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. These concepts may originate in real-world concerns, and the results obtained may later turn out to be useful for practical applications, ...
devoted primarily to the study of the
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s and integer-valued functions. German mathematician Carl Friedrich Gauss (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 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 because the only ways ...
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 of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s) or defined as generalizations of the integers (for example, algebraic integers).
Integers can be considered either in themselves or as solutions to equations ( Diophantine geometry). Questions in number theory 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 (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
) that encode properties of the integers, primes or other number-theoretic objects in some fashion (
analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diric ...
). One may also study
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s in relation to rational numbers, for example, as approximated by the latter ( Diophantine approximation).
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
Sir Thomas Little Heath (; 5 October 1861 – 16 March 1940) was a British civil servant, mathematician, classical scholar, historian of ancient Greek mathematics, translator, and mountaineer. He was educated at Clifton College. Heath transla ...
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 () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
" 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 logic, 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 for ...
, 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 mathematician Giuseppe Peano. These axioms have been used nearly u ...
'', and
computer science
Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied science, practical discipli ...
, as in ''
floating-point arithmetic
In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can be ...
''.) 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 govern ...
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, Sout ...
he title
He or HE may refer to:
Language
* He (pronoun), an English pronoun
* He (kana), the romanization of the Japanese kana へ
* He (letter), the fifth letter of many Semitic alphabets
* He (Cyrillic), a letter of the Cyrillic script called ''He'' in ...
to ''An introduction to arithmetic'', a more novel and in some ways a more appropriate title; but it was pointed out that this might lead to misunderstandings about the content of the book." 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 ( Larsa, Mesopotamia, ca. 1800 BC) contains a list of " Pythagorean triples", that is, integers such that .
The triples are too many and too large to have been obtained by
brute force
Brute Force or brute force may refer to:
Techniques
* Brute force method or proof by exhaustion, a method of mathematical proof
* Brute-force attack, a cryptanalytic attack
* Brute-force search, a computer problem-solving technique
People
* Brut ...
. 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:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
:
which is implicit in routine Old Babylonian exercises. If some other method was used, the triples were first constructed and then reordered by , presumably for actual use as a "table", for example, with a view to applications.
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 objects during the early history of Mesopotamia.
Babylonian astronomy seemed to have focused on a select group of stars and constellations known as Ziqpu stars. These constellations m ...
, 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. 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 that can be called thus—consists of this single, striking fragment, Babylonian algebra (in the secondary-school sense of "
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary a ...
") was exceptionally well developed. Late Neoplatonic sources
Iamblichus
Iamblichus (; grc-gre, Ἰάμβλιχος ; Aramaic: 𐡉𐡌𐡋𐡊𐡅 ''Yamlīḵū''; ) was a Syrian neoplatonic philosopher of Arabic origin. He determined a direction later taken by neoplatonism. Iamblichus was also the biographer of ...
, ''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 ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymou ...
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 (now lost), the ''Catalogue of Geometers''. See also introduction, on Proclus's reliability. state that
Thales
Thales of Miletus ( ; grc-gre, Θαλῆς; ) was a Greek mathematician, astronomer, statesman, and pre-Socratic philosopher from Miletus in Ionia, Asia Minor. He was one of the Seven Sages of Greece. Many, most notably Aristotle, regarded him ...
and Pythagoras traveled and studied in
Egypt
Egypt ( ar, مصر , ), officially the Arab Republic of Egypt, is a transcontinental country spanning the northeast corner of Africa and southwest corner of Asia via a land bridge formed by the Sinai Peninsula. It is bordered by the Mediter ...
.
Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
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
is irrational. Pythagorean mystics gave great importance to the odd and the even.
The discovery that 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 geometry, the spiral of Theodorus (also called ''square root spiral'', ''Einstein spiral'', ''Pythagorean spiral'', or ''Pythagoras's snail'') is a spiral composed of right triangles, placed edge-to-edge. It was named after Theodorus of Cyrene ...
. By revealing (in modern terms) that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history; its proof or its divulgation are sometimes credited to Hippasus, who was expelled or split from the Pythagorean sect. This forced a distinction between ''numbers'' (integers and the rationals—the subjects of arithmetic), on the one hand, and ''lengths'' and ''proportions'' (which we would identify with real numbers, whether rational or not), on the other hand.
The Pythagorean tradition spoke also of so-called polygonal or figurate numbers. While
square number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals ...
s, cubic numbers, 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 numbers. The th triangular number is the number of dots in ...
s, pentagonal numbers, etc., the study of the sums of triangular and pentagonal numbers would prove fruitful in the early modern period (17th to early 19th century).
We know of no clearly arithmetical material in ancient Egyptian 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 ...
sources, though there is some algebra in each. The
Chinese remainder theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
appears as an exercise in '' Sunzi Suanjing'' (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 :
6Now 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 numberexceeds 106, the result is obtained by subtracting 105.
it is the problem that was later solved by
Āryabhaṭa
Aryabhata (ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the '' Aryabhatiya'' (whic ...
below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
Bottom may refer to:
Anatomy and sex
* Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
.)
There is also some numerical mysticism in Chinese mathematics,See, for example, ''Sunzi Suanjing'', Ch. 3, Problem 36, in :
6Now 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 in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''ef'' (pronounced ), and the plural is ''efs''.
Hist ...
8 the eight winds, and 9 the nine divisions
f China under Yu the Great
F, or f, is the sixth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''ef'' (pronounced ), and the plural is ''efs''.
