Prime Numbers Classification and Composite Numbers Factorization

On the ground of Prime Numbers Classification it is generalized approach to Composite Numbers Factorization presented. The resulting task goes to the different Diophantine equations reducing. Some methods of reducing and its practical application, in particular for Fermat numbers, are demonstrated.


Introduction
In simplest presentation the task of the natural numbers factorization determinate as the nontrivial problem of decision for equations in positive numbers X Y = N, X, Y > 1 (1) Till present time we have not generalized algebraic methods for those equations, but we have some calculating algorithms for determination all prime components of number N (look [1,3] and its references). As N rise very big, the calculating problem is growing very hard even for contemporary computing techniques. That situation is remained for numbers sequences, defined algebraic formulas, as Fermat numbers, Mersenne numbers and etc..
As far back Gauss remarked, «… that problem, how to discriminate composite numbers from primes and decompose first's on its prime components belong to the whole arithmetic major problems» [2, p. 496]. Since then on, beside way proposed by Gauss, it was found many algorithms, for that goal intended. There we will consider only second problem -composite numbers factorization. Present state and achievements at that field is reviewed, for example, in [1,3], and it connected with algorithmic theory of numbers and computers programming. Though with hard efforts it was attained decomposing on factors very big numbers, generalized theory of factorization doesn't exist, and we may only talk about phenomenological and particular methods.
In that article the problem of factorization examined on the basis of the prime numbers classification [4]. It permits to reduce factorization to decision of sufficiently simple Diophantine equations.

Two Prime Numbers Products Classification
According to classification, presented in [4, p. 37 in general the problem of factorization we may to solve as the problem of factorization for two prime numbers products. If in result one of components will be composite, then to it we may to apply the same approach.

Factorization and Diophantine Equations Methods of Decision
Then we examine more complex problem, which Shanks solved with his specially developed algorithm and program for simplest computer or programming calculator [5]. The example that Shanks examine N=13290059, our methods may to solve with aid of «pencil and paper».

Fermat Numbers Factorization
One of old classic problem is factorization of Fermat numbers, which have presentation In work [4, p. 38] it was come in sight that components of composite Fermat numbers must be only prime numbers 24m+1 in some quantity and prime numbers 24m+17 in odd quantity. So in that case the problem of factorization is simplified, as for decision it is enough to examine only one equation.
Besides that, it is another property of the components, which we may to use: components of composite n  It is easy to find analogical presentation for right side (28) and end decision Of course, the factorization search in that case is relieved, as the result is known. But our aim is to demonstrate process of decision with elementary methods, which do not used since Fermat time to present. Fermat error suggested (in 1640-1654) that all numbers n F are primes. Euler since century found component 641, that divide 5 F . Next factorization for 6 F was found at 1880 [6, pp. 375-380]. Those give us some foundations to suggest: if the way demonstrated above may be known Fermat and his colleagues, the development of factorization methods went another and more rapid road.

Fermat Numbers Factorization (Continuation)
The problem 6 F factorization is harder in sense of calculations. In general the complicate situation with Fermat numbers consist in its rapid increasing and even for best method of decision get up straight technical complexities in the field of «big numbers». The case with 6 F is transitional as its factorization was found «after several months' labor» [6, p. 377].
After further substitutions (33) to last formulae (32) we have Diophantine equation third degree, where A is variable parameter 3 2 257n +(768A 85)n +(455+A)n+768nr+r=40722652880 − That equation has transformation to 3 2 257n 85n +455n+(768n+1)(An+r)=40722652880 − Now it is great amount of studies, guides and manuals on Diophantine equations, but as we may to see, the approach to factorization presented here was unknown, so corresponding equations have no attention of researchers. The equations of second degree is mentioned in [6, pp. 412-417], as with use of classical methods lead to problem of factorization, but another methods did not studied. Diophantine equation third degree, similar to (34-35) also did not study, but may be classified as inhomogeneous Thue equation ( , ) f x y m = [7]. There authors assert, that their paper «gives in details a practical general method for explicit determination of all solutions of any Thue equation». But now it is unknown Thue equations application for factorization problem.
With another side, elliptic curves third degree application, i.e. Thue equations, is known in factorization algorithms [1, p. 83; 3, p. 319]. However, those connections are outside present article and demands serious researching.

Conclusions
For our mind, we presented new approach to factorization problem. It is surprising, why it was unknown some centuries ago, as demand highly elementary mathematical means and notions. Here we in the main examined examples, that may be decided with «pencil and paper». Diophantine equations, ensuing from presented concrete tasks and its methods of decision are not exhaustive. It needs further development and researching.
Now it is known private effective algorithms and computer programs for factorizations in the wide diapason to big numbers. In that sense the list of references has not fullness and is biased to aim of article. For other well-known material on Factorization we refer to MathWorld -A Wolfram Web Resource.
There is whole factorization for 11 F and all previous Fermat numbers. But 12 F is factored incomplete and composite fragment of it has 1133 decimal digits. So it is interesting whether presented here Diophantine equations methods will be effective for big number problems. However, that needs serious special studying with high quality programming and technical means, and so exceed the limits of present article, as its main aim is only to demonstrate opportunity of new generalized approach for factorization.