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Number theory

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Number theory is a branch of pure 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 prime numbers (the building blocks of integers) 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).

Integers can be considered either in themselves or as solutions to equations (diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (e.g., the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, e.g., as approximated by the latter (diophantine approximation).

The older term for number theory is arithmetic; it was superseded by "number theory" in the nineteenth century, though the adjective arithmetical is still fully current. By 1921, T. L. Heath had to explain: "By arithmetic Plato meant, not arithmetic in our sense, but the science which considers numbers in themselves, in other words, what we mean by the Theory of Numbers."[1] The general public now uses arithmetic to mean elementary calculations, whereas mathematicians use arithmetic as this article shall, viz., as an older synonym for number theory. (The use of the term arithmetic for number theory has regained some ground since Heath's time, arguably in part due to French influence.[2] In particular, arithmetical is preferred as an adjective to number-theoretic. Moreover, "the arithmetic of" is used, whereas "the number theory of" is not; thus, for example, the arithmetic of elliptic curves.)

History

The beginnings

While there are elements of what in retrospect can be seen as number theory in Babylonian and ancient Chinese mathematics (see Plimpton 322 and the Chinese Remainder Theorem, respectively), the history of number theory truly starts with the Greek and Indian traditions.

The irrationality of $\scriptstyle \sqrt{2}$ is credited to the early Pythagoreans.[3] Euclid gave an algorithm for computing the greatest common divisor of two numbers (Euclid's Elements, Prop. VII.2) and a proof that there are infinitely many primes (Elements, Prop. IX.20). Much later in the Hellenistic period, Diophantus studied rational solutions to equations and systems of equations.

Results in number theory within Indian mathematics date from the period that would correspond to the medieval era in Europe. Aryabhata gave an algorithm for solving[4] pairs of congruences $\scriptstyle n\equiv a_1 \text{ mod } m_1$, $\scriptstyle n\equiv a_2 \text{ mod } m_2$, apparently with astronomical applications in mind.[5] Brahmagupta started the systematic study of indefinite quadratic equations, including what would later be misnamed Pell's equation. A general procedure (the chakravala, or "cyclic method") for solving Pell's equation was finally found by Jayadeva (cited in the eleventh century; his work is otherwise lost); the earliest surviving exposition appears in Bhāskara II's Bīja-gaṇita (twelfth century).[6] Unfortunately, these achievements were largely unknown in the West until the late eighteenth century.[7]

Much Greek mathematics and some Indian mathematics was available to Arabic scholars from the early ninth century onwards. Part of the treatise al-Fakhri (by al-Karajī, 953 - ca. 1029) builds on Diophantus's work to some extent.

Modern number theory

Modern number theory begins with Pierre de Fermat, inspired in part by his study of Diophantus. Continuous activity on the subject started almost a century later with Euler.[8] Lagrange provided proofs of some of Fermat's and Euler's key statements. He and Legendre also set the basis of the study of quadratic forms; Legendre was the first to state the law of quadratic reciprocity. In Disquisitiones Arithmeticae, Gauss gave the first valid proof of this law, developed the theory of quadratic forms further, and started the modern study of cyclotomy.

Starting early in the nineteenth century, the following developments gradually took place:

• The rise to self-consciousness of number theory (or higher arithmetic) as a field of study.[9]
• The development of much of modern mathematics necessary for basic modern number theory: complex analysis, group theory, Galois theory -- accompanied by greater rigor in analysis and abstraction in algebra.
• The rough subdivision of number theory into its modern subfields - in particular, analytic and algebraic number theory.

Algebraic number theory may be said to start with the study of reciprocity and cyclotomy, but truly came into its own with the development of abstract algebra and early ideal theory and valuation theory; see below. An obvious conventional starting point for analytic number theory would be Riemann's memoir on the Riemann zeta function (1859); there is also Dirichlet's theorem on arithmetic progressions, which preceded it in the study of the zeta function and even L-functions (for $\scriptstyle Re(s)>1$), or Jacobi's work on the four square theorem, which connected arithmetical questions with elliptic functions. The first use of analytical arguments in number theory goes further back, to Euler.[10]

The history of each subfield is sketched in its own section below. Many of the most interesting questions in each area remain open and are being actively worked on.

Approaches and subfields

Introductory texts and elementary tools

Two of the most popular introductions to the subject are:

• G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 6th ed., rev. by D. R. Heath-Brown and J. H. Silverman, Oxford University Press, Oxford, 2008 (first published in 1938).
• I. M. Vinogradov, Elements of Number Theory, Mineola, NY: Dover Publications, 2003, reprint of the 1954 edition.

