Number theory, or in older usage arithmetic, is a branch of pure
mathematics devoted primarily to the study of the integers. It is
sometimes called "The Queen of Mathematics" because of its
foundational place in the discipline.
Number theorists study prime
numbers as well as the properties of objects made out of integers
(e.g., rational numbers) or defined as generalizations of the integers
(e.g., 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
The older term for number theory is arithmetic. By the early twentieth
century, it had been superseded by "number theory".[note 1] (The word
"arithmetic" is used by the general public to mean "elementary
calculations"; it has also acquired other meanings in mathematical
logic, as in Peano arithmetic, and computer science, as in floating
point arithmetic.) The use of the term arithmetic for number theory
regained some ground in the second half of the 20th century, arguably
in part due to French influence.[note 2] In particular, arithmetical
is preferred as an adjective to number-theoretic.
1.1.1 Dawn of arithmetic
1.1.2 Classical Greece and the early Hellenistic period
1.1.4 Āryabhaṭa, Brahmagupta, Bhāskara
Arithmetic in the Islamic golden age
1.1.6 Western Europe in the Middle Ages
1.2 Early modern number theory
1.2.3 Lagrange, Legendre, and Gauss
1.3 Maturity and division into subfields
2 Main subdivisions
2.1 Elementary tools
2.2 Analytic number theory
Algebraic number theory
2.4 Diophantine geometry
3 Recent approaches and subfields
3.1 Probabilistic number theory
3.3 Computations in number theory
6 See also
10 Further reading
11 External links
Dawn of arithmetic
The first historical find of an arithmetical nature is a fragment of a
table: the broken clay tablet
Plimpton 322 (Larsa, Mesopotamia, ca.
1800 BCE) contains a list of "Pythagorean triples", i.e., integers
displaystyle a^ 2 +b^ 2 =c^ 2
. The triples are too many and too large to have been obtained by
brute force. The heading over the first column reads: "The takiltum of
the diagonal which has been subtracted such that the width..."
Plimpton 322 tablet
The table's layout suggests that it was constructed by means of
what amounts, in modern language, to the identity
displaystyle left( frac 1 2 left(x- frac 1 x right)right)^
2 +1=left( frac 1 2 left(x+ frac 1 x right)right)^ 2 ,
which is implicit in routine Old Babylonian exercises. If some
other method was used, the triples were first constructed and then
, presumably for actual use as a "table", i.e., with a view to
It is not known what these applications may have been, or whether
there could have been any; Babylonian astronomy, for example, truly
came into its own only later. It has been suggested instead that the
table was a source of numerical examples for school problems.[note
While Babylonian number theory—or what survives of Babylonian
mathematics that can be called thus—consists of this single,
striking fragment, Babylonian algebra (in the secondary-school sense
of "algebra") was exceptionally well developed. Late Neoplatonic
sources state that
Pythagoras learned mathematics from the
Babylonians. Much earlier sources state that
Thales and Pythagoras
traveled and studied in Egypt.
Euclid IX 21—34 is very probably Pythagorean; it is very simple
material ("odd times even is even", "if an odd number measures [=
divides] an even number, then it also measures [= divides] half of
it"), but it is all that is needed to prove that
displaystyle sqrt 2
is irrational. Pythagorean mystics gave great importance to the
odd and the even. The discovery that
displaystyle sqrt 2
is irrational is credited to the early Pythagoreans
(pre-Theodorus). By revealing (in modern terms) that numbers could
be irrational, this discovery seems to have provoked the first
foundational crisis in mathematical history; its proof or its
divulgation are sometimes credited to Hippasus, who was expelled or
split from the Pythagorean sect. This forced a distinction between
numbers (integers and the rationals—the subjects of arithmetic), on
the one hand, and lengths and proportions (which we would identify
with real numbers, whether rational or not), on the other hand.
The Pythagorean tradition spoke also of so-called polygonal or
figurate numbers. While square numbers, cubic numbers, etc., are
seen now as more natural than triangular numbers, pentagonal numbers,
etc., the study of the sums of triangular and pentagonal numbers would
prove fruitful in the early modern period (17th to early 19th
We know of no clearly arithmetical material in ancient Egyptian or
Vedic sources, though there is some algebra in both. The Chinese
remainder theorem appears as an exercise  in
Sunzi Suanjing (3rd,
4th or 5th century CE.) (There is one important step glossed over
in Sunzi's solution:[note 4] it is the problem that was later solved
Kuṭṭaka – see below.)
There is also some numerical mysticism in Chinese mathematics,[note 5]
but, unlike that of the Pythagoreans, it seems to have led nowhere.
Like the Pythagoreans' perfect numbers, magic squares have passed from
superstition into recreation.
Classical Greece and the early Hellenistic period
Further information: Ancient Greek mathematics
Aside from a few fragments, the mathematics of Classical Greece is
known to us either through the reports of contemporary
non-mathematicians or through mathematical works from the early
Hellenistic period. In the case of number theory, this means, by
Plato and Euclid, respectively.
