## The Structurization of a Set of Positive Integers and Its Application to the Solution of the Twin Primes Problem

**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.

**Comments:** 25 Pages.

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### Submission history

[v1] 2013-08-05 15:28:53

[v2] 2013-08-07 06:33:23

[v3] 2013-09-20 09:07:16

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