Hardy and Wright's book is a comprehensive classic, though its clarity sometimes suffers due to the authors' insistence on elementary methods.[11] Vinogradov's main attraction consists in its set of problems, which quickly lead to Vinogradov's own research interests; the text itself is very basic and close to minimal.

The term elementary generally denotes a method that does not use complex analysis. For example, the prime number theorem was first proven in 1896, but an elementary proof was found only in 1949. The term is somewhat ambiguous: for example, proofs based on Tauberian theorems are often seen as quite enlightening but not elementary, in spite of using Fourier analysis, rather than complex analysis as such. Here as elsewhere, an elementary proof may be longer and more difficult for most readers than a non-elementary one.

Number theory has the reputation of being a field many of whose results can be stated to the layperson. At the same time, the proofs of these results are not particularly accessible, in part because the range of tools they use is, if anything, unusually broad within mathematics.

Popular choices for a second textbook include Borevich and Shafarevich's Number theory and Serre's Cours d'arithmetique. Textbooks for later stages in one's study tend to branch into analytic and algebraic number theory, among other subfields.

Main fields

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 zeta 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 algebraic properties and algebraic objects of interest in number theory. (Thus, analytic and algebraic number theory can and do overlap: the former is defined by its methods, the latter by its objects of study.) A key topic is that of the 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 $\scriptstyle f(x)=0$ with rational coefficients; for example, every solution $x$ of $\scriptstyle x^5 + (11/2) x^3 - 7 x^2 + 9 = 0$ (say) is an algebraic number. Fields of algebraic numbers are also called algebraic number fields.

It could be argued that the simplest kind of number fields (viz., quadratic fields) 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. (A quadratic field consists of all numbers of the form $\scriptstyle a + b \sqrt{d}$, where $a$ and $b$ are rational numbers and $d$ is a fixed rational number whose square root is not rational.) For that matter, the 11th-century chakravala 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.) The initial impetus for the development of ideal numbers (by Kummer) seems to have come from the study of higher reciprocity laws,[12] i.e., generalisations of quadratic reciprocity.

Number fields are often studied as extensions of smaller number fields: a field L is said to be an extension of a field K if L contains K. (For example, the complex numbers C are an extension of the reals R, and the reals R are an extension of the rationals Q.) 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[13] 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, one of the main current large-scale research plans in mathematics, is sometimes described as an attempt to generalise class field theory to non-abelian extensions of number fields.

Diophantine geometry

The central problem of Diophantine geometry is to determine when a Diophantine equation has solutions, and if it does, how many. The approach taken is to think of the solutions of an equation as a geometric object.

For example, 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. In Diophantine geometry, one asks whether there are any rational points (points all of whose coordinates are rationals) or integral points (points all of whose coordinates are integers) on the curve or surface. If there are any such points, the next step is to ask how many there are and how they are distributed. A basic question in this direction is: are there finitely or infinitely many rational points on a given curve (or surface)? What about integer points?

An example here may be helpful. Consider the equation $x^2+y^2 = 1$; we would like to study its rational solutions, i.e., its solutions $(x,y)$ such that x and y are both rational. This is the same as asking for all integer solutions to $a^2 + b^2 = c^2$; any solution to the latter equation gives us a solution $x = a/c$, $y = b/c$ to the former. It is also the same as asking for all points with rational coordinates on the curve described by $x^2 + y^2 = 1$. (This curve happens to be a circle of radius 1 around the origin.)

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:[14] allow the variables in $f(x,y)=0$ to be complex numbers; then $f(x,y)=0$ defines a 2-dimensional surface in (projective) 4-dimensional space (since two complex variables can be decomposed into four real variables, i.e., four dimensions). 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 $a/q$ (with $gcd(a,q)=1$) 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 geometry, on the other hand, is a contemporary term for much the same domain as that covered by the term diophantine geometry. The term arithmetic geometry is arguably used most often when one wishes to emphasise the connections to modern algebraic geometry (as in, for instance, Faltings' theorem) rather than to techniques in diophantine approximations.

Recent approaches and subfields

The areas below date as such from no earlier than the mid-twentieth century, even if they are based on older material. For example, as is explained below, the matter of algorithms in number theory is very old, in some sense older than the concept of proof; at the same time, the modern study of computational complexity dates only from the 1930s and 1940s.

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.

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 $A$, does it contain many elements in arithmetic progression: $a$, $a+b$, $a+2 b$, $a+3 b$, ... , $a+10b$, 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.

