Number theory is the branch of
pure mathematics concerned with the
properties of
numbers in general, and
integers in particular, as well as the wider
classes of problems that arise from their study.
Number theory may be subdivided into several fields, according to
the methods used and the type of questions investigated. (
See
the list of number theory
topics.)
The terms "
arithmetic" or "the higher
arithmetic" as
nouns are also used to refer to
number theory. These are somewhat older terms, which are no longer
as popular as they once were. However the word "arithmetic" is
popularly used as an
adjective rather than
the more cumbersome phrase "number-theoretic", and also "arithmetic
of" rather than "number theory of". e.g. (
arithmetic geometry,
arithmetic functions,
arithmetic of elliptic
curves).
Fields
Elementary number theory
In
elementary number theory, integers are studied
without use of techniques from other mathematical fields. Questions
of
divisibility, use of the
Euclidean algorithm to compute
greatest common divisors,
integer factorizations into
prime numbers, investigation of
perfect numbers and
congruences belong here. Several
important discoveries of this field are
Fermat's little theorem,
Euler's theorem, the
Chinese remainder theorem and the
law of
quadratic reciprocity.
The properties of
multiplicative
functions such as the
Möbius
function and
Euler's φ
function,
integer sequences,
factorials, and
Fibonacci numbers all also fall into this
area.
Many questions in number theory can be stated in elementary number
theoretic terms, but they may require very deep consideration and
new approaches outside the realm of elementary number theory to
solve. Examples include:
The theory of
Diophantine
equations has even been shown to be
undecidable (see
Hilbert's tenth problem).
Analytic number theory
Analytic number
theory employs the machinery of
calculus and
complex
analysis to tackle questions about integers. The
prime number theorem (PNT) and the
related
Riemann hypothesis are
examples.
Waring's problem
(representing a given integer as a sum of
squares,
cubes etc.), the
twin prime conjecture (finding
infinitely many prime pairs with difference 2) and
Goldbach's conjecture (writing even
integers as sums of two primes) are being attacked with analytical
methods as well.
Proofs of the
transcendence of
mathematical constants, such as
π or
e, are also classified as
analytical number theory. While statements about
transcendental numbers may seem to be
removed from the study of integers, they really study the possible
values of
polynomials with integer
coefficients evaluated at, say,
e; they are also closely
linked to the field of
Diophantine approximation, where
one investigates "how well" a given real number may be approximated
by a
rational one.
Algebraic number theory
In
algebraic number
theory, the concept of a number is expanded to the
algebraic numbers which are
roots of polynomials with
rational coefficients. These domains
contain elements analogous to the integers, the so-called
algebraic integers. In this setting, the
familiar features of the integers (e.g. unique factorization) need
not hold. The virtue of the machinery employed—
Galois theory,
group cohomology,
class field theory,
group representations and
L-functions—is that it allows one to recover that
order partly for this new class of numbers.
Many number theoretic questions are best attacked by studying them
modulo p for all primes
p (see
finite fields). This is called
localization and it leads to the construction of the
p-adic numbers; this field of study is
called
local analysis and it arises
from algebraic number theory.
Geometry of numbers
The
geometry of
numbers incorporates some basic geometric concepts,
such as lattices, into number-theoretic questions. It starts with
Minkowski's theorem about
lattice points in
convex sets, and leads to basic proofs of the
finiteness of the
class number and
Dirichlet's unit theorem,
two fundamental theorems in algebraic number theory.
Combinatorial number theory
Combinatorial number theory deals with number
theoretic problems which involve
combinatorial ideas in their formulations or
solutions.
Paul Erdős is the main
founder of this branch of number theory. Typical topics include
covering system,
zero-sum problems, various
restricted sumsets, and
arithmetic progressions in a set of
integers. Algebraic or analytic methods are powerful in this field.
See also
arithmetic
combinatorics.
Computational number theory
Computational number
theory studies
algorithms
relevant in number theory. Fast algorithms for
prime testing and
integer factorization have important
applications in
cryptography.
Arithmetic algebraic geometry
See
arithmetic geometry.
Arithmetic topology
Arithmetic
topology developed from a series of analogies between
number fields and
3-manifolds;
primes and
knots pointed out by
Barry Mazur and by
Yuri
Manin in the 1960s.
Arithmetic dynamics
Arithmetic
dynamics is a field that emerged in the 1990s that
amalgamates two areas of mathematics,
dynamical systems and number theory.
