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 (such as, for example, 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 suggests 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 numerical 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, "number" meant 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 divisibility, topics that belong unambiguously to number theory and are basic thereto; in particular, he gave the first known proof of the infinitude of primes. (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. Thus, nowadays, we speak of Diophantine equations when we speak of polynomial equations to which rational or integer solutions must be found. In essence, much of Diophantus's opus amounts to worked examples on how to find points with rational coordinates on curves described in the plane or in space.

In sixth-century India, Brahmagupta started the systematic study of indefinite quadratic equations -- in particular, the misnamed Pell equation, in which Archimedes may have first been interested. Later Sanskrit authors would follow Brahmagupta, using his technical terminology. A general method (the cakravāla) for solving Pell's equation was finally found by Jayadeva (cited in the eleventh century; his work is otherwise lost) and Bhāskara (twelfth century).

In the early ninth century, the caliph Al-Ma'mun ordered translations of many Greek mathematical works and at least one Sanskrit work (generally presumed to be Brahmagupta's), thus giving rise to the rich tradition of Islamic mathematics. Diophantus's main work, the Arithmetica, was translated into Arabic in the 10th century; al-Karajī would build on it within a generation. Al-Karajī's contemporary Ibn al-Haytham knew and used what would later be called Wilson's theorem, which, arguably, was thus the first clearly non-trivial result on congruences to prime moduli ever known.

Other than a treatise on squares in arithmetic progression by Fibonacci - who lived and studied in north Africa and Constantinople during his formative years, ca. 1175-1200 - no number theory to speak of was done in western Europe while it went through the Middle Ages. Matters started to change in Europe in the late Rennaissance, thanks to a renewed study of the works of Greek antiquity. The key catalyst was the textual emendation and translation into Latin of Diophantus's Arithmetica.

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 11th-century cakravāla 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. (For example, in the field generated by the rationals and $\scriptstyle {\sqrt {-5}}$, the number $6$ can be factorised both as $\scriptstyle 6=2\cdot 3$ and $\scriptstyle 6=(1+{\sqrt {-5}})(1-{\sqrt {-5}})$; all of $2$, $3$, $\scriptstyle 1+{\sqrt {-5}}$ and $\scriptstyle 1-{\sqrt {-5}}$ are irreducible, and thus, in a naïve sense, analogous to primes among the integers.) 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, extensions 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 how they are distributed. Most importantly: are there finitely or infinitely many rational points on a given curve (or surface)? What about integer points?

The rephrasing of questions on equations in terms of points on curves 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 $\scriptstyle a/q$ (with $\scriptstyle gcd(a,q)=1$) a good approximation to $x$ if $\scriptstyle |x-a/q|<{\frac {1}{q^{c}}}$, 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, together with some rapidly developing new material. Its focus on issues of growth and distribution make the strengthening 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.

Probabilistic number theory

Take a number at random between one and a million. How likely is it to be prime? This is just another way of asking how many primes there are between one and a million. Very well; ask further: how many prime divisors will it have, on average? How many divisors will it have altogether, and with what likelihood? What is the probability that it have many more or many fewer divisors or prime divisors than the average?

Much of probabilistic number theory can be seen as an important special case of the study of variables that are almost, but not quite, mutually independent. For example, the event that a random integer between one and a million be divisible by two and the event that it be divisible by three are almost independent, but not quite.

It is sometimes said that probabilistic combinatorics uses the fact that whatever happens with probability greater than $0$ must happen sometimes; one may say with equal justice that many applications of probabilistic number theory hinge on the fact that whatever is unusual must be rare. If certain algebraic objects (say, rational or integer solutions to certain equations) can be shown to be in the tail of certain sensibly defined distributions, it follows that there must be few of them; this is a very concrete non-probabilistic statement following from a probabilistic one.

Computations in number theory

While the word algorithm goes back only to certain readers of Al-Kwarismi, careful descriptions of methods of solution are older than proofs: such methods - that is, algorithms - are as old as any recognisable mathematics - Egyptian, Babylonian, Vedic, Chinese - whereas proofs appear only with the Greeks.

There are two natural questions: "can we compute this?" and "can we compute it rapidly?". Anybody can test whether a number is prime or, if it is not, split it into prime factors; doing so rapidly is another matter. We now know fast algorithms for testing primality, but, in spite of much work, no truly fast algorithm for factoring.

The difficulty of a computation can be useful: modern protocols for encrypting messages depend on functions that are known to all but whose inverses (a) are known only to one person, or to a few; (b) would take one too long a time to figure on one's own. For example, these functions can be such that their inverses can be computed only if certain large integers are factorised, or discrete logarithms of some sort are taken. While many difficult computational problems outside number theory are known, most working encryption protocols nowadays are based on the difficulty of a few number-theoretical problems.

On a different note - some things may not be computable at all; in fact, this can be proven. For example, Turing showed in 1936 that there is no algorithm for deciding in finite time whether a given algorithm ends in finite time. In 1970, it was proven that there is no algorithm for solving any and all Diophantine equations. That is: there is no universal method for deciding in finite time whether a given polynomial equation with integer coefficients has integer solutions or not.

References

André Weil, Number theory. An approach through history. From Hammurapi to Legendre. Birkhäuser, Boston, MA, 1984.