While Asian mathematics influenced Greek and Hellenistic learning, it
seems to be the case that Greek mathematics is also an indigenous
Eusebius, PE X, chapter 4 mentions of Pythagoras:
"In fact the said Pythagoras, while busily studying the wisdom of each
nation, visited Babylon, and Egypt, and all Persia, being instructed
by the Magi and the priests: and in addition to these he is related to
have studied under the Brahmans (these are Indian philosophers); and
from some he gathered astrology, from others geometry, and arithmetic
and music from others, and different things from different nations,
and only from the wise men of Greece did he get nothing, wedded as
they were to a poverty and dearth of wisdom: so on the contrary he
himself became the author of instruction to the Greeks in the learning
which he had procured from abroad."
Aristotle claimed that the philosophy of
Plato closely followed the
teachings of the Pythagoreans, and Cicero repeats this claim:
Platonem ferunt didicisse Pythagorea omnia ("They say
all things Pythagorean").
Plato had a keen interest in mathematics, and distinguished clearly
between arithmetic and calculation. (By arithmetic he meant, in part,
theorising on number, rather than what arithmetic or number theory
have come to mean.) It is through one of Plato's dialogues—namely,
Theaetetus—that we know that Theodorus had proven that
displaystyle sqrt 3 , sqrt 5 ,dots , sqrt 17
are irrational. Theaetetus was, like Plato, a disciple of
Theodorus's; he worked on distinguishing different kinds of
incommensurables, and was thus arguably a pioneer in the study of
number systems. (
Book X of
Euclid's Elements is described by Pappus as
being largely based on Theaetetus's work.)
Euclid devoted part of his Elements to prime numbers and divisibility,
topics that belong unambiguously to number theory and are basic to it
(Books VII to IX of Euclid's Elements). In particular, he gave an
algorithm for computing the greatest common divisor of two numbers
(the Euclidean algorithm; Elements, Prop. VII.2) and the first known
proof of the infinitude of primes (Elements, Prop. IX.20).
In 1773, Lessing published an epigram he had found in a manuscript
during his work as a librarian; it claimed to be a letter sent by
Archimedes to Eratosthenes. The epigram proposed what has
become known as Archimedes's cattle problem; its solution (absent from
the manuscript) requires solving an indeterminate quadratic equation
(which reduces to what would later be misnamed Pell's equation). As
far as we know, such equations were first successfully treated by the
Indian school. It is not known whether
Archimedes himself had a method
Title page of the 1621 edition of Diophantus's Arithmetica, translated
Latin by Claude Gaspard Bachet de Méziriac.
Very little is known about
Diophantus of Alexandria; he probably lived
in the third century CE, that is, about five hundred years after
Euclid. Six out of the thirteen books of Diophantus's Arithmetica
survive in the original Greek; four more books survive in an Arabic
Arithmetica is a collection of worked-out problems
where the task is invariably to find rational solutions to a system of
polynomial equations, usually of the form
displaystyle f(x,y)=z^ 2
displaystyle f(x,y,z)=w^ 2
. Thus, nowadays, we speak of
Diophantine equations when we speak of
polynomial equations to which rational or integer solutions must be
One may say that
Diophantus was studying rational points — i.e.,
points whose coordinates are rational — on curves and algebraic
varieties; however, unlike the Greeks of the Classical period, who did
what we would now call basic algebra in geometrical terms, Diophantus
did what we would now call basic algebraic geometry in purely
algebraic terms. In modern language, what
Diophantus did was to find
rational parametrizations of varieties; that is, given an equation of
the form (say)
displaystyle f(x_ 1 ,x_ 2 ,x_ 3 )=0
, his aim was to find (in essence) three rational functions
displaystyle g_ 1 ,g_ 2 ,g_ 3
such that, for all values of
displaystyle x_ i =g_ i (r,s)
gives a solution to
displaystyle f(x_ 1 ,x_ 2 ,x_ 3 )=0.
Diophantus also studied the equations of some non-rational curves, for
which no rational parametrisation is possible. He managed to find some
rational points on these curves (elliptic curves, as it happens, in
what seems to be their first known occurrence) by means of what
amounts to a tangent construction: translated into coordinate geometry
(which did not exist in Diophantus's time), his method would be
visualised as drawing a tangent to a curve at a known rational point,
and then finding the other point of intersection of the tangent with
the curve; that other point is a new rational point. (
resorted to what could be called a special case of a secant
Diophantus was concerned largely with rational solutions, he
assumed some results on integer numbers, in particular that every
integer is the sum of four squares (though he never stated as much
Āryabhaṭa, Brahmagupta, Bhāskara
While Greek astronomy probably influenced Indian learning, to the
point of introducing trigonometry, it seems to be the case that
Indian mathematics is otherwise an indigenous tradition; in
particular, there is no evidence that
Euclid's Elements reached India
before the 18th century.
Āryabhaṭa (476–550 CE) showed that pairs of simultaneous
displaystyle nequiv a_ 1 bmod m _ 1
displaystyle nequiv a_ 2 bmod m _ 2
could be solved by a method he called kuṭṭaka, or pulveriser;
this is a procedure close to (a generalisation of) the Euclidean
algorithm, which was probably discovered independently in India.
