On the Elementary Solution of Goldbach's Binary Conjecture via 𝕂-Indices of primes: The Lacunary Case

In this paper, we present a solution to a famous open problem of number theory - Goldbach’s binary conjecture. The proof of Goldbach’s binary conjecture is elementary and based on a combinatorial covering argument. We show that the proof of Goldbach’s binary conjecture is reduced to the proof of a conjecture on the covering for the set of natural numbers except 1 by means of the set of sums of pairs of natural numbers, each of which corresponds to a prime or twin primes. We construct a Generating set 𝕂 of integers and prove a Covering lemma (for the special "lacunary" case), which shows that the set 𝕂 is an additive basis of order 2 for the set of natural numbers ℕ except 1. The proof of this lemma (for the special case "lacuna" of configuration for pairs (k,N-k) ) proceeds by contradiction, using Bertrand's Postulate (the Bertrand–Chebyshev theorem) to rule out the existence of a counterexample. From this lemma (in such restriction noted), Goldbach’s binary conjecture follows directly. The approach does not rely on analytic methods or heavy machinery. The proof covers the lacunary configuration, in which all 𝕂-indices in the interval N/2,N are absent. The general case is treated in a separate work.