</td><td>5</td><td>997</td><td>5</td></tr><tr><td>1004</td><td>1001</td><td>997</td><td>7</td><td>997</td><td>7</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>
<table><tbody><tr><td>1006</td><td>1003</td><td>997</td><td>9</td><td>983</td><td>23</td></tr><tr><td>1008</td><td>1005</td><td>997</td><td>11</td><td>997</td><td>11</td></tr><tr><td>1010</td><td>1007</td><td>997</td><td>13</td><td>997</td><td>13</td></tr><tr><td>1012</td><td>1009</td><td>1009</td><td>3</td><td>1009</td><td>3</td></tr><tr><td>1014</td><td>1011</td><td>1009</td><td>5</td><td>1009</td><td>5</td></tr><tr><td>1016</td><td>1013</td><td>1013</td><td>3</td><td>1013</td><td>3</td></tr><tr><td>1018</td><td>1015</td><td>1013</td><td>5</td><td>1013</td><td>5</td></tr><tr><td>1020</td><td>1017</td><td>1013</td><td>7</td><td>1013</td><td>7</td></tr><tr><td>And so on..................</td><td>..................................................</td><td>..................................................</td><td>..................................................</td><td>..................................................</td><td>..................................................</td></tr><tr><td>5000</td><td>4997</td><td>4993</td><td>7</td><td>4993</td><td>7</td></tr><tr><td>5002</td><td>4999</td><td>4999</td><td>3</td><td>4999</td><td>3</td></tr><tr><td>5004</td><td>5001</td><td>4999</td><td>5</td><td>4999</td><td>5</td></tr><tr><td>5006</td><td>5003</td><td>5003</td><td>3</td><td>5003</td><td>3</td></tr><tr><td>5008</td><td>5005</td><td>5003</td><td>5</td><td>5003</td><td>5</td></tr><tr><td>5010</td><td>5007</td><td>5003</td><td>7</td><td>5003</td><td>7</td></tr><tr><td>5012</td><td>5009</td><td>5009</td><td>3</td><td>5009</td><td>3</td></tr><tr><td>5014</td><td>5011</td><td>5011</td><td>3</td><td>5011</td><td>3</td></tr><tr><td>5016</td><td>5013</td><td>5011</td><td>5</td><td>5011</td><td>5</td></tr><tr><td>5018</td><td>5015</td><td>5011</td><td>7</td><td>5011</td><td>7</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>5020</td><td>5017</td><td>5011</td><td>9</td><td>5009</td><td>11</td></tr><tr><td>And so on..................</td><td>..................................................</td><td>..................................................</td><td>..................................................</td><td>..................................................</td><td>..................................................</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>10000</td><td>9997</td><td>9973</td><td>27</td><td>9941</td><td>59</td></tr><tr><td>10002</td><td>9999</td><td>9973</td><td>29</td><td>9973</td><td>29</td></tr><tr><td>10004</td><td>10001</td><td>9973</td><td>31</td><td>9973</td><td>31</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>10006</td><td>10003</td><td>9973</td><td>33</td><td>9923</td><td>83</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>10008</td><td>10005</td><td>9973</td><td>35</td><td>9967</td><td>41</td></tr><tr><td>10010</td><td>10007</td><td>10007</td><td>3</td><td>10007</td><td>3</td></tr><tr><td>10012</td><td>10009</td><td>10009</td><td>3</td><td>10009</td><td>3</td></tr><tr><td>10014</td><td>10011</td><td>10009</td><td>5</td><td>10009</td><td>5</td></tr></tbody></table>
