Proof of the Binary Goldbach Conjecture

In this article the proof of the binary Goldbach conjecture via Chen’s weak conjecture are established (any integer greater than three is the sum and the difference of two positive primes). To this end, a "localised" algorithm is developed for the construction of two recurrent sequences of extreme Goldbach decomponents (U2n and (V2n), ((U2n) dependent of (V2n)) verifying: for any integer \( n \ge 2 \)(U2n) and (V2n) are positive primes and U2n + V2n = 2n. To form them, a third sequence of primes (W2n) is defined for any integer \( n \ge 3 \) by W2n = Sup \( (p \in P : p \le 2n - 3) \), \( P \)denoting the set of positive primes. The Goldbach conjecture has been proved for all even integers 2n between 4 and 4.1018 and in the neighbourhoodof 10100,10200and 10300 for intervals of amplitude 109. The table of extreme Goldbach decomponents, compiled using the programs in Appendix 15 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 = 105000. Algorithms for locating Goldbach's decomponentss for very large values of 2n are also proposed. 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 (Wq2n) defined by: Wq2n = Sup \( (p \in P : p \le 2n - q) \) for any odd positive prime q, and a further proof by Euclidean divisions of 2n by its two assumed extreme Goldbach decomponents is announced by identifying uniqueness, coincidence and consistency of the two operations. Next, a majorization of U2n by n0.525, 0.7 ln2.2(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. In Aditional notes, we provide heuristic estimates for Goldbach's comet and presented a graphical synthesis using a reversible Goldbach tree (parallel algorithm).