TheInfoList

Number theory (or arithmetic or higher arithmetic in older usage) is a branch of
pure mathematics Pure mathematics is the study of mathematical concepts independently of any application outside mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...
devoted primarily to the study of the
integer An integer (from the Latin Latin (, or , ) is a classical language A classical language is a language A language is a structured system of communication Communication (from Latin ''communicare'', meaning "to share" or "to ...
s and integer-valued functions. German mathematician
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of m ...

(1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s as well as the properties of mathematical objects made out of integers (for example,
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s) or defined as generalizations of the integers (for example,
algebraic integer In algebraic number theory Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory. Algebraic number theory is a branch of number theory that uses the techniques of abstrac ...
s). Integers can be considered either in themselves or as solutions to equations (
Diophantine geometry In mathematics, Diophantine geometry is the study of points of algebraic varieties with coordinates in the integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be wr ...
). Questions in number theory are often best understood through the study of analytical objects (for example, the
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter The Greek alphabet has been used to write the Greek language Greek (modern , romanized: ''Elliniká'', Ancient Greek, ancient , ''Hellēnikḗ'') is ...

) that encode properties of the integers, primes or other number-theoretic objects in some fashion (
analytic number theory 300px, Riemann zeta function ''ζ''(''s'') in the complex plane. The color of a point ''s'' encodes the value of ''ζ''(''s''): colors close to black denote values close to zero, while hue encodes the value's Argument (complex analysis)">argument ...
). One may also study
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s in relation to rational numbers, for example, as approximated by the latter (
Diophantine approximation In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Fried ...
). The older term for number theory is ''arithmetic''. By the early twentieth century, it had been superseded by "number theory".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." (The word "
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
" is used by the general public to mean " elementary calculations"; it has also acquired other meanings in
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal sys ...
, as in ''
Peano arithmetic In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian people, Italian mathematician Giuseppe Peano. These axioms have been ...
'', and
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of computation, automation, a ...
, as in ''
floating point arithmetic In computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes and development of both computer hardware , hardware and soft ...
''.) 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.Take, for example, . In 1952,
Davenport Davenport may refer to: Places Australia *Davenport, Northern Territory, a locality *Hundred of Davenport, cadastral unit in South Australia **Davenport, South Australia, suburb of Port Augusta **District Council of Davenport, former local governme ...

still had to specify that he meant ''The Higher Arithmetic''.
Hardy Hardy may refer to: People * Hardy (surname) * Hardy (given name) * Hardy (singer), American singer-songwriter Places Antarctica * Mount Hardy, Enderby Land * Hardy Cove, Greenwich Island * Hardy Rocks, Biscoe Islands Australia * Hardy, South A ...
and Wright wrote in the introduction to '' An Introduction to the Theory of Numbers'' (1938): "We proposed at one time to change he titleto ''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."
In particular, ''arithmetical'' is commonly preferred as an adjective to ''number-theoretic''.

History

Origins

Dawn of arithmetic

The earliest historical find of an arithmetical nature is a fragment of a table: the broken clay tablet
Plimpton 322 Plimpton 322 is a Babylonia Babylonia () was an Ancient history, ancient Akkadian language, Akkadian-speaking state (polity), state and cultural area based in central-southern Mesopotamia (present-day Iraq and Syria). A small Amorites, Amori ...

( Larsa, Mesopotamia, ca. 1800 BCE) contains a list of " Pythagorean triples", that is, integers $\left(a,b,c\right)$ such that $a^2+b^2=c^2$. 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..." The table's layout suggests that it was constructed by means of what amounts, in modern language, to the
identity Identity may refer to: Social sciences * Identity (social science), personhood or group affiliation in psychology and sociology Group expression and affiliation * Cultural identity, a person's self-affiliation (or categorization by others ...
:$\left\left(\frac \left\left(x - \frac\right\right)\right\right)^2 + 1 = \left\left(\frac \left\left(x + \frac \right\right)\right\right)^2,$ which is implicit in routine
Old Babylonian Old Babylonian may refer to: *the period of the First Babylonian dynasty (20th to 16th centuries BC) *the historical stage of the Akkadian language of that time See also

*Old Assyrian (disambiguation) {{disambig ...
exercises. If some other method was used, the triples were first constructed and then reordered by $c/a$, presumably for actual use as a "table", for example, with a view to applications. It is not known what these applications may have been, or whether there could have been any;
Babylonian astronomy Babylonian astronomy was the study or recording of celestial object In astronomy, an astronomical object or celestial object is a naturally occurring physical entity, association, or structure that exists in the observable universe. In ...
, 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.. This is controversial. See
Plimpton 322 Plimpton 322 is a Babylonia Babylonia () was an Ancient history, ancient Akkadian language, Akkadian-speaking state (polity), state and cultural area based in central-southern Mesopotamia (present-day Iraq and Syria). A small Amorites, Amori ...

