Since the elements are closed, the set Θ = {6λ ± 1; λ ∈ N }, is a semigroup with respect to the operation of multiplication. The paper focuses on presenting even numbers ζ > 8 in the form of sums of two elements: θ1 = 6λ1 ± 1 and θ2 = 6λ2 ± 1 from the set Θ. Any even number ζ > 8 is comparable with one of the numbers m = (0, 2, -2), according to (mod 6). According to the remnants listed m , even numbers ζ > 8 are divided into 3 types. Each type has its own way of presenting sums in the form of two elements from the set Θ. For any even number ζ > 8 on the segment [1 ÷ ν] there is always at least a pair of numbers (λ1, λ2) ∈ [1 ÷ ν], that both are elements from the union of sets: the parameters of the prime numbers (twins) and the parameters (composite and prime) of numbers Θ. A variant of the solution of Goldbach - Euler conjecture for even numbers ζ > 8 is given on the set of primes P. Goldbach - Euler conjecture is also solvable in the set of prime numbers (twins), if the parameters of numbers θ1 and θ2, i.e. λ1 and λ2 belong to the set N \ FN , where FN is the set of parameters of the composite numbers Θ on the segment [1 ÷ ν].
BINARY GOLDBACH-EULER PROBLEM IN THE SET Θ = {6k ±1/k∈N}