Āryabhaṭa seems to have had in mind applications to astronomical
Brahmagupta (628 CE) started the systematic study of indefinite
quadratic equations—in particular, the misnamed Pell equation, in
Archimedes may have first been interested, and which did not
start to be solved in the West until the time of Fermat and Euler.
Later Sanskrit authors would follow, using Brahmagupta's technical
terminology. A general procedure (the chakravala, or "cyclic method")
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
Indian mathematics remained largely unknown in Europe until the late
Brahmagupta and Bhāskara's work was
translated into English in 1817 by Henry Colebrooke.
Arithmetic in the Islamic golden age
Mathematics in medieval Islam
Al-Haytham seen by the West: frontispice of Selenographia, showing
Alhasen [sic] representing knowledge through reason, and Galileo
representing knowledge through the senses.
In the early ninth century, the caliph
Al-Ma'mun ordered translations
of many Greek mathematical works and at least one Sanskrit work (the
Sindhind, which may  or may not be Brahmagupta's
Brāhmasphuṭasiddhānta). Diophantus's main work, the Arithmetica,
was translated into Arabic by
Qusta ibn Luqa (820–912). Part of the
treatise al-Fakhri (by al-Karajī, 953 – ca. 1029) builds on it to
some extent. According to Rashed Roshdi, Al-Karajī's contemporary Ibn
al-Haytham knew what would later be called Wilson's theorem.
Western Europe in the Middle Ages
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 during the Middle Ages. Matters
started to change in Europe in the late Renaissance, thanks to a
renewed study of the works of Greek antiquity. A catalyst was the
textual emendation and translation into
Latin of Diophantus's
Arithmetica (Bachet, 1621, following a first attempt by Xylander,
Early modern number theory
Pierre de Fermat
Pierre de Fermat
Pierre de Fermat (1601–1665) never published his writings; in
particular, his work on number theory is contained almost entirely in
letters to mathematicians and in private marginal notes. He wrote
down nearly no proofs in number theory; he had no models in the
area. He did make repeated use of mathematical induction,
introducing the method of infinite descent.
One of Fermat's first interests was perfect numbers (which appear in
Euclid, Elements IX) and amicable numbers;[note 6] this led him to
work on integer divisors, which were from the beginning among the
subjects of the correspondence (1636 onwards) that put him in touch
with the mathematical community of the day. He had already studied
Bachet's edition of
Diophantus carefully; by 1643, his interests
had shifted largely to Diophantine problems and sums of squares
(also treated by Diophantus).
Fermat's achievements in arithmetic include:
Fermat's little theorem
Fermat's little theorem (1640), stating that, if a is not
divisible by a prime p, then
displaystyle a^ p-1 equiv 1 bmod p .
If a and b are coprime, then
displaystyle a^ 2 +b^ 2
is not divisible by any prime congruent to −1 modulo 4; and
every prime congruent to 1 modulo 4 can be written in the form
displaystyle a^ 2 +b^ 2
. These two statements also date from 1640; in 1659, Fermat stated
to Huygens that he had proven the latter statement by the method of
infinite descent. Fermat and Frenicle also did some work (some of
it erroneous) on other quadratic forms.
Fermat posed the problem of solving
displaystyle x^ 2 -Ny^ 2 =1
as a challenge to English mathematicians (1657). The problem was
solved in a few months by Wallis and Brouncker. Fermat considered
their solution valid, but pointed out they had provided an algorithm
without a proof (as had Jayadeva and Bhaskara, though Fermat would
never know this). He states that a proof can be found by descent.
Fermat developed methods for (doing what in our terms amounts to)
finding points on curves of genus 0 and 1. As in Diophantus, there are
many special procedures and what amounts to a tangent construction,
but no use of a secant construction.
Fermat states and proves (by descent) in the appendix to Observations
Diophantus (Obs. XLV) that
displaystyle x^ 4 +y^ 4 =z^ 4
has no non-trivial solutions in the integers. Fermat also mentioned
to his correspondents that
displaystyle x^ 3 +y^ 3 =z^ 3
has no non-trivial solutions, and that this could be proven by
descent. The first known proof is due to Euler (1753; indeed by
Fermat's claim ("Fermat's last theorem") to have shown there are no
displaystyle x^ n +y^ n =z^ n
displaystyle ngeq 3
appears only in his annotations on the margin of his copy of
The interest of
Leonhard Euler (1707–1783) in number theory was
first spurred in 1729, when a friend of his, the amateur[note 8]
Goldbach, pointed him towards some of Fermat's work on the
subject. This has been called the "rebirth" of modern number
theory, after Fermat's relative lack of success in getting his
contemporaries' attention for the subject. Euler's work on number
theory includes the following:
Proofs for Fermat's statements. This includes Fermat's little theorem
(generalised by Euler to non-prime moduli); the fact that
displaystyle p=x^ 2 +y^ 2
if and only if
displaystyle pequiv 1 bmod 4
; initial work towards a proof that every integer is the sum of four
squares (the first complete proof is by
Joseph-Louis Lagrange (1770),
soon improved by Euler himself); the lack of non-zero integer
displaystyle x^ 4 +y^ 4 =z^ 2
(implying the case n=4 of Fermat's last theorem, the case n=3 of
which Euler also proved by a related method).