. Robson's article is written polemically with a view to "perhaps ..knocking limpton 322off its pedestal" ; 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 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".
While Babylonian number theory—or what survives of
Babylonian mathematics Babylonian mathematics (also known as ''Assyro-Babylonian mathematics'') denotes the mathematics developed or practiced by the people of Mesopotamia, from the days of the early Sumerians to the centuries following the fall of Babylon in 539 BC. Bab ...
that can be called thus—consists of this single, striking fragment, Babylonian algebra (in the secondary-school sense of "
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

") was exceptionally well developed. Late Neoplatonic sources
Iamblichus Iamblichus (; grc-gre, Ἰάμβλιχος ; Safaitic Safaitic ( ''Ṣafāʾiyyah'') is a variety of the South Semitic script used by the nomads of the basalt desert of southern Syria and northern Jordan, the so-called Ḥarrah, to carve ro ...

, ''Life of Pythagoras'',(trans., for example, ) cited in . See also Porphyry, ''Life of Pythagoras'', paragraph 6, in Van der Waerden sustains the view that Thales knew Babylonian mathematics.
state that
Pythagoras Pythagoras of Samos, or simply ; in Ionian Greek () was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graec ...

learned mathematics from the Babylonians. Much earlier sourcesHerodotus (II. 81) and Isocrates (''Busiris'' 28), cited in: . On Thales, see Eudemus ap. Proclus, 65.7, (for example, ) cited in: . Proclus was using a work by
Eudemus of Rhodes Eudemus of Rhodes ( grc-gre, Εὔδημος) was an ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roughly d ...
(now lost), the ''Catalogue of Geometers''. See also introduction, on Proclus's reliability.
state that
Thales Thales of Miletus ( ; el, Θαλῆς Thales of Miletus ( ; el, Θαλῆς Thales of Miletus ( ; el, Θαλῆς (ὁ Μιλήσιος), ''Thalēs''; ) was a Greek mathematician A mathematician is someone who uses an extensive kn ...

and
Pythagoras Pythagoras of Samos, or simply ; in Ionian Greek () was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graec ...

traveled and studied in
Egypt Egypt ( ar, مِصر, Miṣr), officially the Arab Republic of Egypt, is a transcontinental country This is a list of countries located on more than one continent A continent is one of several large landmasses. Generally identi ...

.
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

IX 21–34 is very probably Pythagorean;, cited in: . it is very simple material ("odd times even is even", "if an odd number measures dividesan even number, then it also measures divideshalf of it"), but it is all that is needed to prove that $\sqrt$ is
irrational Irrationality is cognition Cognition () refers to "the mental action or process of acquiring knowledge and understanding through thought, experience, and the senses". It encompasses many aspects of intellectual functions and processes such as: ...
. Pythagorean mystics gave great importance to the odd and the even. The discovery that $\sqrt$ is irrational is credited to the early Pythagoreans (pre- Theodorus).Plato, ''Theaetetus'', p. 147 B, (for example, ), cited in : "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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

.
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 Hippasus of Metapontum Metapontum or Metapontium ( grc, Μεταπόντιον, Metapontion) was an important city of Magna Graecia, situated on the gulf of Taranto, Tarentum, between the river Bradanus and the Casuentus (modern Basento). It wa ...
, 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 In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space t ...
or
figurate numbersThe term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polygonal ...
. While
square number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...
s,
cubic number In arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', ' ...
s, etc., are seen now as more natural than
triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and Cube (algebra)#In integers, cube numbers. The th triangular number ...
s,
pentagonal number 181px, A visual representation of the first six pentagonal numbers A pentagonal number is a figurate number that extends the concept of triangular number, triangular and square numbers to the pentagon, but, unlike the first two, the patterns involv ...

s, etc., the study of the sums of triangular and pentagonal numbers would prove fruitful in the early modern period (17th to early 19th century). We know of no clearly arithmetical material in
ancient Egyptian Ancient Egypt was a civilization of Ancient history, ancient North Africa, concentrated along the lower reaches of the Nile, Nile River, situated in the place that is now the country Egypt. Ancient Egyptian civilization followed prehistori ...
or
Vedic upright=1.2, The Vedas are ancient Sanskrit texts of Hinduism. Above: A page from the ''Atharvaveda''. The Vedas (, , ) are a large body of religious texts originating in ancient India. Composed in Vedic Sanskrit, the texts constitute the ol ...