Classically, discrete dynamics refers to the study of the
iteration of self-maps of the
complex plane or
real
line. Arithmetic dynamics is the study of the number-theoretic
properties of integer, rational, -adic, and/or algebraic points
under repeated application of a
polynomial or
rational function.
Modular forms
Modular forms are (complex)
analytic functions on the
upper half-plane satisfying a certain kind
of
functional equation and
growth condition. The theory of modular forms therefore belongs to
complex analysis but the main
importance of the theory has traditionally been in its connections
with number theory. Modular forms appear in other areas, such as
algebraic topology and
string theory.
History
Greek number theory
Number
theory was a favorite study among the Greek mathematicians of the late
Hellenistic period (3rd century AD) in Alexandria, Egypt, who were
aware of the Diophantine
equation concept in numerous special cases. The first
Greek mathematician to study these equations was
Diophantus.
Diophantus also looked for a method of finding integer solutions to
linear indeterminate equations, equations
that lack sufficient information to produce a single discrete set
of answers. The equation x + y = 5 is such an equation. Diophantus
discovered that many indeterminate equations can be reduced to a
form where a certain category of answers is known even though a
specific answer is not.
Classical Indian number theory
Diophantine equations were
extensively studied by mathematicians in medieval India, who were
the first to systematically investigate methods for the
determination of integral solutions of Diophantine equations.
Aryabhata (499) gave the first explicit
description of the general integral solution of the linear
Diophantine equation ay + bx = c, which occurs in his text
Aryabhatiya. This
kuttaka algorithm is considered
to be one of the most significant contributions of Aryabhata in
pure mathematics, which found solutions to Diophantine equations by
means of
continued fractions. The
technique was applied by Aryabhata to give integral solutions of
simultaneous linear Diophantine equations, a problem with important
applications in astronomy. He also found the general solution to
the
indeterminate linear equation using this method.
Brahmagupta in 628 handled more
difficult Diophantine equations. He used the
chakravala method to solve
quadratic Diophantine equations,
including forms of
Pell's equation,
such as 61x^2 + 1 = y^2. His
Brahma Sphuta Siddhanta was
translated into
Arabic in 773 and was
subsequently translated into
Latin in 1126.
The
equation 61x^2 + 1 = y^2 was later posed as a problem in 1657 by
the French
mathematician Pierre de
Fermat. The general solution to this particular form of
Pell's equation was found over 70 years later by
Leonhard Euler, while the general solution to
Pell's equation was found over 100 years later by
Joseph Louis Lagrange in 1767.
Meanwhile, many centuries ago, the general solution to Pell's
equation was recorded by
Bhaskara II in
1150, using a modified version of Brahmagupta's
chakravala
method, which he also used to find the general solution to other
indeterminate quadratic equations and quadratic Diophantine
equations. Bhaskara's
chakravala method for finding the
general solution to Pell's equation was much simpler than the
method used by Lagrange over 600 years later. Bhaskara also found
solutions to other indeterminate quadratic,
cubic,
quartic, and higher-order
polynomial equations.
Narayana Pandit further improved on the
chakravala method and found more general solutions to
other indeterminate quadratic and higher-order polynomial
equations.
Islamic number theory
From the 9th century,
Islamic
mathematics had a keen interest in number theory. The first of
these mathematicians was
Thabit ibn
Qurra, who discovered an algorithm which allowed pairs of
amicable numbers to be found, that
is two numbers such that each is the sum of the proper divisors of
the other. In the 10th century,
Al-Baghdadi looked at a slight variant of Thabit
ibn Qurra's method.
In the 10th century,
al-Haitham seems to
have been the first to attempt to classify all even
perfect numbers (numbers equal to the sum of
their proper divisors) as those of the form 2^{k-1}(2^k - 1) where
2^k - 1 is prime. Al-Haytham is also the first person to state
Wilson's theorem, namely that if p
is prime then 1+(p-1)! is divisible by p. It is unclear whether he
knew how to prove this result. It is called Wilson's theorem
because of a comment made by
Edward
Waring in 1770 that
John
Wilson had noticed the result. There is no evidence that Wilson
knew how to prove it and most certainly Waring did not. Lagrange
gave the first proof in 1771.