Pell's equation, first misnamed by Euler. He wrote on the link
between continued fractions and Pell's equation.
First steps towards analytic number theory. In his work of sums of
four squares, partitions, pentagonal numbers, and the distribution of
prime numbers, Euler pioneered the use of what can be seen as analysis
(in particular, infinite series) in number theory. Since he lived
before the development of complex analysis, most of his work is
restricted to the formal manipulation of power series. He did,
however, do some very notable (though not fully rigorous) early work
on what would later be called the Riemann zeta function.
Quadratic forms. Following Fermat's lead, Euler did further research
on the question of which primes can be expressed in the form
displaystyle x^ 2 +Ny^ 2
, some of it prefiguring quadratic reciprocity. 
Diophantine equations. Euler worked on some
Diophantine equations of
genus 0 and 1. In particular, he studied Diophantus's work; he
tried to systematise it, but the time was not yet ripe for such an
endeavour – algebraic geometry was still in its infancy. He did
notice there was a connection between Diophantine problems and
elliptic integrals, whose study he had himself initiated.
Lagrange, Legendre, and Gauss
Carl Friedrich Gauss's Disquisitiones Arithmeticae, first edition
Joseph-Louis Lagrange (1736–1813) was the first to give full proofs
of some of Fermat's and Euler's work and observations – for
instance, the four-square theorem and the basic theory of the misnamed
"Pell's equation" (for which an algorithmic solution was found by
Fermat and his contemporaries, and also by Jayadeva and Bhaskara II
before them.) He also studied quadratic forms in full generality (as
displaystyle mX^ 2 +nY^ 2
) — defining their equivalence relation, showing how to put them in
reduced form, etc.
Adrien-Marie Legendre (1752–1833) was the first to state the law of
quadratic reciprocity. He also conjectured what amounts to the prime
number theorem and Dirichlet's theorem on arithmetic progressions. He
gave a full treatment of the equation
displaystyle ax^ 2 +by^ 2 +cz^ 2 =0
 and worked on quadratic forms along the lines later developed
fully by Gauss. In his old age, he was the first to prove
"Fermat's last theorem" for
(completing work by Peter Gustav Lejeune Dirichlet, and crediting
both him and Sophie Germain).
Carl Friedrich Gauss
Disquisitiones Arithmeticae (1798), Carl Friedrich Gauss
(1777–1855) proved the law of quadratic reciprocity and developed
the theory of quadratic forms (in particular, defining their
composition). He also introduced some basic notation (congruences) and
devoted a section to computational matters, including primality
tests. The last section of the Disquisitiones established a link
between roots of unity and number theory:
The theory of the division of the circle...which is treated in sec. 7
does not belong by itself to arithmetic, but its principles can only
be drawn from higher arithmetic.
In this way,
Gauss arguably made a first foray towards both Évariste
Galois's work and algebraic number theory.
Maturity and division into subfields
Peter Gustav Lejeune Dirichlet
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.
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
The rough subdivision of number theory into its modern subfields—in
particular, analytic and algebraic number theory.
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. A conventional starting point for analytic number
Dirichlet's theorem on arithmetic progressions (1837),
 whose proof introduced
L-functions and involved some asymptotic
analysis and a limiting process on a real variable. The first use
of analytic ideas in number theory actually goes back to Euler
(1730s),  who used formal power series and non-rigorous (or
implicit) limiting arguments. The use of complex analysis in number
theory comes later: the work of
Bernhard Riemann (1859) on the zeta
function is the canonical starting point; Jacobi's four-square
theorem (1839), which predates it, belongs to an initially different
strand that has by now taken a leading role in analytic number theory
The history of each subfield is briefly addressed in its own section
below; see the main article of each subfield for fuller treatments.
Many of the most interesting questions in each area remain open and
are being actively worked on.
The term elementary generally denotes a method that does not use
complex analysis. For example, the prime number theorem was first
proven using complex analysis in 1896, but an elementary proof was
found only in 1949 by Erdős and Selberg. The term is somewhat
ambiguous: for example, proofs based on complex Tauberian theorems
(e.g. Wiener–Ikehara) 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
Analytic number theory
Main article: Analytic number theory
Riemann zeta function
Riemann zeta function ζ(s) in the complex plane. The color of a point
s gives the value of ζ(s): dark colors denote values close to zero
and hue gives the value's argument.
The action of the modular group on the upper half plane. The region in
grey is the standard fundamental domain.
Analytic number theory
Analytic number theory may be defined
in terms of its tools, as the study of the integers by means of tools
from real and complex analysis; or
in terms of its concerns, as the study within number theory of
estimates on size and density, as opposed to identities.
Some subjects generally considered to be part of analytic number
theory, e.g., sieve theory,[note 9] are better covered by the second
rather than the first definition: some of sieve theory, for instance,
uses little analysis,[note 10] yet it does belong to analytic number
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). The theory of
modular forms (and, more generally, automorphic forms) also occupies
an increasingly central place in the toolbox of analytic number
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 (generalizations of prime numbers 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 generalizations of the Riemann zeta
function, a key analytic object at the roots of the subject. This
is an example of a general procedure in analytic number theory:
deriving information about the distribution of a sequence (here, prime
ideals or prime numbers) from the analytic behavior of an
appropriately constructed complex-valued function.