sources, though there is some algebra in each. The
Chinese remainder theorem In number theory, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number ...
appears as an exercise in ''
Sunzi Suanjing ''Sunzi Suanjing'' () was a mathematical treatise A treatise is a formal and systematic written discourse on some subject, generally longer and treating it in greater depth than an essay, and more concerned with investigating or exposing the p ...
'' (3rd, 4th or 5th century CE).The date of the text has been narrowed down to 220–420 CE (Yan Dunjie) or 280–473 CE (Wang Ling) through internal evidence (= taxation systems assumed in the text). See . (There is one important step glossed over in Sunzi's solution:''Sunzi Suanjing'', Ch. 3, Problem 26, in :
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 exceeds 106, the result is obtained by subtracting 105.
it is the problem that was later solved by
Āryabhaṭa Aryabhata (, ISO 15919, ISO: ) or Aryabhata I (476–550 Common Era, CE) was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the ''Āryabhaṭīya'' (which me ...
's
KuṭṭakaKuṭṭaka is an algorithm for finding integer solutions of linear Diophantine equations. A linear Diophantine equation is an equation of the form ''ax'' + ''by'' = ''c'' where ''x'' and ''y'' are variable (mathematics), unknown quantities and ''a'' ...
– see
below Below may refer to: *Earth *Ground (disambiguation) *Soil *Floor *Bottom (disambiguation) *Less than *Temperatures below freezing *Hell or underworld People with the surname *Fred Below (1926–1988), American blues drummer *Fritz von Below (1853 ...
.) There is also some numerical mysticism in Chinese mathematics,See, for example, ''Sunzi Suanjing'', Ch. 3, Problem 36, in :
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'': Male.
''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
f the Dipper F, or f, is the sixth letter Letter, letters, or literature may refer to: Characters typeface * Letter (alphabet) A letter is a segmental symbol A symbol is a mark, sign, or word that indicates, signifies, or is understood as repr ...
8 the eight winds, and 9 the nine divisions f China under Yu the Great If the remainder is odd, he sexis male and if the remainder is even, he sexis female.
This is the last problem in Sunzi's otherwise matter-of-fact treatise.
but, unlike that of the Pythagoreans, it seems to have led nowhere. Like the Pythagoreans'
perfect number In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number ...
s,
magic squares rotation and reflection) non-trivial case of a magic square, order 3 In recreational mathematics Recreational mathematics is mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), ...
have passed from superstition into
recreation Recreation is an activity of leisure Leisure has often been defined as a quality of experience or as free time. Free time is spent away from , , , , and , as well as necessary activities such as and ing. Leisure as an experience usuall ...
.

Classical Greece and the early Hellenistic period

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 and large, ''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 tradition.
Eusebius Eusebius of Caesarea (; grc-gre, Εὐσέβιος τῆς Καισαρείας, ''Eusébios tés Kaisareías''; AD 260/265 – 339/340), also known as Eusebius Pamphili (from the grc-gre, Εὐσέβιος τοῦ Παμϕίλου) ...

, PE X, chapter 4 mentions of
Pythagoras Pythagoras of Samos, or simply ; in Ionian Greek () was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymous founder of Pythagoreanism. His political and religious teachings were well known in Magna Graec ...

:
"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 Plato learned all things Pythagorean").
Plato Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was an Classical Athens, Athenian philosopher during the Classical Greece, Classical period in Ancient Greece, founder of the Platonist school of thought and the Platoni ...

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 $\sqrt, \sqrt, \dots, \sqrt$ 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 The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematics, mathematical treatise consisting of 13 books attributed to the ancient Greek mathematics, Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a colle ...
is described by Pappus as being largely based on Theaetetus's work.)
Euclid Euclid (; grc-gre, Εὐκλείδης Euclid (; grc, Εὐκλείδης – ''Eukleídēs'', ; fl. 300 BC), sometimes called Euclid of Alexandria to distinguish him from Euclid of Megara, was a Greek mathematician, often referre ...

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 The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematics, mathematical treatise consisting of 13 books attributed to the ancient Greek mathematics, Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a colle ...
). In particular, he gave an algorithm for computing the greatest common divisor of two numbers (the
Euclidean algorithm In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an effi ...
; ''Elements'', Prop. VII.2) and the first known proof of the infinitude of primes (''Elements'', Prop. IX.20). In 1773,
LessingLessing is a German surname of Slavic origin, originally ''Lesnik'' meaning "woodman". Lessing may refer to: A German family of writers, artists, musicians and politicians who can be traced back to a Michil Lessigk mentioned in 1518 as being a line ...

published an
epigram An epigram is a brief, interesting, memorable, and sometimes surprising or satirical statement. The word is derived from the Ancient Greek, Greek "inscription" from "to write on, to inscribe", and the literary device has been employed for o ...
he had found in a manuscript during his work as a librarian; it claimed to be a letter sent by
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

to
Eratosthenes Eratosthenes of Cyrene (; grc-gre, Ἐρατοσθένης ;  – ) was a Greek polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose knowledge spans a ...

. 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 Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2-ny^2=1 where ''n'' is a given positive nonsquare integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is co ...

). 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 of solution.

Diophantus

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 ''Arithmetica'' ( grc-gre, Ἀριθμητικά) is an Ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roug ...