<table><tbody><tr><td>10016</td><td>10013</td><td>10009</td><td>7</td><td>10009</td><td>7</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>10018</td><td>10015</td><td>10009</td><td>9</td><td>10007</td><td>11</td></tr><tr><td>10020</td><td>10017</td><td>10009</td><td>11</td><td>10009</td><td>11</td></tr><tr><td>And so on</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>2n-M</td><td>(2n-3)-M</td><td>W<sub>2n</sub> - M</td><td>T<sub>2n</sub></td><td>V<sub>2n</sub> - M</td><td>U<sub>2n</sub></td></tr><tr><td>+1000</td><td>+997</td><td>+993</td><td>7</td><td>+993</td><td>7</td></tr><tr><td>Rt)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+1002</td><td>+999</td><td>+993</td><td>9</td><td>+931</td><td>71</td></tr><tr><td>+1004</td><td>+1001</td><td>+993</td><td>11</td><td>+993</td><td>11</td></tr><tr><td>+1006</td><td>+1003</td><td>+993</td><td>13</td><td>+993</td><td>13</td></tr><tr><td>Rt)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+1008</td><td>+1005</td><td>+993</td><td>15</td><td>+919</td><td>89</td></tr><tr><td>+1010</td><td>+1007</td><td>+993</td><td>17</td><td>+993</td><td>17</td></tr><tr><td>+1012</td><td>+1009</td><td>+993</td><td>19</td><td>+993</td><td>19</td></tr><tr><td>+1014</td><td>+1011</td><td>+1011</td><td>3</td><td>+1011</td><td>3</td></tr><tr><td>+1016</td><td>+1013</td><td>+1011</td><td>5</td><td>+1011</td><td>5</td></tr><tr><td>+1018</td><td>+1015</td><td>+1011</td><td>7</td><td>+1011</td><td>7</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+1020</td><td>+1017</td><td>+1011</td><td>9</td><td>+931</td><td>89</td></tr><tr><td>And so on</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>2n-R</td><td>(2n-3)-R</td><td>W<sub>2n</sub> - R</td><td>T<sub>2n</sub></td><td>V<sub>2n</sub> - R</td><td>U<sub>2n</sub></td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+1000</td><td>+997</td><td>+979</td><td>21</td><td>+903</td><td>97</td></tr><tr><td>+1002</td><td>+999</td><td>+979</td><td>23</td><td>+979</td><td>23</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+1004</td><td>+1001</td><td>+979</td><td>25</td><td>+951</td><td>53</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+1006</td><td>+1003</td><td>+979</td><td>27</td><td>+903</td><td>103</td></tr><tr><td>+1008</td><td>+1005</td><td>+979</td><td>29</td><td>+979</td><td>29</td></tr></tbody></table>
<table><tbody><tr><td>+1010</td><td>+1007</td><td>+979</td><td>31</td><td>+979</td><td>31</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+1012</td><td>+1009</td><td>+979</td><td>33</td><td>+951</td><td>61</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+1014</td><td>+1011</td><td>+979</td><td>35</td><td>+781</td><td>233</td></tr><tr><td>+1016</td><td>+1013</td><td>+979</td><td>37</td><td>+979</td><td>37</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+1018</td><td>+1015</td><td>+979</td><td>39</td><td>+951</td><td>67</td></tr><tr><td>+1020</td><td>+1017</td><td>+1017</td><td>3</td><td>+1017</td><td>3</td></tr><tr><td>And so on..................</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>2n-G</td><td>(2n-3)-G</td><td>W<sub>2n</sub>-G</td><td>T<sub>2n</sub></td><td>V<sub>2n</sub>-G</td><td>U<sub>2n</sub></td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+10000</td><td>+9997</td><td>+9631</td><td>369</td><td>+7443</td><td>2557</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+10002</td><td>+9999</td><td>+9631</td><td>371</td><td>+9259</td><td>743</td></tr><tr><td>+10004</td><td>+10001</td><td>+9631</td><td>373</td><td>+9631</td><td>373</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+10006</td><td>+10003</td><td>+9631</td><td>375</td><td>+8583</td><td>1423</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+10008</td><td>+10005</td><td>+9631</td><td>377</td><td>+6637</td><td>3371</td></tr><tr><td>+10010</td><td>+10007</td><td>+9631</td><td>379</td><td>+9631</td><td>379</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+10012</td><td>+10009</td><td>+9631</td><td>381</td><td>+8583</td><td>1429</td></tr><tr><td>+10014</td><td>+10011</td><td>+9631</td><td>383</td><td>+9631</td><td>383</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+10016</td><td>+10013</td><td>+9631</td><td>385</td><td>+9259</td><td>757</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+10018</td><td>+10015</td><td>+9631</td><td>387</td><td>+4491</td><td>5527</td></tr><tr><td>+10020</td><td>+10017</td><td>+9631</td><td>389</td><td>+9631</td><td>389</td></tr><tr><td>And so on..................</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>2n-S</td><td>(2n-3)-S</td><td>W<sub>2n</sub>-S</td><td>T<sub>2n</sub></td><td>V<sub>2n</sub>-S</td><td>U<sub>2n</sub></td></tr></tbody></table>
