PROOF OF THE BINARY GOLDBACH CONJECTURE

Published October 19, 2024 | Version v1

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In this paper, a "local" algorithm is determined for the construction of two recurrent sequences of positive primes ( 𝑈2𝑛) and (

𝑉2𝑛), (( 𝑈2𝑛 ) dependent of ( 𝑉2𝑛) ), such that for each integer n ≥ 2, their sum is equal to 2n . To form this, a third sequence of

primes ( 𝑊2𝑛) is defined for any integer n ≥ 3 by : 𝑊2𝑛 = Sup( p ∈ 𝒫 : p ≤ 2n - 3 ) , where 𝒫 is the infinite set of primes. The

Goldbach conjecture has been proved for all even integers 2n between 4 and 4.1018. . In the table of terms of Goldbach

sequences given in appendix 10 , values of the order of 2n = 101000 are reached. This " finite ascent and descent " method

proves the binary Goldbach conjecture ; an analogous proof by recurrence is established and an increase in 𝑈2𝑛 by

0.7(ln(2𝑛)) 2.2

is justified. Moreover, the Lagrange-Lemoine-Levy conjecture and its generalization, the Bezout-Goldbach

conjecture, are proven by the same type of procedure.

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