Authors: Alexander Fedorov
One of causes why Goldbach's binary problem was unsolved over a long period is that binary representations of even integer 2n (BR2n) in the view of a sum of two odd primes(VSTOP) are considered separately from other BR2n. By purpose of this work is research of connections between different types of BR2n. For realization of this purpose by author was developed the "Arithmetic of binary representations of even positive integer 2n" (ABR2n). In ABR2n are defined four types BR2n. As shown in ABR2n all types BR2n are connected with each other by relations which represent distribution of prime and composite positive integers less than 2n between them. On the basis of this relations (axioms ABR2n) are deduced formulas for computation of the number of BR2n (NBR2n) for each types. In ABR2n also is defined and computed Average value of the number of binary sums are formed from odd prime and composite positive integers $ < 2n $ (AVNBS). Separately AVNBS for prime and AVNBS for composite positive integers. We also deduced formulas for computation of deviation NBR2n from AVNBS. It was shown that if $n$ go to infinity then NBR2n go to AVNBS that permit to apply formulas for AVNBS to computation of NBR2n. At the end is produced the proof of the Goldbach's binary problem with help of ABR2n. For it apply method of a proof by contradiction in which we make an assumption that for any 2n not exist BR2n in the VSTOP then make computations at this conditions then we come to contradiction. Hence our assumption is false and forall $2n > 2$ exist BR2n in the VSTOP.
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