Algebraic number theory
Algebraic number theory
An algebraic number is any complex number that is a solution to some
with rational coefficients; for example, every solution
displaystyle x^ 5 +(11/2)x^ 3 -7x^ 2 +9=0
(say) is an algebraic number. Fields of algebraic numbers are also
called algebraic number fields, or shortly number fields. Algebraic
number theory studies algebraic number fields. Thus, analytic and
algebraic number theory can and do overlap: the former is defined by
its methods, the latter by its objects of study.
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
displaystyle a+b sqrt d
are rational numbers and
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
displaystyle sqrt -5
, the number
can be factorised both as
displaystyle 6=2cdot 3
displaystyle 6=(1+ sqrt -5 )(1- sqrt -5 )
; all of
displaystyle 1+ sqrt -5
displaystyle 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,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[note 11] 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.
An example of an active area of research in algebraic number theory is
Iwasawa theory. 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.
Main article: 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
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
An example here may be helpful. Consider the Pythagorean equation
displaystyle x^ 2 +y^ 2 =1
; we would like to study its rational solutions, i.e., its solutions
such that x and y are both rational. This is the same as asking for
all integer solutions to
displaystyle a^ 2 +b^ 2 =c^ 2
; any solution to the latter equation gives us a solution
to the former. It is also the same as asking for all points with
rational coordinates on the curve described by
displaystyle x^ 2 +y^ 2 =1
. (This curve happens to be a circle of radius 1 around the origin.)
Two examples of an elliptic curve, i.e., a curve of genus 1 having at
least one rational point. (Either graph can be seen as a slice of a
torus in four-dimensional space.)
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
is a polynomial in two variables—turns out to depend crucially on
the genus of the curve. The genus can be defined as follows:[note 12]
allow the variables in
to be complex numbers; then
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
. Other geometrical notions turn out to be just as crucial.
There is also the closely linked area of Diophantine approximations:
given a number
, 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 good approximation to
displaystyle x-a/q< frac 1 q^ c
is large.) This question is of special interest if
is an algebraic number. If
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. This question is also of special
interest in transcendental number theory: if a number can be better
approximated than any algebraic number, then it is a transcendental
number. It is by this argument that π and e have been shown to be
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's theorem) rather than to techniques in Diophantine
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 computability dates only from the 1930s and 1940s,
and computational complexity theory from the 1970s.
Probabilistic number theory
Main article: 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. 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 will 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
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.
At times, a non-rigorous, probabilistic approach leads to a number of
heuristic algorithms and open problems, notably Cramér's conjecture.
Arithmetic combinatorics and Additive number theory
Let A be a set of N integers. Consider the set A + A = m + n m, n
∈ 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, for example, does A resemble an
If we begin from a fairly "thick" infinite set
, does it contain many elements in arithmetic progression:
displaystyle a+b,a+2b,a+3b,ldots ,a+10b
, say? Should it be possible to write large integers as sums of
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
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 accounts in part for its developing links with ergodic
theory, finite group theory, model theory, and other fields. The term
additive combinatorics is also used; however, the sets
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
may be compared.
Computations in number theory
Main article: Computational 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. An interesting early case is that of what we now
call the Euclidean algorithm. In its basic form (namely, as an
algorithm for computing the greatest common divisor) it appears as
Proposition 2 of
Book VII in Elements, together with a proof of
correctness. However, in the form that is often used in number theory
(namely, as an algorithm for finding integer solutions to an equation
, or, what is the same, for finding the quantities whose existence is
assured by the Chinese remainder theorem) it first appears in the
Āryabhaṭa (5th–6th century CE) as an algorithm called
kuṭṭaka ("pulveriser"), without a proof of correctness.
There are two main questions: "can we compute this?" and "can we
compute it rapidly?". Anyone 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 (both theoretical and practical), 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 factorized. 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 in some instances. For instance, in 1970, it
was proven, as a solution to Hilbert's 10th problem, that there is no
Turing machine which can solve all Diophantine equations. In
particular, this means that, given a computably enumerable set of
axioms, there are
Diophantine equations for which there is no proof,
starting from the axioms, of whether the set of equations has or does
not have integer solutions. (We would necessarily be speaking of
Diophantine equations for which there are no integer solutions, since,
Diophantine equation with at least one solution, the solution
itself provides a proof of the fact that a solution exists. We cannot
prove, of course, that a particular
Diophantine equation is of this
kind, since this would imply that it has no solutions.)
This section needs expansion with:
Modern applications of
Number theory. You can help by adding to it.
Leonard Dickson (1874–1954) said "Thank God that
number theory is unsullied by any application". Such a view is no
longer applicable to number theory. In 1974,
Donald Knuth said
"...virtually every theorem in elementary number theory arises in a
natural, motivated way in connection with the problem of making
computers do high-speed numerical calculations". Elementary number
theory is taught in discrete mathematics courses for computer
scientists; on the other hand, number theory also has applications to
the continuous in numerical analysis. As well as the well-known
applications to cryptography, there are also applications to many
other areas of mathematics.[specify]
American Mathematical Society
American Mathematical Society awards the
Cole Prize in Number
Theory. Moreover number theory is one of the three mathematical
subdisciplines rewarded by the Fermat Prize.