'' survive in the original Greek and four more survive in an Arabic translation. The ''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 $f\left(x,y\right)=z^2$ or $f\left(x,y,z\right)=w^2$. Thus, nowadays, we speak of ''Diophantine equations'' when we speak of polynomial equations to which rational or integer solutions must be found. One may say that Diophantus was studying rational points, that is, points whose coordinates are rational—on
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...

s and
algebraic varieties Algebraic varieties are the central objects of study in algebraic geometry Algebraic geometry is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures ...
; 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) $f\left(x_1,x_2,x_3\right)=0$, his aim was to find (in essence) three
rational functions In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
$g_1, g_2, g_3$ such that, for all values of $r$ and $s$, setting $x_i = g_i\left(r,s\right)$ for $i=1,2,3$ gives a solution to $f\left(x_1,x_2,x_3\right)=0.$ 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 curve In , an elliptic curve is a , , of one, on which there is a specified point ''O''. An elliptic curve is defined over a ''K'' and describes points in ''K''2, the of ''K'' with itself. If the field's is different from 2 and 3, then the curv ...
s, 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. (Diophantus also resorted to what could be called a special case of a secant construction.) While 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 explicitly).

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;Any early contact between Babylonian and Indian mathematics remains conjectural . in particular, there is no evidence that Euclid's Elements reached India before the 18th century.
Āryabhaṭa Aryabhata (, ISO 15919, ISO: ) or Aryabhata I (476–550 Common Era, CE) was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the ''Āryabhaṭīya'' (which me ...
(476–550 CE) showed that pairs of simultaneous congruences $n\equiv a_1 \bmod m_1$, $n\equiv a_2 \bmod m_2$ could be solved by a method he called ''kuṭṭaka'', or ''pulveriser''; this is a procedure close to (a generalisation of) the
Euclidean algorithm In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an effi ...
, which was probably discovered independently in India. Āryabhaṭa seems to have had in mind applications to astronomical calculations.
Brahmagupta Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, doc ...

(628 CE) started the systematic study of indefinite quadratic equations—in particular, the misnamed
Pell equation Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structur ...

, in which
Archimedes Archimedes of Syracuse (; grc, ; ; ) was a Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Eu ...

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") 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 Bhāskara (c. 1114–1185) also known as Bhāskarācārya ("Bhāskara, the teacher"), and as Bhāskara II to avoid confusion with Bhāskara I, was an Indian people, Indian Indian mathematicians, mathematician and astronomer. He was born in Bija ...
's Bīja-gaṇita (twelfth century). Indian mathematics remained largely unknown in Europe until the late eighteenth century; Brahmagupta and Bhāskara's work was translated into English in 1817 by Henry Colebrooke.

Arithmetic in the Islamic golden age

In the early ninth century, the caliph
Al-Ma'mun Abu al-Abbas Abdallah ibn Harun al-Rashid ( ar, أبو العباس عبد الله بن هارون الرشيد, Abū al-ʿAbbās ʿAbd Allāh ibn Hārūn ar-Rashīd; 14 September 786 – 9 August 833), better known by his regnal name al-Ma'mu ...
ordered translations of many Greek mathematical works and at least one Sanskrit work (the ''Sindhind'', which may or may not, and , cited in . be
Brahmagupta Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, doc ...

's
Brāhmasphuṭasiddhānta The ''Brāhmasphuṭasiddhānta'' ("Correctly Established siddhanta, Doctrine of Brahma", abbreviated BSS) is the main work of Brahmagupta, written c. 628. This text of mathematical astronomy contains significant mathematical content, including a g ...
). Diophantus's main work, the ''Arithmetica'', was translated into Arabic by
Qusta ibn Luqa Qusta ibn Luqa (820–912) (Costa ben Luca, Constabulus) was a Syrian Melkite Christian physician A physician (American English), medical practitioner (English in the Commonwealth of Nations, Commonwealth English), medical doctor, or simply ...

(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 Ḥasan Ibn al-Haytham (Latinized Latinisation or Latinization can refer to: * Latinisation of names, the practice of rendering a non-Latin name in a Latin style * Latinisation in the Soviet Union, the campaign in the USSR during the 1920s and ...

knew what would later be called
Wilson's theorem In number theory, Wilson's theorem states that a natural number In mathematics, the natural numbers are those used for counting (as in "there are ''six'' coins on the table") and total order, ordering (as in "this is the ''third'' largest city ...
.

Western Europe in the Middle Ages

Other than a treatise on squares in arithmetic progression by
Fibonacci Fibonacci (; also , ; – ), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician A mathematician is someone who uses an extensive knowledge of mathem ...

—who traveled and studied in north Africa and Constantinople—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 The Renaissance ( , ) , from , with the same meanings. is a period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in ...

, thanks to a renewed study of the works of Greek antiquity. A catalyst was the textual emendation and translation into Latin of Diophantus' ''
Arithmetica ''Arithmetica'' ( grc-gre, Ἀριθμητικά) is an Ancient Greek Ancient Greek includes the forms of the Greek language used in ancient Greece and the classical antiquity, ancient world from around 1500 BC to 300 BC. It is often roug ...

''.

Early modern number theory

Fermat

Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of suc ...

