Authors: Alexander Fedorov
One of causes why Twin Primes problem was unsolved over a long period is that pairs of Twin Primes (PTP) are considered separately from other pairs of Twin Numbers (PTN). By purpose of this work is research of connections between different types of PTN. For realization of this purpose by author was developed the "Arithmetic of pairs of Twin Numbers" (APTN). In APTN are defined three types PTN. As shown in APTN all types PTN are connected with each other by relations which represent distribution of prime and composite positive integers less than n between them. On the basis of this relations (axioms APTN) are deduced formulas for computation of the number of PTN (NPTN) for each types. In APTN also is defined and computed Average value of the number of pairs are formed from odd prime and composite positive integers $ < n $ . Separately AVNPP for prime and AVNPC for composite positive integers. We also deduced formulas for computation of deviation NPTN from AVNPP and AVNPC. It was shown that if $n$ go to infinity then NPTN go to AVNPC or AVNPP respectively that permit to apply formulas for AVNPP and AVNPC to computation of NPTN. At the end is produced the proof of the Twin Primes problem with help of APTN. It is shown that if n go to infinity then NPTP go to infinity.
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