Number theory portal
Algebraic function field
^ Already in 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." (Heath 1921, p. 13)
^ Take, e.g. Serre 1973. In 1952, Davenport still had to specify that
he meant The Higher Arithmetic. Hardy and Wright wrote in the
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." (Hardy & Wright 2008)
^ Robson 2001, p. 201. This is controversial. See Plimpton 322.
Robson's article is written polemically (Robson 2001, p. 202)
with a view to "perhaps [...] knocking [Plimpton 322] off its
pedestal" (Robson 2001, p. 167); at the same time, it settles to
the conclusion that
[...] the question "how was the tablet calculated?" does not have to
have the same answer as the question "what problems does the tablet
set?" The first can be answered most satisfactorily by reciprocal
pairs, as first suggested half a century ago, and the second by some
sort of right-triangle problems (Robson 2001, p. 202).
Robson takes issue with the notion that the scribe who produced
Plimpton 322 (who had to "work for a living", and would not have
belonged to a "leisured middle class") could have been motivated by
his own "idle curiosity" in the absence of a "market for new
mathematics".(Robson 2001, pp. 199–200)
^ Sunzi Suanjing, Ch. 3, Problem 26, in Lam & Ang 2004,
 Now there are an unknown number of things. If we count by threes,
there is a remainder 2; if we count by fives, there is a remainder 3;
if we count by sevens, there is a remainder 2. Find the number of
things. Answer: 23.
Method: If we count by threes and there is a remainder 2, put down
140. If we count by fives and there is a remainder 3, put down 63. If
we count by sevens and there is a remainder 2, put down 30. Add them
to obtain 233 and subtract 210 to get the answer. If we count by
threes and there is a remainder 1, put down 70. If we count by fives
and there is a remainder 1, put down 21. If we count by sevens and
there is a remainder 1, put down 15. When [a number] exceeds 106, the
result is obtained by subtracting 105.
^ See, e.g., Sunzi Suanjing, Ch. 3, Problem 36, in Lam & Ang 2004,
 Now there is a pregnant woman whose age is 29. If the gestation
period is 9 months, determine the sex of the unborn child. Answer:
Method: Put down 49, add the gestation period and subtract the age.
From the remainder take away 1 representing the heaven, 2 the earth, 3
the man, 4 the four seasons, 5 the five phases, 6 the six pitch-pipes,
7 the seven stars [of the Dipper], 8 the eight winds, and 9 the nine
divisions [of China under Yu the Great]. If the remainder is odd, [the
sex] is male and if the remainder is even, [the sex] is female.
This is the last problem in Sunzi's otherwise matter-of-fact treatise.
^ Perfect and especially amicable numbers are of little or no interest
nowadays. The same was not true in medieval times – whether in the
West or the Arab-speaking world – due in part to the importance
given to them by the Neopythagorean (and hence mystical) Nicomachus
(ca. 100 CE), who wrote a primitive but influential "Introduction to
Arithmetic". See van der Waerden 1961, Ch. IV.
^ Here, as usual, given two integers a and b and a non-zero integer m,
displaystyle aequiv b bmod m
(read "a is congruent to b modulo m") to mean that m divides
a − b, or, what is the same, a and b leave the same
residue when divided by m. This notation is actually much later than
Fermat's; it first appears in section 1 of Gauss's Disquisitiones
Fermat's little theorem
Fermat's little theorem is a consequence of the fact
that the order of an element of a group divides the order of the
group. The modern proof would have been within Fermat's means (and was
indeed given later by Euler), even though the modern concept of a
group came long after Fermat or Euler. (It helps to know that inverses
exist modulo p (i.e., given a not divisible by a prime p, there is an
integer x such that
displaystyle xaequiv 1 bmod p
); this fact (which, in modern language, makes the residues mod p into
a group, and which was already known to Āryabhaṭa; see above) was
familiar to Fermat thanks to its rediscovery by Bachet (Weil 1984,
p. 7). Weil goes on to say that Fermat would have recognised that
Bachet's argument is essentially Euclid's algorithm.
^ Up to the second half of the seventeenth century, academic positions
were very rare, and most mathematicians and scientists earned their
living in some other way (Weil 1984, pp. 159, 161). (There were
already some recognisable features of professional practice, viz.,
seeking correspondents, visiting foreign colleagues, building private
libraries (Weil 1984, pp. 160–161). Matters started to shift in
the late 17th century (Weil 1984, p. 161); scientific academies
were founded in England (the Royal Society, 1662) and France (the
Académie des sciences, 1666) and Russia (1724). Euler was offered a
position at this last one in 1726; he accepted, arriving in St.