(1607–1665) never published his writings; in particular, his work on number theory is contained almost entirely in letters to mathematicians and in private marginal notes. In his notes and letters, he scarcely wrote any proofs - he had no models in the area. Over his lifetime, Fermat made the following contributions to the field: * One of Fermat's first interests was
perfect number In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number ...
s (which appear in Euclid, ''Elements'' IX) and amicable numbers;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 Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 AD) was an important ancient mathematician best known for his works ''Introduction to Arithmetic'' and ''Manual of Harmonics'' in Ancient Greek, Greek. He was born in Gerasa ...
(ca. 100 CE), who wrote a primitive but influential "
Introduction to Arithmetic The book ''Introduction to Arithmetic'' ( grc-gre, Ἀριθμητικὴ εἰσαγωγή, ''Arithmetike eisagoge'') is the only extant work on mathematics by Nicomachus Nicomachus of Gerasa ( grc-gre, Νικόμαχος; c. 60 – c. 120 ...
". See .
these topics led him to work on integer
divisor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s, 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. * In 1638, Fermat claimed, without proof, that all whole numbers can be expressed as the sum of four squares or fewer. *
Fermat's little theorem Fermat's little theorem states that if is a prime number, then for any integer , the number is an integer multiple of . In the notation of modular arithmetic, this is expressed as :a^p \equiv a \pmod p. For example, if = 2 and = 7, then 27 = ...
(1640): if ''a'' is not divisible by a prime ''p'', then $a^ \equiv 1 \bmod p.$Here, as usual, given two integers ''a'' and ''b'' and a non-zero integer ''m'', we write $a \equiv b \bmod m$ (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 Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of m ...

's
Disquisitiones Arithmeticae Title page of the first edition The (Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through ...
. Fermat's little theorem is a consequence of the
fact A fact is something that is true True most commonly refers to truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things ...
that the
order Order, ORDER or Orders may refer to: * Orderliness Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...
of an element of a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
divides the
order Order, ORDER or Orders may refer to: * Orderliness Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...
of the
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
. 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'', that is, given ''a'' not divisible by a prime ''p'', there is an integer ''x'' such that $x a \equiv 1 \bmod p$); this fact (which, in modern language, makes the residues mod ''p'' into a group, and which was already known to
Āryabhaṭa Aryabhata (, ISO 15919, ISO: ) or Aryabhata I (476–550 Common Era, CE) was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the ''Āryabhaṭīya'' (which me ...
; see above) was familiar to Fermat thanks to its rediscovery by Bachet . Weil goes on to say that Fermat would have recognised that Bachet's argument is essentially Euclid's algorithm.
* If ''a'' and ''b'' are coprime, then $a^2 + b^2$ 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 $a^2 + b^2$. 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. * In 1657, Fermat posed the problem of solving $x^2 - N y^2 = 1$ as a challenge to English mathematicians. 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 wasn't aware of this). He stated that a proof could be found by infinite descent. * Fermat stated and proved (by infinite descent) in the appendix to ''Observations on Diophantus'' (Obs. XLV) that $x^ + y^ = z^$ has no non-trivial solutions in the integers. Fermat also mentioned to his correspondents that $x^3 + y^3 = z^3$ has no non-trivial solutions, and that this could also be proven by infinite descent. The first known proof is due to Euler (1753; indeed by infinite descent). * Fermat claimed (
Fermat's Last Theorem In number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is the queen of the sciences—and number ...
) to have shown there are no solutions to $x^n + y^n = z^n$ for all $n\geq 3$; this claim appears in his annotations in the margins of his copy of Diophantus.

Euler

The interest of
Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ) ...

(1707–1783) in number theory was first spurred in 1729, when a friend of his, the amateurUp 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 . (There were already some recognisable features of professional ''practice'', viz., seeking correspondents, visiting foreign colleagues, building private libraries . Matters started to shift in the late 17th century ; scientific academies were founded in England (the
Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society A learned society (; also known as a learned academy, scholarly society, or academic association) is an organization that exis ...
, 1662) and France (the
Académie des sciences The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV of France, Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French Scientific met ...
, 1666) and
Russia Russia ( rus, link=no, Россия, Rossiya, ), or the Russian Federation, is a country spanning Eastern Europe Eastern Europe is the eastern region of . There is no consistent definition of the precise area it covers, partly because th ...
(1724). Euler was offered a position at this last one in 1726; he accepted, arriving in St. Petersburg in 1727 ( and ). 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 ; cited in ). Notice, however, that Goldbach published some works on mathematics and sometimes held academic positions.
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 Fermat's little theorem states that if is a prime number, then for any integer , the number is an integer multiple of . In the notation of modular arithmetic, this is expressed as :a^p \equiv a \pmod p. For example, if = 2 and = 7, then 27 = ...
(generalised by Euler to non-prime moduli); the fact that $p = x^2 + y^2$ if and only if $p\equiv 1 \bmod 4$; initial work towards a proof that every integer is the sum of four squares (the first complete proof is by
Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaPell's equation Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2-ny^2=1 where ''n'' is a given positive nonsquare integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is co ...

