In this article the proof of the binary Goldbach conjecture is established (Any integer greater than one is the mean arithmetic of two positive primes) . To this end, Chen’s weak conjecture is proved (Any even integer greater than one is the difference of two positive primes) and a "localised" algorithm is developed for the construction of two recurrent sequences of primes ( U_2n ) and ( V_2n ), ( ( U_2n ) dependent of ( V_2n) ) such that for any integer n ≥ 2 their sum is equal to 2n : ( U_2n ) and ( V_2n ) are extreme Goldbach decomponents. To form them, a third sequence of primes ( W_2n) is defined for any integer n ≥ 3 by V_2n = Sup ( p ∈ P : p ≤ 2n - 3 ) , P denoting the set of positive primes. The Goldbach conjecture has been prove d for all even integers 2n between 4 and 4.1018. and in the neighbourhood of 10^100 , 10^200 and 10^300 for intervals of amplitude 10^9 . The table of extreme Goldbach decomposants, compiled using the programs in Appendix 14 and written with the Maxima and Maple scientific computing software, as well as files from ResearchGate, Internet Archive, and the OEIS, reaches values of the order of 2n = 10^5000. In addition, a global proof by strong recurrence "finite ascent and descent method" on all the Goldbach decomponents is provided by using sequences of primes (��2_q2n) defined by :��_q2n = Sup ( p ∈ P : p ≤ 2n - q ) for any odd positive prime q , and a majorization of U_2n by n^0.525 , 0.7ln_2.2(n) with probability one and 5 ln1.3(n) on average for any integer n large enough is justified.. Finally, the Lagrange-Lemoine-Levy (3L) conjecture and its generalization called "Bachet-Bézout-Goldbach"(BBG) conjecture are proven by the same type of method.