Petersburg in 1727 (Weil 1984, p. 163 and Varadarajan 2006,
p. 7). In this context, the term amateur usually applied to
Goldbach is well-defined and makes some sense: he has been described
as a man of letters who earned a living as a spy (Truesdell 1984,
p. xv); cited in Varadarajan 2006, p. 9). Notice, however,
that Goldbach published some works on mathematics and sometimes held
Sieve theory figures as one of the main subareas of analytic number
theory in many standard treatments; see, for instance, Iwaniec &
Kowalski 2004 or Montgomery & Vaughan 2007
^ This is the case for small sieves (in particular, some combinatorial
sieves such as the Brun sieve) rather than for large sieves; the study
of the latter now includes ideas from harmonic and functional
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 Évariste 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.)
^ It may be useful to look at an example here. Say we want to study
displaystyle y^ 2 =x^ 3 +7
. We allow x and y to be complex numbers:
displaystyle (a+bi)^ 2 =(c+di)^ 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.
^ Long 1972, p. 1.
^ Neugebauer & Sachs 1945, p. 40. The term takiltum is
problematic. Robson prefers the rendering "The holding-square of the
diagonal from which 1 is torn out, so that the short side comes
up...".Robson 2001, p. 192
^ Robson 2001, p. 189. Other sources give the modern formula
displaystyle (p^ 2 -q^ 2 ,2pq,p^ 2 +q^ 2 )
. Van der Waerden gives both the modern formula and what amounts to
the form preferred by Robson.(van der Waerden 1961, p. 79)
^ van der Waerden 1961, p. 184.
^ Neugebauer (Neugebauer 1969, pp. 36–40) discusses the table
in detail and mentions in passing Euclid's method in modern notation
(Neugebauer 1969, p. 39).
^ Friberg 1981, p. 302.
^ van der Waerden 1961, p. 43.
^ Iamblichus, Life of Pythagoras,(trans. e.g. Guthrie 1987) cited in
van der Waerden 1961, p. 108. See also Porphyry, Life of
Pythagoras, paragraph 6, in Guthrie 1987 Van der Waerden (van der
Waerden 1961, pp. 87–90) sustains the view that
^ Herodotus (II. 81) and Isocrates (Busiris 28), cited in: Huffman
2011. On Thales, see Eudemus ap. Proclus, 65.7, (e.g. Morrow 1992,
p. 52) cited in: O'Grady 2004, p. 1.
Proclus was using a
Eudemus of Rhodes (now lost), the Catalogue of Geometers. See
also introduction, Morrow 1992, p. xxx on Proclus's reliability.
^ Becker 1936, p. 533, cited in: van der Waerden 1961,
^ Becker 1936.
^ van der Waerden 1961, p. 109.
^ Plato, Theaetetus, p. 147 B, (e.g. Jowett 1871), cited in von Fritz
2004, p. 212: "Theodorus was writing out for us something about
roots, such as the roots of three or five, showing that they are
incommensurable by the unit;..." See also Spiral of Theodorus.
^ von Fritz 2004.
^ Heath 1921, p. 76.
^ Sunzi Suanjing, Chapter 3, Problem 26. This can be found in Lam
& Ang 2004, pp. 219–220, which contains a full translation
of the Suan Ching (based on Qian 1963). See also the discussion in Lam
& Ang 2004, pp. 138–140.
^ The date of the text has been narrowed down to 220–420 AD (Yan
Dunjie) or 280–473 AD (Wang Ling) through internal evidence (=
taxation systems assumed in the text). See Lam & Ang 2004,
^ Boyer & Merzbach 1991, p. 82.
^ Metaphysics, 1.6.1 (987a)
^ Tusc. Disput. 1.17.39.
^ Vardi 1998, p. 305–319.
^ Weil 1984, pp. 17–24.
^ a b Plofker 2008, p. 119.
^ Any early contact between Babylonian and Indian mathematics remains
conjectural (Plofker 2008, p. 42).
^ Mumford 2010, p. 387.
^ Āryabhaṭa, Āryabhatīya, Chapter 2, verses 32–33, cited in:
Plofker 2008, pp. 134–140. See also Clark 1930,
pp. 42–50. A slightly more explicit description of the
kuṭṭaka was later given in Brahmagupta, Brāhmasphuṭasiddhānta,
XVIII, 3–5 (in Colebrooke 1817, p. 325, cited in Clark 1930,
^ Mumford 2010, p. 388.
^ Plofker 2008, p. 194.
^ Plofker 2008, p. 283.
^ Colebrooke 1817.
^ Colebrooke 1817, p. lxv, cited in Hopkins 1990, p. 302.
See also the preface in Sachau 1888 cited in Smith 1958, pp. 168
^ Pingree 1968, pp. 97–125, and Pingree 1970,
pp. 103–123, cited in Plofker 2008, p. 256.
^ Rashed 1980, p. 305–321.
^ Weil 1984, pp. 45–46.
^ Weil 1984, p. 118. This was more so in number theory than in
other areas (remark in Mahoney 1994, p. 284). Bachet's own proofs
were "ludicrously clumsy" (Weil 1984, p. 33).
^ Mahoney 1994, pp. 48, 53–54. The initial subjects of Fermat's
correspondence included divisors ("aliquot parts") and many subjects
outside number theory; see the list in the letter from Fermat to
Roberval, 22.IX.1636, Tannery & Henry 1891, Vol. II, pp. 72, 74,
cited in Mahoney 1994, p. 54.
^ a b Weil 1984, pp. 1–2.
