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 of 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. If one begins to axiomatize the fundamental properties of various number systems one soon develops 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 field of algebraic number theory studies the properties of fields of algebraic numbers containing rings of integers with properties similar to those of the ordinary 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 in the study of various mathematical objects and systems where there is overlap between number theory and certain other fields of mathematics. Number theory overlaps with many other fields, including group theory, abstract algebra in general, function theory, algebraic geometry, including elliptic curves and various parts of analysis and 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, nowadays lies at the interface of algebraic geometry and number theory. The ABC conjecture and Beale prize problem are examples of such problems.

Another important number system is the p-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, two numbers are considered to be close together if a high power of a prime $p$ divides their difference. The study of such completions involves topological techniques, in particular topological groups, rings and modules.

More advanced areas of number theory include Iwasawa theory which studies towers of algebraic number fields and relates their properties to L-series. Other areas of number theory which relate to L-series include a famous problem called the Riemann hypothesis about the zeroes of a function called the Riemann zeta function.