Number Theory & Combinatorics

Both number theory and combinatorics are part of what is called discrete mathematics, which has important applications in computer science and information technology, as well as an intrinsic elegance and fascination for mathematicians, professionals and amateurs alike.

Number theory originated as the study of the structure and properties of the ordinary integers, but nowadays has expanded into the study of analogous properties of other (possibly non-commutative) rings. The methods employed are sometimes algebraic (e.g. group theory, ring theory and field theory, especially Galois theory), sometimes analytic (e.g. complex variable theory, Fourier analysis), sometimes geometric (e.g. algebraic geometry of curves and higher-dimensional varieties, Diophantine geometry), sometimes probabilistic (e.g. additive number theory) and sometimes combinatorial (e.g. graph theory, generating functions).

In combinatorics one is usually concerned with a finite set with some additional structure (e.g. a projective geometry, a graph or a block-design), and seeks to relate it to some already-known set of the same kind, or perhaps to show that certain structures can (or cannot) be imposed on a given set. Another type of question is the enumeration of particular kinds of structures (e.g. how many connected graphs are there on n vertices?).

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