# Combinatorial Number Theory

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## Combinatorial Number Theory

Combinatorial number theory is a new mathematical subject, that unifies combinatorics, number - and function theory to one new mathematical field. This is done by establishing 'combinatorial series', '- products', '- operators' and other mathematical objects, whereby the three problems (specific to combinatorial series and - products):

1. convergency
2. compactification
3. representation

occur, that are solved by:

1. the introduction of the combinatorial operator
2. the fundamental lemma (of combinatorial number theory)
3. the introduction of combinatorial functions
4. new convergency theorems

The focus lies on the categorization of types of combinatorial series, which are divided into two groups:

1. the natural occuring -
2. and the artificial constructed ones

which are further divided into three subgroups:

1. without interior functions
2. with logarithmic interior functions
3. and with arbitrary interior - and - exterior functions

In combinatorial number functions theory, an expansion of the combinatorial number theory, it is shown, that for an algorithm to sum up those series successful, the interior function has to have the same group homomorphism as the logarithmic - and the exterior - has to fullfil a duality.

## Combinatorial Number Functions Theory

The combinatorial number functions theory is a highly important extension of the combinatorial number theory, that expands the (algebraic) range of the combinatorial series and ${\displaystyle -}$ products from real ${\displaystyle -}$ to complex numbers. This allows for the construction of completely new (combinatorial) clusters, but it makes a cluster characterization necessary, which is quite severe and extensive. It also analyzes the creation of patterns and the minimal conditions needed to be provided by both, the interior ${\displaystyle -}$ and exterior function, in order to be able to sum up the combinatorial series. This leads to a characterization of combinatorial number functions theory, which in return provides new series. As integrals root in sums, their properties are examined as well, delivering an answers to the question, why complex integrals have the properties of path independence. At the end stands a classification of the different integrals, products and series.

## Chaotic Combinatorial Number Functions Theory

The chaotic combinatorial number functions theory is an expansion of the combinatorial number theory, that unifies combinatorics, number ${\displaystyle -}$, function ${\displaystyle -}$ and chaos theory. It deals with the question, what pattern can actually be summed up.

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## Combinatorial Functional Theory

The combinatorial functional theory utilizes concepts of combinatorial number theory and expands them onto functionals, respectively operators. Many aspects of the subject are unclear at the moment, especially to its extent and development. For further information see combinatorial von Neumann series.