'', first misnamed by Euler.. Euler was generous in giving credit to others , not always correctly. He wrote on the link between continued fractions and Pell's equation. *''First steps towards
analytic number theory 300px, Riemann zeta function ''ζ''(''s'') in the complex plane. The color of a point ''s'' encodes the value of ''ζ''(''s''): colors close to black denote values close to zero, while hue encodes the value's Argument (complex analysis)">argument ...
.'' In his work of sums of four squares, partitions, pentagonal numbers, and the
distributionDistribution may refer to: Mathematics *Distribution (mathematics) Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distr ...
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 Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Der ...
, most of his work is restricted to the formal manipulation of
power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \cdots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a const ...
. He did, however, do some very notable (though not fully rigorous) early work on what would later be called the
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter The Greek alphabet has been used to write the Greek language Greek (modern , romanized: ''Elliniká'', Ancient Greek, ancient , ''Hellēnikḗ'') is ...

. *''Quadratic forms''. Following Fermat's lead, Euler did further research on the question of which primes can be expressed in the form $x^2 + N y^2$, some of it prefiguring
quadratic reciprocity In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...
. *''Diophantine equations''. Euler worked on some Diophantine equations of genus 0 and 1. In particular, he studied
Diophantus Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the autho ...
'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 integral In integral calculus In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integra ...
s, whose study he had himself initiated.

Lagrange, Legendre, and Gauss

Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiafour-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 form In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s in full generality (as opposed to $m X^2 + n Y^2$)—defining their equivalence relation, showing how to put them in reduced form, etc.
Adrien-Marie Legendre Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named a ...
(1752–1833) was the first to state the law of quadratic reciprocity. He also conjectured what amounts to the
prime number theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
and
Dirichlet's theorem on arithmetic progressions In number theory, Dirichlet's theorem, also called the Dirichlet prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater ...
. He gave a full treatment of the equation $a x^2 + b y^2 + c z^2 = 0$ 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 $n=5$ (completing work by
Peter Gustav Lejeune Dirichlet Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study ...

, and crediting both him and
Sophie Germain Marie-Sophie Germain (; 1 April 1776 – 27 June 1831) was a French mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as qu ...

). In his ''
Disquisitiones Arithmeticae Title page of the first edition The (Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through ...
'' (1798),
Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician This is a List of German mathematician A mathematician is someone who uses an extensive knowledge of m ...

(1777–1855) proved the law of
quadratic reciprocity In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...
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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis) ...
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 Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by Nth root, radical ...
's work and
algebraic number theory Algebraic number theory is a branch of number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of devoted primarily to the study of the s and . German mathematician (1777–1855) said, "Mathematics is th ...
.

Maturity and division into subfields

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 Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Der ...
,
group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ...
,
Galois theory In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
—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 In mathematics, a root of unity, occasionally called a Abraham de Moivre, de Moivre number, is any complex number that yields 1 when exponentiation, raised to some positive integer power . Roots of unity are used in many branches of mathematic ...
, but truly came into its own with the development of
abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
and early ideal theory and valuation theory; see below. A conventional starting point for analytic number theory is
Dirichlet's theorem on arithmetic progressions In number theory, Dirichlet's theorem, also called the Dirichlet prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater ...
(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 Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics ...
(1859) on the
zeta function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

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 (
modular form In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s). 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.

Main subdivisions

Elementary number theory

The term ''
elementary In computational complexity theory, the complexity class ELEMENTARY of elementary recursive functions is the union of the classes : \begin \mathsf & = \bigcup_ k\mathsf \\ & = \mathsf\left(2^n\right)\cup\mathsf\left(2^\right)\ ...
'' generally denotes a method that does not use
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Der ...
. For example, the
prime number theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
was first proven using complex analysis in 1896, but an elementary proof was found only in 1949 by and . The term is somewhat ambiguous: for example, proofs based on complex
Tauberian theoremIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
s (for example, 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 mathematics.

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, for example,
sieve theory The fine mesh strainer, also known as the sift, commonly known as sieve, is a device for separating wanted elements from unwanted material or for characterizing the particle size distribution of a sample, typically using a woven screen such as ...
,Sieve theory figures as one of the main subareas of analytic number theory in many standard treatments; see, for instance, or are better covered by the second rather than the first definition: some of sieve theory, for instance, uses little analysis,This is the case for small sieves (in particular, some combinatorial sieves such as the
Brun sieveIn the field of number theory, the Brun sieve (also called Brun's pure sieve) is a technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by Congruence relation#Modular arithmetic ...
) rather than for
large sieve The large sieve is a method (or family of methods and related ideas) in analytic number theory. It is a type of Sieve theory, sieve where up to half of all residue classes of numbers are removed, as opposed to small sieves such as the Selberg sieve ...
s; the study of the latter now includes ideas from
harmonic A harmonic is any member of the harmonic series Harmonic series may refer to either of two related concepts: *Harmonic series (mathematics) *Harmonic series (music) {{Disambig .... The term is employed in various disciplines, including music ...
and
functional analysis 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. Functional analysis is a branch of mathemat ...
.
yet it does belong to analytic number theory. The following are examples of problems in analytic number theory: the
prime number theorem In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
, the
Goldbach conjecture Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struc ...
(or the
twin prime conjecture A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin pri ...
, or the
Hardy–Littlewood conjecture A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin pri ...
s), the
Waring problem In number theory, Waring's problem asks whether each natural number ''k'' has an associated positive integer ''s'' such that every natural number is the sum of at most ''s'' natural numbers raised to the power ''k''. For example, every natural numb ...
and the
Riemann hypothesis In mathematics, the Riemann hypothesis is a conjecture In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), ...
. 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 form In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s (and, more generally,
automorphic forms Image:Dedekind Eta.jpg, 500px, The Dedekind eta-function is an automorphic form in the complex plane. In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or c ...
) also occupies an increasingly central place in the toolbox of analytic number theory. One may ask analytic questions about
algebraic number An algebraic number is any complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, ev ...
s, 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 number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s 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 function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
s, which are generalizations of the
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter The Greek alphabet has been used to write the Greek language Greek (modern , romanized: ''Elliniká'', Ancient Greek, ancient , ''Hellēnikḗ'') is ...