Amicable numbers played a large role in Islamic mathematics. In the
13th century,
Persian mathematician
Al-Farisi gave a new proof of
Thabit ibn Qurra's theorem, introducing
important new ideas concerning factorisation and combinatorial
methods. He also gave the pair of amicable numbers 17296, 18416
which have been attributed to Euler, but we know that these were
known earlier than al-Farisi, perhaps even by Thabit ibn Qurra
himself. In the 17th century,
Muhammad Baqir Yazdi gave the pair of
amicable numbers 9,363,584 and 9,437,056 still many years before
Euler's contribution.
Early European number theory
Number theory began in
Europe in the 16th and
17th centuries, with
Vieta,
Bachet de Meziriac, and
especially
Fermat, whose
infinite descent method was the first
general proof of diophantine questions.
Fermat's Last Theorem was posed as a
problem in 1637, a proof of which wasn't found until 1994. Fermat
also posed the equation 61x^2 + 1 = y^2 as a problem in 1657.
In the eighteenth century, Euler and Lagrange made important
contributions to number theory. Euler did some work on
analytic number theory, and found a
general solution to the equation 61x^2 + 1 = y^2. Lagrange found a
solution to the more general Pell's equation.
Euler and Lagrange
solved these Pell equations by means of continued fractions, though this was more
difficult than the Indian chakravala method.
Beginnings of modern number theory
Around the beginning of the nineteenth century books of
Legendre (1798), and
Gauss put together the first systematic
theories in Europe. Gauss's
Disquisitiones Arithmeticae
(1801) may be said to begin the modern theory of numbers.
The formulation of the theory of
congruences starts with Gauss's
Disquisitiones. He introduced the notation
- a \equiv b \pmod c,
and explored most of the field.
Chebyshev published in 1847 a work in
Russian on the subject, and in France
Serret popularised it.
Besides summarizing previous work,
Legendre
stated the
law of quadratic
reciprocity. This law, discovered by
induction and enunciated by Euler,
was first proved by Legendre in his
Théorie des Nombres
(1798) for special cases. Independently of Euler and Legendre,
Gauss discovered the law about 1795, and was the first to give a
general proof. The following have also contributed to the subject:
Cauchy;
Dirichlet whose
Vorlesungen über
Zahlentheorie is a classic;
Jacobi, who introduced the
Jacobi symbol;
Liouville,
Zeller,
Eisenstein,
Kummer, and
Kronecker.
The theory extends to include
cubic and
quartic reciprocity, (Gauss, Jacobi who
first proved the law of cubic reciprocity, and Kummer).
To Gauss is also due the representation of numbers by binary
quadratic forms.
Prime number theory
A recurring and productive theme in number theory is the study of
the distribution of prime numbers.
Carl Friedrich Gauss conjectured an
asymptotic rule for such
behaviour (the
prime number
theorem) as a teenager.
Chebyshev (1850) gave useful bounds for
the number of primes between two given limits. Riemann introduced
complex analysis into the theory of
the
Riemann zeta function.
This led to a relation between the zeros of the zeta function and
the distribution of primes, eventually leading to a proof of
prime number theorem
independently by
Hadamard and
de la Vallée
Poussin in 1896. However, an elementary proof was given later
by
Paul Erdős and
Atle Selberg in 1949. Here elementary means
that it does not use techniques of complex analysis; however, the
proof is still very ingenious and difficult. The
Riemann hypothesis, which would give much
more accurate information, is still an open question.
Nineteenth-century developments
Cauchy,
Poinsot (1845), and notably
Hermite have added to the subject. In the
theory of ternary forms,
Eisenstein has been a leader, and to
him and
H. J. S. Smith is also due a noteworthy advance in the
theory of forms in general. Smith gave a complete classification of
ternary quadratic forms, and extended Gauss's researches concerning
real
quadratic forms to complex
forms. The investigations concerning the representation of numbers
by the sum of 4, 5, 6, 7, 8 squares were advanced by Eisenstein and
the theory was completed by Smith.
Dirichlet was the first to lecture upon
the subject in a German university. Among his contributions is the
proof of
Fermat's Last
Theorem:
- x^n+y^n \neq z^n, (x,y,z \neq 0, n > 2)
for the cases
n = 5 and
n = 14 (Euler and
Legendre had already proved the cases
n = 3 and
n
= 4 and therefore by implication, all multiples of 3 and 4). Among
the later French writers are
Borel;
Poincaré, whose memoirs are
numerous and valuable;
Tannery, and
Stieltjes. Among the leading
contributors in Germany were
Leopold
Kronecker,
Ernst Kummer,
Ernst Christian Julius
Schering,
Paul Bachmann, and
Dedekind. In Austria
Stolz's
Vorlesungen über allgemeine
Arithmetik (1885-86), and in England
Mathews' Theory of Numbers (Part I,
1892) were scholarly general works.
