Number theory

Number theory is a branch of mathematics devoted primarily to the study of the integers. Any attempt to such a study naturally leads to an examination of the properties of that which integers are made of (namely, prime numbers) as well as the properties of objects made out of integers (such as rational numbers) or defined as generalisations of the integers (algebraic integers).

Origins

See: History of number theory

Given an equation or equations, can we find solutions that are integers? This is one of the basic questions of number theory. In some cultures, integer lengths were held to have religious significance. For example, some early Indian texts enjoin the reader to build altars in such a way that certain distances are integers - and this, in modern language, is the same as having to give integer solutions to some equations. On the other hand, integers are easy to write down, manipulate and experiment with; thus, for example, the Babylonian tablet Plimpton 322, which reveals that Babylonians knew how to construct "Pythagorean triples", is nothing other than a table of integer solutions.

What about finding rational solutions to equations? One can make a distinction between rational and irrational solutions only if one knows that not all numbers are rational; this was first shown by the Pythagoreans. Rather than "not all numbers are rational", they might have said that not all lengths can be expressed as ratios, i.e., rationals; for the Greeks, a number was an integer or a rational. The discovery of irrationals is said to have come as a shock to the Pythagoreans, for whom numbers - that is, rationals and integers - were supposed to be the key to harmony and the universe.

One may perhaps say, then, that the roots of number theory lie in the number mysticism of the ancients and in the curiosity of habitual calculators.

Pure mathematics, in the sense of a formal pursuit with its own goals, dates from Classical Greece. Hellenistic mathematicians had a keen interest in what would later be called number theory: Euclid devoted part of his Elements to prime numbers and factorization. (Some questions on divisibility and congruences were studied elsewhere in antiquity; see the Chinese remainder theorem). Five centuries and a half after Euclid, Diophantus would devote himself to the study of rational solutions to equations.

In the next thousand years, Islamic mathematics dealt with some questions related to congruences, while Indian mathematicians of the classical period found the first systematic method for finding integer solutions to quadratic equations in the case in which such a problem is difficult (see Pell's equation).

Number theory started to flower in western Europe thanks to a renewed study of the works of Greek antiquity. Fermat's careful reading of Diophantus's Arithmetica resulted spurred him to many new results and conjectures around which further research in the field crystallised.

Subfields

Analytic number theory

Analytic number theory is generally held to denote the study of problems in number theory by analytic means, i.e., by the tools of calculus. Some would emphasize the use of complex analysis: the study of the Riemann zeta function and other L-functions can be seen as the epitome of analytic number theory. At the same time, the subfield is often held to cover studies of elementary problems by elementary means, e.g., the study of the divisors of a number without the use of analysis, or the application of sieve methods. Most generally, a problem in number theory is said to be analytic if it involves statements on quantity or distribution, and if the ordering of the objects studied (e.g., the primes) is crucial. At least two different senses of the word analytic are thus being conflated in common parlance.

The following are examples of problems in analytic number theory: the prime number theorem, the Goldbach conjecture (or the twin prime conjecture, or the Hardy-Littlewood conjectures), the Waring problem and the Riemann Hypothesis. Some of the most important tools of analytic number theory are the circle method, sieve methods and L-functions (or, rather, the study of their properties).