, 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

An ''algebraic number'' is any complex number that is a solution to some polynomial equation $f\left(x\right)=0$ with rational coefficients; for example, every solution $x$ of $x^5 + \left(11/2\right) x^3 - 7 x^2 + 9 = 0$ (say) is an algebraic number. Fields of algebraic numbers are also called ''
algebraic number field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s'', or shortly ''
number field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
s''. 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 Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considered ...
and
norms Norm, the Norm or NORM may refer to: In academic disciplines * Norm (geology), an estimate of the idealised mineral content of a rock * Norm (philosophy), a standard in normative ethics that is prescriptive rather than a descriptive or explanato ...
in quadratic fields. (A ''quadratic field'' consists of all numbers of the form $a + b \sqrt$, 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 The ''chakravala'' method ( sa, चक्रवाल विधि) is a cyclic algorithm of an algorithm (Euclid's algorithm) for calculating the greatest common divisor (g.c.d.) of two numbers ''a'' and ''b'' in locations named A and B. The alg ...
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 $\sqrt$, the number $6$ can be factorised both as $6 = 2 \cdot 3$ and $6 = \left(1 + \sqrt\right) \left( 1 - \sqrt\right)$; all of $2$, $3$, $1 + \sqrt$ and $1 - \sqrt$ 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, that is, generalisations of
quadratic reciprocity In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...
. 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
The Galois group of an extension ''L/K'' 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 Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by Nth root, radical ...
; 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.)
Gal(''L''/''K'') of ''L'' over ''K'' is an
abelian group In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
—are relatively well understood. Their classification was the object of the programme of
class field theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, which was initiated in the late 19th century (partly by and Eisenstein) and carried out largely in 1900–1950. An example of an active area of research in algebraic number theory is
Iwasawa theory In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), math ...
. The
Langlands program In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
, 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (geo ...
, a
surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ...
or some other such object in ''n''-dimensional space. In Diophantine geometry, one asks whether there are any ''
rational points In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field (mathematics), field. If the field is not mentioned, the field of rational numbers is generally understood. If th ...
'' (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 if there are finitely or infinitely many rational points on a given curve (or surface). In the $x^2+y^2 = 1,$ we would like to study its rational solutions, that is, its solutions $\left(x,y\right)$ 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 $f\left(x,y\right)=0$, 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:If we want to study the curve $y^2 = x^3 + 7$. We allow ''x'' and ''y'' to be complex numbers: $\left(a + b i\right)^2 = \left(c + d i\right)^3 + 7$. 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 In geometry, a torus (plural tori, colloquially donut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanarity, coplanar with the circle. If the axis of revolution does not to ...

, loosely speaking, the surface of a doughnut (somewhat stretched). A doughnut has one hole; hence the genus is 1.
allow the variables in $f\left(x,y\right)=0$ to be complex numbers; then $f\left(x,y\right)=0$ defines a 2-dimensional surface in (projective) 4-dimensional space (since two complex variables can be decomposed into four real variables, that is, four dimensions). If we count the number of (doughnut) holes in the surface; we call this number the ''genus'' of $f\left(x,y\right)=0$. Other geometrical notions turn out to be just as crucial. There is also the closely linked area of Diophantine approximations: given a number $x$, then finding 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 $\gcd\left(a,q\right)=1$) a good approximation to $x$ if $, x-a/q, <\frac$, 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 200px, A cuboid demonstrating the dimensions length, width">length.html" ;"title="cuboid demonstrating the dimensions length">cuboid demonstrating the dimensions length, width, and height. Height is measure of vertical distance, either vertical ...

