or the book by André Weil, see Number Theory: An Approach Through
History from Hammurapi to Legendre.
Not to be confused with Numerology.
The distribution of prime numbers, a central point
of study in number theory, illustrated by an Ulam spiral. It shows the
conditional independence between being prime and being a value of certain quadratic
polynomials.
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Number theory is a branch of pure mathematics devoted primarily to the
study of the integers and arithmetic functions. Number theorists study prime
numbers as well as the properties of mathematical objects constructed from
integers (for example, rational numbers), or defined as generalizations of the
integers (for example, algebraic integers).
Integers can be considered either in themselves or as solutions to equations
(Diophantine geometry). Questions in number theory can often be understood
through the study of analytical objects, such as 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, as for instance how irrational
numbers can be approximated by fractions (Diophantine approximation).
Number theory is one of the oldest branches of mathematics alongside
geometry. One quirk of number theory is that it deals with statements that are
simple to understand but are very difficult to solve. Examples of this
are Fermat's Last Theorem, which was proved 358 years after the original
formulation, and Goldbach's conjecture, which remains unsolved since the
18th century. German mathematician Carl Friedrich Gauss (1777–1855) said,
"Mathematics is the queen of the sciences—and number theory is the queen
of mathematics."[1] It was regarded as the example of pure mathematics with
no applications outside mathematics until the 1970s, when it became known
that prime numbers would be used as the basis for the creation of public-key
cryptography algorithms.
History
[edit]
Number theory is the branch of mathematics that studies integers and
their properties and relations.[2] The integers comprise a set that extends the
set of natural numbers to include number and the negation of
natural numbers . Number theorists study prime numbers as well as the
properties of mathematical objects constructed from integers (for
example, rational numbers), or defined as generalizations of the integers (for
example, algebraic integers).[3][4]
Number theory is closely related to arithmetic and some authors use the
terms as synonyms.[5] However, the word "arithmetic" is used today to mean
the study of numerical operations and extends to the real numbers.[6] In a
more specific sense, number theory is restricted to the study of integers and
focuses on their properties and relationships.[7] Traditionally, it is known as
higher arithmetic.[8] By the early twentieth century, the term number theory had
been widely adopted.[note 1] The term number means whole numbers, which
refers to either the natural numbers or the integers.[9][10][11]
Elementary number theory studies aspects of integers that can be
investigated using elementary methods such as elementary proofs.[12] Analytic
number theory, by contrast, relies on complex numbers and techniques from
analysis and calculus.[13] Algebraic number theory employs algebraic
structures such as fields and rings to analyze the properties of and relations
between numbers. Geometric number theory uses concepts from geometry to
study numbers.[14] Further branches of number theory are probabilistic number
theory,[15] combinatorial number theory,[16] computational number theory,[17] and
applied number theory, which examines the application of number theory to
science and technology.[18]
Origins
[edit]
Ancient Mesopotamia
[edit]
Plimpton 322 tablet
The earliest historical find of an arithmetical nature is a fragment of a
table: Plimpton 322 (Larsa, Mesopotamia, c. 1800 BC), a broken clay tablet,
contains a list of "Pythagorean triples", that is, integers such that .
The triples are too numerous 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..."[19]
The table's layout suggests that it was constructed by means of what
amounts, in modern language, to the identity[20]
which is implicit in routine Old Babylonian exercises. If some other method
was used, the triples were first constructed and then reordered by ,
presumably for actual use as a "table", for example, with a view to
applications.[21]
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 many centuries later. It has been suggested instead that the table was a
source of numerical examples for school problems.[22][note 2] Plimpton 322 tablet
is the only surviving evidence of what today would be called number theory
within Babylonian mathematics, though a kind of Babylonian algebra was
much more developed.[23]
Ancient Greece
[edit]
Further information: Ancient Greek mathematics
Although other civilizations probably influenced Greek mathematics at the
beginning,[24] all evidence of such borrowings appear relatively late,[25][26] and it
is likely that Greek arithmētikḗ (the theoretical or philosophical study of
numbers) is an indigenous tradition. Aside from a few fragments, most of what
is known about Greek mathematics in the 6th to 4th centuries BC
(the Archaic and Classical periods) comes through either the reports of
contemporary non-mathematicians or references from mathematical works in
the early Hellenistic period.[27] In the case of number theory, this means
largely Plato, Aristotle, and Euclid.
Plato had a keen interest in mathematics, and distinguished clearly
between arithmētikḗ and calculation (logistikē). Plato reports in his
dialogue Theaetetus that Theodorus had proven that are
irrational. Theaetetus, a disciple of Theodorus's, worked on distinguishing
different kinds of incommensurables, and was thus arguably a pioneer in the
study of number systems. Aristotle further claimed that the philosophy of Plato
closely followed the teachings of the Pythagoreans,[28] and Cicero repeats this
claim: Platonem ferunt didicisse Pythagorea omnia ("They say Plato learned
all things Pythagorean").[29]
Euclid devoted part of his Elements (Books VII–IX) to topics that belong to
elementary number theory, including prime numbers and divisibility.[30] He
gave an algorithm, the Euclidean algorithm, for computing the greatest
common divisor of two numbers (Prop. VII.2) and a proof implying the
infinitude of primes (Prop. IX.20). There is also older material likely based on
Pythagorean teachings (Prop. IX.21–34), such as "odd times even is even"
and "if an odd number measure