Computations in number theory

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

There are two main 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 (e.g., RSA) depend on functions that are known to all, but whose inverses (a) are known only to a chosen few, and (b) would take one too long a time to figure out on one's own. For example, these functions can be such that their inverses can be computed only if certain large integers are factorised. 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 instance, 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. There are thus some problems in number theory that will never be solved. We even know the shape of some of them, viz., Diophantine equations in nine variables; we simply do not know, and cannot know, which coefficients give us equations for which the following two statements are both true: there are no solutions, and we shall never know that there are no solutions.

References

1. Sir Thomas Heath, A History of Greek Mathematics, vol. 1, Dover, 1981, p. 13.
2. Take, e.g., Serre's A Course in Arithmetic (1970; translated into English in 1973). In 1952, Davenport still had to specify that he meant The Higher Arithmetic. Hardy and Wright wrote in the introduction to An Introduction to the Theory of Numbers (1938): "We proposed at one time to change [the title] to An introduction to arithmetic, a more novel and in some ways a more appropriate title; but it was pointed out that this might lead to misunderstandings about the content of the book."
3. Plato, Theaetetus, p. 147 B, cited in: Kurt von Fritz, "The discovery of incommensurability by Hippasus of Metapontum", p. 212, in: J. Christianidis (ed.), Classics in the History of Greek Mathematics, Kluwer, 2004. Plato reports on further work by Theodorus on irrationality.
4. Āryabhaṭa, Āryabhatīya, Chapter 2, verses 32-33, cited in: K. Plofker, Mathematics in India, Princeton University Press, 2008, pp. 134-140. See also W. E. Clark, The Āryabhaṭīya of Āryabhaṭa: An ancient Indian work on Mathematics and Astronomy, University of Chicago Press, 1930, pp. 42-50. A slightly more explicit description of the kuṭṭaka was later given in Brahmagupta, Brāhmasphuṭasiddhānta, XVIII, 3-5 (in Colebrooke, Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bháscara, London, 1817, p. 325, cited in: Clark, op. cit., p. 42).
5. K. Plofker, Mathematics in India, Princeton University Press, 2008, p. 119.
6. Plofker, op. cit., p. 194
7. Plofker, op. cit., p. 283
8. A. Weil, 'Number theory: an approach through history - from Hammurapi to Legendre, Birkhäuser, 1984, pp. 1-2.
9. See the discussion in section 5 of C. Goldstein and N. Schappacher, "A book in search of a discipline (1801-1860)', in C. Goldstein, N. Schappacher and J. Schwermer (eds.), "The shaping of arithmetic after C. F. Gauss's Disquisitiones Arithmeticae", Springer, 2007. Early signs of self-consciousness are present already in letters by Fermat: thus his remarks on what number theory is, and how "Diophantus's work [...] does not really belong to [it]" (quoted in A. Weil, op. cit., p. 25).
10. H. Iwaniec and E. Kowalski, Analytic number theory, AMS Colloquium Pub., Vol. 53, 2004, p. 1.
11. T. M. Apostol, Review of An introduction to the theory of numbers, Mathematical Reviews, MR0568909.
12. H. M. Edwards, Fermat's Last Theorem: a genetic introduction to algebraic number theory, Springer Verlag, 1977, p. 79.
13. The Galois group of an extension K/L consists of the operations (isomorphisms) that send elements of L to other elements of L while leaving all elements of K fixed. Thus, for instance, Gal(C/R) consists of two elements: the identity element (taking every element x+iy of C to itself) and complex conjugation (the map taking each element x+iy to x-iy). The Galois group of an extension tells us many of its crucial properties. The study of Galois groups started with Evariste Galois; in modern language, the main outcome of his work is that an equation f(x)=0 can be solved by radicals (that is, x can be expressed in terms of the four basic operations together with square roots, cubic roots, etc.) if and only if the extension of the rationals by the roots of the equation f(x)=0 has a Galois group that is solvable in the sense of group theory. ("Solvable", in the sense of group theory, is a simple property that can be checked easily for finite groups.)
14. It may be useful to look at an example here. Say we want to study the curve $y^2 = x^3 + 7$. We allow x and y to be complex numbers: $(a + b i)^2 = (c + d i)^3 + 7$. This is, in effect, a set of two equations on four variables, since both the real and the imaginary part on each side must match. As a result, we get a surface (two-dimensional) in four dimensional space. After we choose a convenient hyperplane on which to project the surface (meaning that, say, we choose to ignore the coordinate a), we can plot the resulting projection, which is a surface in ordinary three-dimensional space. It then becomes clear that the result is a torus, i.e., the surface of a doughnut (somewhat stretched). A doughnut has one hole; hence the genus is 1.