) turn out to be critical both in Diophantine geometry and in the study of Diophantine approximations. This question is also of special interest in
transcendental number theory Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of suc ...
: if a number can be better approximated than any algebraic number, then it is a
transcendental number In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. It is by this argument that

and e have been shown to be transcendental. Diophantine geometry should not be confused with the
geometry of numbersGeometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice (group), lattice in \mathbb R^n, and the study of these lattices provides fundame ...
, which is a collection of graphical methods for answering certain questions in algebraic number theory. ''Arithmetic geometry'', however, 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 In arithmetic geometry, the Mordell conjecture is the conjecture made by that a curve of genus Genus (plural genera) is a taxonomic rank Taxonomy (general) is the practice and science of classification of things or concepts, including the p ...
) rather than to techniques in Diophantine approximations.

Other subfields

The areas below date 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 Computability is the ability to solve a problem in an effective manner. It is a key topic of the field of computability theory Computability theory, also known as recursion theory, is a branch of mathematical logic Mathematical logic, also calle ...
dates only from the 1930s and 1940s, and
computational complexity theory Computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by ...
from the 1970s.

Probabilistic number theory

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 Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * 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. At times, a non-rigorous, probabilistic approach leads to a number of
heuristic A heuristic (; ), or heuristic technique, is any approach to problem solving or self-discovery that employs a practical method that is not guaranteed to be Mathematical optimisation, optimal, perfect, or Rationality, rational, but is nevertheless ...
algorithms and open problems, notably
Cramér's conjectureIn number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, is an estimate for the size of prime gap, gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always sma ...
.

Arithmetic combinatorics

If we begin from a fairly "thick" infinite set $A$, does it contain many elements in arithmetic progression: $a$, $a+b, a+2 b, a+3 b, \ldots, a+10b$, 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 Additive number theory is the subfield of number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued funct ...
'' (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 numbersGeometry of numbers is the part of number theory which uses geometry for the study of algebraic numbers. Typically, a ring of algebraic integers is viewed as a lattice (group), lattice in \mathbb R^n, and the study of these lattices provides fundame ...
'', 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 Ergodic theory ( Greek: ' "work", ' "way") is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
, finite group theory,
model theory In mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical p ...
, and other fields. 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 A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic identi ...
, for which the multiplication symbol, not the addition symbol, is traditionally used; they can also be subsets of
rings Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to: *Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck Ring may also refer to: Sounds * Ri ...
, in which case the growth of $A+A$ and $A$·$A$ may be compared.

Computational number theory

While the word ''algorithm'' goes back only to certain readers of
al-Khwārizmī Muḥammad ibn Mūsā al-Khwārizmī ( fa, محمد بن موسی خوارزمی, Moḥammad ben Musā Khwārazmi; ), Arabization, Arabized as al-Khwarizmi and formerly Latinisation of names, Latinized as ''Algorithmi'', was a Persians, Persian ...
, 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 early case is that of what we now call the
Euclidean algorithm In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an effi ...
. In its basic form (namely, as an algorithm for computing the
greatest common divisor In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is ...

) 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 $a x + b y = c$, or, what is the same, for finding the quantities whose existence is assured by the
Chinese remainder theorem In number theory, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number ...
) it first appears in the works of
Āryabhaṭa Aryabhata (, ISO 15919, ISO: ) or Aryabhata I (476–550 Common Era, CE) was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the ''Āryabhaṭīya'' (which me ...
(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 (for example, RSA) depend on functions that are known to all, but whose inverses are known only to a chosen few, and 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. 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 A Turing machine is a mathematical model of computation Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are ...

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, given a Diophantine equation with at least one solution, the solution itself provides a proof of the fact that a solution exists. We cannot prove that a particular Diophantine equation is of this kind, since this would imply that it has no solutions.)

Applications

The number-theorist
Leonard Dickson Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as ...
(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 Donald Ervin Knuth ( ; born January 10, 1938) is an American computer scientist A computer scientist is a person A person (plural people or persons) is a being that has certain capacities or attributes such as reason, morality, consciousnes ...
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 Discrete mathematics is the study of mathematical structures In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geom ...
courses for
computer scientist A computer scientist is a person who has acquired the knowledge of computer science, the study of the theoretical foundations of information and computation and their application. Computer scientists typically work on the theoretical side of c ...

s; on the other hand, number theory also has applications to the continuous in
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical analysis ...
. As well as the well-known applications to
cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia ''-logy'' is a suffix in the English language, used with words originally adapted from Ancient Greek ending in (''- ...

, there are also applications to many other areas of mathematics.

Prizes

The
American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematics, mathematical research and scholarship, and serves the national and international community through its publicatio ...
awards the '' Cole Prize in Number Theory''. Moreover, number theory is one of the three mathematical subdisciplines rewarded by the ''
Fermat Prize The Fermat prize of mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and ...
''.

*
Algebraic function field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
*
Finite field In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
*
p-adic number group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analys ...

Sources

* * (Subscription needed) * *
1968 edition
at archive.org * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
Volume 1Volume 2Volume 3Volume 4 (1912)
* For other editions, see Iamblichus#List of editions and translations * This Google books preview of ''Elements of algebra'' lacks Truesdell's intro, which is reprinted (slightly abridged) in the following book: * * * * * *