<table><tbody><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+20000</td><td>+19997</td><td>+18031</td><td>1969</td><td>+17409</td><td>2591</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+20002</td><td>+19999</td><td>+18031</td><td>1971</td><td>+17409</td><td>2593</td></tr><tr><td>+20004</td><td>+20001</td><td>+18031</td><td>1973</td><td>+18031</td><td>1973</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+20006</td><td>+20003</td><td>+18031</td><td>1975</td><td>+16663</td><td>3343</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+20008</td><td>+20005</td><td>+18031</td><td>1977</td><td>+16941</td><td>3067</td></tr><tr><td>+20010</td><td>+20007</td><td>+18031</td><td>1979</td><td>+18031</td><td>1979</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+20012</td><td>+20009</td><td>+18031</td><td>1981</td><td>+5671</td><td>14341</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+20014</td><td>+20011</td><td>+18031</td><td>1983</td><td>+4101</td><td>15913</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+20016</td><td>+20013</td><td>+18031</td><td>1985</td><td>+3229</td><td>16787</td></tr><tr><td>+20018</td><td>+20015</td><td>+18031</td><td>1987</td><td>+18031</td><td>1987</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+20020</td><td>+20017</td><td>+18031</td><td>1989</td><td>+16941</td><td>3079</td></tr><tr><td>And so on..................</td><td>..................................................</td><td>..................................................</td><td>..................................................</td><td>..................................................</td><td>..................................................</td></tr><tr><td>2n-T</td><td>(2n-3)-T</td><td>W<sub>2n</sub>-T</td><td>T<sub>2n</sub></td><td>V<sub>2n</sub> - T</td><td>U<sub>2n</sub></td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+40000</td><td>+39997</td><td>+29737</td><td>10263</td><td>+21567</td><td>18433</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+40002</td><td>+39999</td><td>+29737</td><td>10265</td><td>+22273</td><td>17729</td></tr><tr><td>+40004</td><td>+40001</td><td>+29737</td><td>10267</td><td>+29737</td><td>10267</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+40006</td><td>+40003</td><td>+29737</td><td>10269</td><td>+21567</td><td>18439</td></tr><tr><td>+40008</td><td>+40005</td><td>+29737</td><td>10271</td><td>+29737</td><td>10271</td></tr><tr><td>+40010</td><td>+40007</td><td>+29737</td><td>10273</td><td>+29737</td><td>10273</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+40012</td><td>+40009</td><td>+29737</td><td>10275</td><td>+10401</td><td>29611</td></tr></tbody></table>
<table><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+40014</td><td>+40011</td><td>+29737</td><td>10277</td><td>-56003</td><td>96017</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+40016</td><td>+40013</td><td>+29737</td><td>10279</td><td>+27057</td><td>12959</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+40018</td><td>+40015</td><td>+29737</td><td>10281</td><td>+25947</td><td>14071</td></tr><tr><td>Ret)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>+40020</td><td>+40017</td><td>+29737</td><td>10283</td><td>+24493</td><td>15527</td></tr></table>

