Number theory

Number theory is a branch of mathematics devoted primarily to the study of the integers. Any attempt to conduct 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

The dawn of arithmetic

Given an equation or equations, can we find solutions that are integers? This is one of the basic questions of number theory. (Number theory is a modern term; until not too long ago, the expressions arithmetic or higher arithmetic were used instead.) 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.

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.

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 saying that "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.

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 first flowered in western Europe in the late Rennaissance, thanks to a renewed study of the works of Greek antiquity. Fermat's careful reading of Diophantus's Arithmetica 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. A problem in number theory can be said to be analytic simply if it involves statements on quantity or distribution, or if the ordering of the objects studied (e.g., the primes) is crucial. Several different senses of the word analytic are thus conflated in the designation analytic number theory as it is commonly used.

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).

One may ask analytic questions about algebraic numbers, and use analytic means to answer such questions; it is thus that algebraic and analytic number theory intersect. For example, one may define prime ideals (generalisations of prime numbers living in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question can be answered by means of an examination of Dedekind L functions, which are generalisations of the Riemann zeta function, an all-important analytic object that controls the distribution of prime numbers.

Algebraic number theory

Algebraic number theory studies fields of algebraic numbers, which are generalisations of the rational numbers. (Briefly, an algebraic number is any complex number that is a solution to some polynomial equation with rational coefficients.) Fields of algebraic numbers are also called number fields.

It could be argued that the simplest kind of number fields (viz., those of degree two over the rationals) were already studied by Gauss, as the discussion of quadratic forms in Disquisitiones arithmeticae can be restated in terms of ideals and norms in quadratic fields. For that matter, the 12th-century cakravala method amounts - in modern terms - to an algorithm for finding the units of a real quadratic number field. However, neither Bhāskara nor Gauss knew of number fields as such.

The grounds of the subject as we know it were set in the late nineteenth century, when ideal numbers, the theory of ideals and valuation theory were developed; these are three complementary ways of dealing with the lack of unique factorisation in algebraic number fields. A failure of awareness of this lack had led to an early erroneous "proof" of Fermat's Last Theorem by G. Lamé; the realisation that this proof was erroneous made others study the consequences of this lack, and ways in which it could be alleviated.

Number fields are often studied as extensions of smaller number fields: a number field L is said to be an extension of a number field K if L contains K. Classifying the possible extensions of a given number field is a difficult and partially open problem. Abelian extensions -- that is, L of K such that the Galois group Gal(L/K) of L over K is an abelian group -- are relatively well understood. Their classification was the object of the programme of class field theory, which was initiated in the late 19th century (partly by Kronecker and Eisenstein) and carried out largely in 1900--1950.

The Langlands program is sometimes described as an attempt to generalise class field theory to non-abelian extensions of number fields.

Diophantine Geometry

Consider an equation or system of equations. Does it have rational or integer solutions, and if so, how many? This is the central question of Diophantine geometry.

We may think of this question in the following graphic way. An equation in two variables defines a curve in the plane; more generally, an equation, or system of equations, in two or more variables defines a curve, a surface or some other such object in n-dimensional space. We are asking whether there are any rational points (points all of whose coordinates are rationals) or integer points (points all of whose coordinates are integers) on the curve or surface. If there are any such points on the curve or surface, we may ask how many there are, and, whether there are finitely or infinitely many, how they are distributed.

This rephrasing turns out to be felicitous. The finiteness or not of the number of rational or integer points on an algebraic curve - that is, rational or integer solutions to an equation $f(x,y)=0$, where $f$ is a polynomial in two variables - turns out to depend crucially on the genus of the curve. The genus can be defined as follows: allow the variables in $f(x,y)=0$ to be complex numbers; then $f(x,y)=0$ defines a 2-dimensional surface in 4-dimensional surface; count the number of (doughnut) holes in the surface; call this number the genus of $f(x,y)=0$. Other geometrical notions turn out to be just as crucial.

There is also the closely linked area of diophantine approximations: given a number $x$, how well can it be approximated by rationals? (We are looking for approximations that are good relative to the amount of space that it takes to write the rational: call $a/q$ $(gcd(a,q)=1)$ a good approximation to $x$ if $|x-a/q|<{\frac {1}{q^{C}}}$, where $C$ is large.) This question is of special interest if $x$ is an algebraic number. If $x$ cannot be well approximated, then some equations do not have integer or rational solutions. Moreover, several concepts (especially that of height) turn out to be crucial both in diophantine geometry and in the study of diophantine approximations.

Diophantine geometry should not be confused with the geometry of numbers, which is a collection of graphical methods for answering certain questions in algebraic number theory.

Arithmetic Combinatorics

Let $A$ be a set of integers. Consider the set $A+A$ consisting of all sums of two elements of $A$. Is $A+A$ much larger than A? Barely larger? If $A+A$ is barely larger than $A$, must $A$ have plenty of arithmetic structure - e.g., does it look like an arithmetic progression?

If we begin from a fairly "thick" infinite set $A$ (say, the primes), does it contain many elements in arithmetic progression: $a$, $a+b$, $a+2b$, $a+3b$, ... , $a+10b$, say? Should it be possible to write large integers as sums of elements of $A$?

These questions are characteristic of arithmetic combinatorics. This is a presently coalescing field; it subsumes additive number theory (which concerns itself with certain very specific sets $A$ of arithmetic significance, such as the primes or the squares) and, arguably, some of the geometry of numbers. Its connection with issues of growth and distribution make the development of links with ergodic theory likely. The term additive combinatorics is also used; however, the sets $A$ being studied need not be sets of integers, but rather subsets of non-commutative groups, for which the multiplication symbol, not the addition symbol, is traditionally used; they can also be subsets of rings, in which case the growth of $A+A$ and $A$·$A$ may be compared.