Fillings method in number theory

The concept of an integer with the complete right should be named one of the first abstract object scientific cognition of the world, a tool of which has become the mathematics. The representation about prime number, allocated by distinctive and well by known features has occurred also in antiquity. The further researches have delivered problems of prime numbers distribution, but the progress in the sanction some of them was scheduled in comparison recently.

The successes in an essential degree have appeared connected with development of the sieving process, at one time offered by Eratosthenes. At the same time the series of problems, despite of elementary mathematical statements, has not yielded all efforts and progress to new results was essentially slowed. To the present time is not new directions, leading though presumably to the for a long time scheduled purposes.

The reason of a created situation can be only one: the sieving process as any method has restricted capacity, adequate to limiting level of achievement and conclusions. And this potential, as far as demonstrate real development, is practically exhausted. The absence of fresh ideas in so before a fruitful direction of the number theory also testifies to a limit of research opportunities of the profound antiquity method.

The sieving process has a series essential and in essence ineradicable defects, among which very weak interdependence between base (prime) numbers, forming a sieve. Some characteristics of distribution of primes in a sequence natural is sharp depend from a set of forming values. Therefore there was a idea to study them on classes of sets, being not limited only primes. This idea has resulted to creation of a quite new method of the number theory -fillings method [1,2].

Beginning of the mathematics development as integer arithmetics is impossible to separate from a problem of divisibility. Probably, definition of prime number was known long before Euclid, but only he proved by the simple and elegant theorem the Cantorian equipotency of countable sets of natural and prime numbers. The representation of actual infinity, necessary for realization of this fact, was produced by Phales school considerably earlier.

The concepts of countability, potential and actual infinity, series of methods and operations over finite and unlimited numerical objects, though and were received in half-intuitional level, have nevertheless found reasonably qualified base for use. Boiling joy of seizing by infinity axiom and unrestrained flight of the idea have not allowed to pay attention for a computability problem, but this permanent sin there were at soul of the mathematics (and not only mathematics) and now.

Ancient Greek approach to registration and definition of number -even integer and infinite, contained in core the ineradicable factor of absolute unrestriction. Naturally, such significant fact has an effect in speed creation of numerical objects of actual infinity. At all development the mathematics (and the science as a whole) have remained at the achieved stage. In particular, one of examples of a extreme degree unreasonable idealization of a theory has acted Cantorian theory of sets.

Considerable achievement of a modern science including the mathematics can not, nevertheless, to hide sorrowful facts of deep failures, hopeless deadlocks and insoluble contradictions. Their initial source is the only infinity axiom, transformed a zone of knowledge to field of an authority of an infinity quantifier. The consequences of this phenomenon reach much further problems only sciences.

However all offered development are oriented to the conventional schemes [3] and will be conducted without references to non-foundation, indefiniteness and unprovableness of infinity axiom. Though the series determining and known rules, in particular connected with concept of set of all primes, does not make sense outside of mentioned axiom at all its numerous defects. The ideas, connected with determining, central and paramount role of infinity axiom, are considered and carefully justified in work [4][5][6].

The Euclidean theorem about infinity of primes set can be considered as the first research, the provable statement of a modern stage of consciousness -stage of infinity axiom. In this sense the theorem has acted as the first-born of a present condition of the science. In a maximum degree is fair, that those there was the theorem about numbers. The modern stage of development and level cognition was determined earlier as time by creation and adaptation of unrestriction idea.

A following major studied stage of primes distribution was the sieve of Eratosthenes, realizing finite algorithm of reception all primes, smaller p 2 n , if series primes down to p n-1 inclusive is known. It sieve will hereinafter appear great-parent of updating and basis for theoretical research of the various characteristics and features of primes distribution accommodation in the natural

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