## § 11 Perspectives and generalizations.

11.1 We can consider studying other Goldbach sequences ($G'_{2n}$) and ($G''_{2n}$) independent of ($G_{2n}$) using the increasing sequences of prime numbers ($W'_{2n}$), (see paragraph § 8) and ($W''_{2n}$) defined by:

For any integer $n \ge 3$, $W''_{2n} = \text{Sup}(p \in \text{IP}: p \le f(n))$, $f$ being a function defined on the interval $I=[3;+\infty[$ and satisfying the following conditions:

$f$ is strictly increasing on the interval $I$, $\lim_{x \to +\infty} f(x) = +\infty$, $f(3)=3$ and $\forall x \in I$, $f(x) \le 2x - 3$.

Thus, we can choose for example, one of the following functions defined on $I$.

a) $f: x \to ax+3-3a$; ($a \in \text{IR}: 0<a \le 2$).

b) $g: x \to [2\sqrt{x} - 2, \sqrt{6} + 3]$, ($[.]$ designating the integer part of the real number $x$).

c) $h: x \to \ln(x)+3-\ln(3)$.

11.2 Using this method, it would be interesting to study the Schnirelmann density [13] of certain prime numbers like 3, 5, 7, 11, ..., in the sequence ($U_{2n}$) and in arithmetic sequences (Dirichlet's theorem) for $n \in [K_N; P_N]$ as a function of $N$.

11.3 It is possible to exceed the values indicated in the Table ($2n=10^{1000}$) by optimizing this algorithm, by using supercomputers and more efficient scientific calculation software as Maple or PARI/GP.

11.4 Diophantine equations and conjectures of the same nature (Lagrange-Lemoine-Levy's conjecture, for example) can be addressed with similar processes and algorithms. To validate the Lagrange conjecture, we can study the following sequences of prime numbers ($WL_{2n}$), ($VL_{2n}$) and ($UL_{2n}$) defined by: For any integer $n \ge 3$, $WL_{2n} = \text{Sup}(p \in \text{IP}: p \le n - 1)$ and,

1) If $TL_{2n} = (2n+1 - 2.WL_{2n})$ is a prime number, then we put: $VL_{2n} = WL_{2n}$ and $UL_{2n} = TL_{2n}$.

2) If $TL_{2n}$ is a composite number, then there exists an integer $k$, ($1 \le k \le n - 3$) such that: $UL_{2(n-k)} + 2k$ is a prime number and we put: $VL_{2n} = VL_{2(n-k)}$ and $UL_{2n} = UL_{2(n-k)} + 2k$.

3) With the same type of process, one can validate generalized Bezout-Goldbach's conjectures of the form:

a) Given two integers K and Q odd and coprime, for any integer $n$ such that: ($2n \ge 3(K+Q)$), there exist two prime numbers $U'''_{2n}$ and $V'''_{2n}$ verifying: $K.U'''_{2n} + Q.V'''_{2n} = 2n$.

b) Given two integers K and Q of different parity and coprime, for any integer $n$ such that: ($2n \ge 3(K+Q)$), there exist two prime numbers $U'''_{2n}$ and $V'''_{2n}$ verifying: $K.U'''_{2n} + Q.V'''_{2n} = 2n + 1$.

## § 12 Conclusion.

12.1 We have explicitly constructed a unique Goldbach sequence ($G_{2n}$) = ($U_{2n}; V_{2n}$) such that $U_{2n}$ and $V_{2n}$ are prime numbers satisfying: $U_{2n} + V_{2n} = 2n$ for any integer $n \ge 2$, using a simple and very efficient unconventional "local recursive" algorithm.

12.2 We broke the record held by Silva [15] on a personal computer, and we were able to reach values of the order of $2n=10^{1000}$ with a reasonable calculation time (less than three hours for the evaluation of ten values of $U_{2n}$ and $V_{2n}$).
12.3 For a given integer $n$, the evaluation of the terms $U_{2n}$ and $V_{2n}$ does not require the calculation of all the previous terms $U_{2k}$ and $V_{2k}$, ($1 \le k < n-1$). You just need to know the prime numbers $p_i$ and the terms $V_{2r}$ in a neighborhood of $2n$, (on the interval $J = [2n-K1.log(2n);2n]$). This property allows a fast calculation of $U_{2n}$ and $V_{2n}$ even for values of $2n$ of the order of $10^{1000}$.

