Number theory
Number theory is the branch of pure mathematics devoted to the study of the properties of numbers.
Introduction
Carl Friedrich Gauss, one of the greatest mathematicians to have ever lived, said, "Mathematics is the queen of the sciences and number theory is the queen of mathematics".
Number theory is often described as an eclectic collection of problems with simple statements and complicated solutions. The most famous example perhaps is Fermat's Last Theorem which simply states that the cube of a whole number cannot be the sum of two cubes, nor the fourth power the sum of fourth powers and likewise for all powers higher than two. It was stated by Pierre de Fermat around 1630 and was finally resolved in 1994 by Andrew Wiles with some help from Richard Taylor in a set of two highly complicated papers which were built upon decades of research by numerous mathematicians and which, at the time, few people understood due to the exotic nature of the techniques employed.
Number theory begins with arithmetic, the study of the natural numbers 1,2,3,... Important problems concern properties of various subsets of the natural numbers such as the prime numbers, figurate numbers, perfect numbers, perfect powers or of sequences of natural numbers, such as the Fibonacci or Lucas sequences. Particularly important to arithmetic is modulo arithmetic involving congruences between whole numbers.
If one allows zero and negative whole numbers, one obtains the integers. As one begins to axiomatize the fundamental properties of various number systems one quickly ends up studying various abstract algebraic systems which encapsulate important properties of those number systems and so abstract algebra becomes an important tool in number theory. For example, the ring of integers has the rational numbers as its field of fractions. Important sequences of rational numbers include Farey sequences and Bernoullli numbers.
Adding in the concept of convergence to a limit, one fills the `holes' in the rational number system with irrational numbers, thus obtaining the real number system. At this point, one must use tools from analysis to deal with issues of convergence. Typical problems involve the approximation of real numbers by rationals and determining whether various real numbers are irrational or rational.
Solutions to polynomial equations are known as algebraic numbers. However solutions don't necessarily lie in the real numbers and one again extends the number system to include complex numbers so that all polynomial equations can be solved. Complex numbers which are not the roots of polynomials are known as transcendental numbers. The subfield of number theory known as algebraic number theory studies the properties of fields of algebraic numbers containing rings of integers with properties similar to those of the ordinary ring of (rational) integers contained within the field of rational numbers. Typical number theoretic problems include determining whether certain numbers are transcendental or not.
Certain generalizations of the complex numbers exist, including the Hamiltonian quaternions and the octonians. These arise naturally in the study of certain mathematical objects and systems in other fields of mathematics.
Modern day number theory overlaps with many other fields and subfields of mathematics, including group theory and more generally abstract algebra, function theory, algebraic geometry, elliptic curves, modular forms and various parts of analysis. At the interface with each such field, numerous important problems exist. For example, the study of Diophantine equations, such as those which occur in Fermat's last theorem, lies at the interface of algebraic geometry and number theory. The ABC conjecture and the Beale prize problem are other famous examples of number theoretic problems which have an algebro-geometric side.
Another important number system is the -adic numbers. These arise when one tries to complete the rational number system with respect to a different topology than that which leads to the real numbers. In particular, in the -adic topology two numbers are considered to be `close' together if a high power of a prime divides their difference. The study of such completions of the rationals involves topological techniques, in particular topological groups, rings and modules.
More specialised areas of number theory include Iwasawa theory which studies towers of algebraic number fields and relates their properties to L-series. A famous number theoretical problem which relates to L-series in a different way is the Riemann hypothesis about the zeroes of a function called the Riemann zeta function. This is one of the Millenium Prize Problems for which the Clay Mathematics Institute has offered one million dollars for a correct solution. The Riemann hypothesis encodes important information about the prime numbers and many generalisations exist with number theoretical, computational, geometric and even physical applications.
Coding theory is a modern area of mathematics involving a lot of number theory and which has applications in the world of computing and data communication. It lies at the interface of mathematics and computer science. Cryptography and cryptanalysis are two other such fields, which relate to the making and breaking of codes, respectively, and again many of the important techniques rely heavily on number theory.