^ Weil 1984, p. 53.
^ Tannery & Henry 1891, Vol. II, p. 209, Letter XLVI from Fermat
to Frenicle, 1640, cited in Weil 1984, p. 56
^ Tannery & Henry 1891, Vol. II, p. 204, cited in Weil 1984,
p. 63. All of the following citations from Fermat's Varia Opera
are taken from Weil 1984, Chap. II. The standard Tannery & Henry
work includes a revision of Fermat's posthumous Varia Opera
Mathematica originally prepared by his son (Fermat 1679).
^ Tannery & Henry 1891, Vol. II, p. 213.
^ Tannery & Henry 1891, Vol. II, p. 423.
^ Weil 1984, pp. 80, 91–92.
^ Weil 1984, p. 92.
^ Weil 1984, Ch. II, sect. XV and XVI.
^ Tannery & Henry 1891, Vol. I, pp. 340–341.
^ Weil 1984, p. 115.
^ Weil 1984, pp. 115–116.
^ Weil 1984, pp. 2, 172.
^ Varadarajan 2006, p. 9.
^ Weil 1984, p. 2 and Varadarajan 2006, p. 37
^ Varadarajan 2006, p. 39 and Weil 1984, pp. 176–189
^ Weil 1984, pp. 178–179.
^ Weil 1984, p. 174. Euler was generous in giving credit to
others (Varadarajan 2006, p. 14), not always correctly.
^ Weil 1984, p. 183.
^ Varadarajan 2006, pp. 45–55; see also chapter III.
^ Varadarajan 2006, pp. 44–47.
^ Weil 1984, pp. 177–179.
^ Edwards 1983, pp. 285–291.
^ Varadarajan 2006, pp. 55–56.
^ Weil 1984, pp. 179–181.
^ a b Weil 1984, p. 181.
^ Weil 1984, pp. 327–328.
^ Weil 1984, pp. 332–334.
^ Weil 1984, pp. 337–338.
^ Goldstein & Schappacher 2007, p. 14.
^ From the preface of Disquisitiones Arithmeticae; the translation is
taken from Goldstein & Schappacher 2007, p. 16
^ See the discussion in section 5 of Goldstein & Schappacher 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
Weil 1984, p. 25).
^ a b Apostol 1976, p. 7.
^ Davenport & Montgomery 2000, p. 1.
^ See the proof in Davenport & Montgomery 2000, section 1
^ Iwaniec & Kowalski 2004, p. 1.
^ Varadarajan 2006, sections 2.5, 3.1 and 6.1.
^ Granville 2008, pp. 322–348.
^ See the comment on the importance of modularity in Iwaniec &
Kowalski 2004, p. 1
^ Goldfeld 2003.
^ See, e.g., the initial comment in Iwaniec & Kowalski 2004,
^ Granville 2008, section 1: "The main difference is that in algebraic
number theory [...] one typically considers questions with answers
that are given by exact formulas, whereas in analytic number theory
[...] one looks for good approximations."
^ See the remarks in the introduction to Iwaniec & Kowalski 2004,
p. 1: "However much stronger...".
^ Granville 2008, section 3: "[Riemann] defined what we now call the
Riemann zeta function
Riemann zeta function [...] Riemann's deep work gave birth to our
^ See, e.g., Montgomery & Vaughan 2007, p. 1.
^ CITEREFMilne2014, p. 2.
^ Edwards 2000, p. 79.
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This article incorporates material from the
Number theory", which is licensed under the Creative Commons
Attribution-ShareAlike 3.0 Unported License but not under the GFDL.
Two of the most popular introductions to the subject are:
G. H. Hardy; E. M. Wright (2008) . An introduction to the theory
of numbers (rev. by D. R. Heath-Brown and J. H. Silverman, 6th ed.).
Oxford University Press. ISBN 978-0-19-921986-5. Retrieved
Vinogradov, I. M. (2003) . Elements of
Number Theory (reprint of
the 1954 ed.). Mineola, NY: Dover Publications.
Hardy and Wright's book is a comprehensive classic, though its clarity
sometimes suffers due to the authors' insistence on elementary methods
(Apostol n.d.). 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. Other popular
first introductions are:
Ivan M. Niven; Herbert S. Zuckerman;
Hugh L. Montgomery
Hugh L. Montgomery (2008) .
An introduction to the theory of numbers (reprint of the 5th edition
1991 ed.). John Wiley & Sons. ISBN 978-81-265-1811-1.
Kenneth H. Rosen (2010). Elementary
Number Theory (6th ed.). Pearson
Education. ISBN 978-0-321-71775-7. Retrieved 2016-02-28.
Popular choices for a second textbook include:
Borevich, A. I.; Shafarevich, Igor R. (1966).
Number theory. Pure and
Applied Mathematics. 20. Boston, MA: Academic Press.
ISBN 978-0-12-117850-5. MR 0195803.
Serre, Jean-Pierre (1996) . A course in arithmetic. Graduate
texts in mathematics. 7. Springer. ISBN 978-0-387-90040-7.
Hazewinkel, Michiel, ed. (2001) , "
Number theory", Encyclopedia
of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic
Publishers, ISBN 978-1-55608-010-4
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