Genocchi,
Sylvester, and
J. W.
L. Glaisher have also added to the
theory.
Late nineteenth- and early twentieth-century developments
It was the time of major advancements in number theory due to the
work of
Axel Thue on
diophantine equations, of
David Hilbert in
algebraic number theory (he also
proved the
Waring's
prime number conjecture), and to the creation of
geometric number theory by
Hermann Minkowski, but also thanks to
Adolf Hurwitz,
Georgy F. Voronoy,
Waclaw Sierpinski,
Derrick Norman Lehmer and several
others.
Twentieth-century developments
Major figures in twentieth-century number theory include
Hermann Weyl,
Nikolai Chebotaryov,
Emil Artin,
Erich
Hecke,
Helmut Hasse,
Alexander Gelfond,
Yuri Linnik,
Paul
Erdős,
Gerd Faltings,
G. H. Hardy,
Edmund
Landau,
Louis Mordell,
John Edensor Littlewood,
Ivan Niven,
Srinivasa Ramanujan,
André Weil,
Ivan
Vinogradov,
Atle Selberg,
Carl Ludwig Siegel,
Igor Shafarevich,
John
Tate,
Robert Langlands,
Goro Shimura,
Kenkichi Iwasawa,
Jean-Pierre Serre,
Pierre Deligne,
Enrico Bombieri,
Alan Baker,
Peter Swinnerton-Dyer,
Bryan Birch,
Vladimir Drinfeld,
Laurent Lafforgue,
Andrew Wiles, and
Richard Taylor.
Milestones in twentieth-century number theory include the proof of
Fermat's Last Theorem by
Andrew Wiles in 1994 and the proof of
the related
Taniyama–Shimura
conjecture in 1999.
Applied number theory
The book "Number theory for computing" says that number theory has
been applied to: "physics, chemistry, biology, computing,
engineering, coding and cryptography, random number generation,
acoustics, communications, graphic design and even music and
business." It also says that
Shiing-Shen Chern "considers number theory
as a branch of applied mathematics because of its strong
applicability in other fields."
Residue number system
A
residue number system (RNS)
represents a large integer using a set of smaller integers, so that
computation may be performed more efficiently. It relies on the
Chinese remainder theorem
of
modular arithmetic for its
operation. RNS have applications in the field of digital computer
arithmetic. By decomposing in this a large integer into a set of
smaller integers, a large calculation can be performed as a series
of smaller calculations that can be performed independently and in
parallel. Because of this, it is particularly popular in hardware
implementations.
Integer programming
The
LLL algorithm is used in
integer linear programming.
Quotations
- "Mathematics is the queen of the sciences and number theory is
the queen of mathematics." — Gauss
- "God invented the integers; all else is the work of man." —
Kronecker
- "Number is the within of all things." — Attributed to Pythagoras
Notes
- Number theory for computing
- Quoted in Gauss zum Gedächtniss (1856) by Wolfgang
Sartorius von Waltershausen
- "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist
Menschenwerk" Heinrich Weber: Leopold Kronecker.
Jahresberichte D.M.V 2 (1893) 5-31
- L.A. Michael: The Principles of Existence & Beyond (Dec
2007) ISBN 1847991998, ISBN 978-1847991997
References & further reading
- Smith, David. History of Modern Mathematics (1906) (adapted
public domain text)
- Dutta, Amartya Kumar (2002). 'Diophantine equations: The
Kuttaka', Resonance - Journal of Science
Education.
- O'Connor, John J. and Robertson, Edmund F. (2004). 'Arabic/Islamic mathematics', MacTutor History of
Mathematics archive.
- O'Connor, John J. and Robertson, Edmund F. (2004). 'Index of Ancient Indian mathematics',
MacTutor
History of Mathematics archive.
- O'Connor, John J. and Robertson, Edmund F. (2004). 'Numbers and Number Theory Index', MacTutor History of
Mathematics archive.
- Important
publications in number theory
External links