12.4 Therefore, the strong EULER-GOLDBACH conjecture is true.

### References.

[1] Bombieri and Davenport, "Small differences between prime numbers", Proc. Roy. Soc. Ser. A293 (1966), p. 1-18.

[2] Jing Run Chen, "On the representation of a large even integer as the sum of a prime and the product of at most two primes". Kexue Tongbao 17 (1966), p. 385-386 (Chinese).

[3] Harald Cramer, "On the order of magnitude of the difference between consecutive prime numbers", Acta Arithmetica vol. 2, (1986), p.23-46.

[4] Erdos, "On a new method in elementary number theory which leads to an elementary proof of the prime number theorem", Proc. Natl. Acad. Sci. USA 36, p. 374-384 (1949).

[5] Euclid, (trad. Bernard Vitrac), «Euclid's Elements», Ed. PUF Paris, vol.2, p. 444-446 and p. 339-341, (1994).

[6] J. Hadamard, «On the zeros of the function ζ(s) of Riemann». C. R. 122, p.1470-1473 (1896), and «On the distribution of zeros of the function ζ(s) and its arithmetical consequences». S. M. F. Bull. 24, p. 199-220 (1896).

[7] Hardy & Wright, "An introduction to the Theory of numbers", Oxford: Oxford University Press 621 p. (2008).

[8] Helfgott & Platt, "The ternary Goldbach conjecture", Gaz. Math. Soc. Math. Fr. 140, p. 5-18 (2014). "The weak Goldbach conjecture", Gac. R. Soc. Mat. Esp. 16, No. 4, 709-726 (2013). "Numerical verification of the ternary Goldbach conjecture up to 8,875. 10³⁰.", Exp. Math. 22, No. 4, 406-409 (2013).

[9] Iwaniec & Pintz, "Primes in short intervals", Monatsh. Math. 98, p. 115-143 (1984).

[10] E. Landau, "Handbuch der Lehre von der Verteilung der Primzahlen", vol. 1 und vol. 2 (1909), published by Chelsea Publishing Company (1953).

[11] J. Littlewood, "Sur la distribution des nombres premiers", CRAS Paris, vol. 158, (1914), p. 1869-1875.

[12] Ramaré & Saouter, "Short effective intervals containing primes", J. Number theory, 98, No. 1, p.10-33 (2003).

[13] Schnirelmann & Mann, "Schnirelmann density", Wikipédia, (on line, internet) and "A proof of the fundamental theorem on the density of sums of sets of positive integers", Annals of Math., 2ème série, vol. 43, no3, (1942), p. 523-527.

[14] Selberg, "Lectures on sieves", in Collected papers II, Springer-Verlag, Berlin, (1991), p. 65-247.

[15] Silva, Herzog & Pardi, "Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4.10¹⁸". Math. Comput. 83, No. 288, p. 2033-2060 (2014).

[16] T. Tao, "Every odd number greater than 1 is the sum of at most five primes", Math. Comput. 83, No. 286, p.997-1038 (2014).

[17] Tchebychev, "Mémoire sur les nombres premiers" J. math. pures et appliquées, 1ère série, t.17, (1852), p. 366-390 and p. 381-382.

[18] Jean Charles de La Vallée-Poussin, «Analytical research on the theory of prime numbers.», Brux. S. sc. 21 B, (1896), p. 183-256, 281-362, 363-397, vol.21 B, p. 351-368

[19] Vinogradov, "Representation of an odd number as a sum of three primes". Dokl. Akad.Nauk. SSR, 15:291-294, (1937).

[20] Zhang, "Bounded gaps between primes", Ann. Math. (2) 179, No. 3, p.1121-1174 (2014).

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