Hist ...
If the remainder is odd,
he sex
He or HE may refer to:
Language
* He (pronoun), an English pronoun
* He (kana), the romanization of the Japanese kana へ
* He (letter), the fifth letter of many Semitic alphabets
* He (Cyrillic), a letter of the Cyrillic script called ''He'' in ...
is male and if the remainder is even,
he sex
He or HE may refer to:
Language
* He (pronoun), an English pronoun
* He (kana), the romanization of the Japanese kana へ
* He (letter), the fifth letter of many Semitic alphabets
* He (Cyrillic), a letter of the Cyrillic script called ''He'' in ...
is 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, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.
T ...
s,
magic squares
In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number o ...
have passed from superstition into
recreation
Recreation is an activity of leisure, leisure being discretionary time. The "need to do something for recreation" is an essential element of human biology and psychology. Recreational activities are often done for enjoyment, amusement, or pleasur ...
.
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 Classical antiquity, the Hellenistic period covers the time in Mediterranean history after Classical Greece, between the death of Alexander the Great in 323 BC and the emergence of the Roman Empire, as signified by the Battle of Actium in 3 ...
. In the case of number theory, this means, by and large, ''
Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first institution ...
'' 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 ; 260/265 – 30 May 339), also known as Eusebius Pamphilus (from the grc-gre, Εὐσέβιος τοῦ Παμφίλου), was a Greek historian of Christianity, exegete, and Christian ...
, PE X, chapter 4 mentions of
Pythagoras
Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samos, Samian, or simply ; in Ionian Greek; ) was an ancient Ionians, Ionian Ancient Greek philosophy, Greek philosopher and the eponymou ...
:
"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 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 are irrational.
Theaetetus Theaetetus (Θεαίτητος) is a Greek name which could refer to:
* Theaetetus (mathematician) (c. 417 BC – 369 BC), Greek geometer
* ''Theaetetus'' (dialogue), a dialogue by Plato, named after the geometer
* Theaetetus (crater), a lunar imp ...
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 mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postulat ...
is described by Pappus as being largely based on Theaetetus's work.)
Euclid devoted part of his ''Elements'' to prime numbers and divisibility, topics that belong unambiguously to number theory and are basic to it (Books VII to IX of Euclid's Elements). In particular, he gave an algorithm for computing the greatest common divisor of two numbers (the
Euclidean algorithm
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
Euclid's theorem is a fundamental statement in number theory that asserts that there are Infinite set, infinitely many prime number, prime numbers. It was first proved by Euclid in his work ''Euclid's Elements, Elements''. There are several proofs ...
(''Elements'', Prop. IX.20).
In 1773,
Lessing Lessing 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 lin ...
published an
epigram
An epigram is a brief, interesting, memorable, and sometimes surprising or satirical statement. The word is derived from the Greek "inscription" from "to write on, to inscribe", and the literary device has been employed for over two mille ...
he had found in a manuscript during his work as a librarian; it claimed to be a letter sent by
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
to
Eratosthenes
Eratosthenes of Cyrene (; grc-gre, Ἐρατοσθένης ; – ) was a Greek polymath: a mathematician, geographer, poet, astronomer, and music theorist. He was a man of learning, becoming the chief librarian at the Library of Alexandria ...
. The epigram proposed what has become known as
Archimedes's cattle problem; its solution (absent from the manuscript) requires solving an indeterminate quadratic equation (which reduces to what would later be misnamed Pell's equation). As far as we know, such equations were first successfully treated by the Indian school. It is not known whether Archimedes himself had a method of solution.
Diophantus
Very little is known about Diophantus of Alexandria; he probably lived in the third century AD, 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 text on mathematics written by the mathematician Diophantus () in the 3rd century AD. It is a collection of 130 algebraic problems giving numerical solutions of determinate e ...
'' 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 or . 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 (ge ...
s and algebraic varieties; however, unlike the Greeks of the Classical period, who did what we would now call basic algebra in geometrical terms, Diophantus did what we would now call basic algebraic geometry in purely algebraic terms. In modern language, what Diophantus did was to find rational parametrizations of varieties; that is, given an equation of the form (say)
, his aim was to find (in essence) three rational functions such that, for all values of and , setting
for gives a solution to
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 mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
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
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...
, 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 (476–550 AD) showed that pairs of simultaneous congruences , could be solved by a method he called ''kuṭṭaka'', or ''pulveriser''; this is a procedure close to (a generalisation of) the Euclidean algorithm, which was probably discovered independently in India. Āryabhaṭa seems to have had in mind applications to astronomical calculations.
Brahmagupta (628 AD) started the systematic study of indefinite quadratic equations—in particular, the misnamed Pell equation, in which
Archimedes
Archimedes of Syracuse (;; ) was a Greek mathematician, physicist, engineer, astronomer, and inventor from the ancient city of Syracuse in Sicily. Although few details of his life are known, he is regarded as one of the leading scientists ...
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
Jayadeva (; born ), also spelt Jaideva, was a Sanskrit poet during the 12th century. He is most known for his epic poem ''Gita Govinda'' which concentrates on Krishna's love with the '' gopi'', Radha, in a rite of spring. This poem, which presen ...
(cited in the eleventh century; his work is otherwise lost); the earliest surviving exposition appears in
Bhāskara II
Bhāskara II (c. 1114–1185), also known as Bhāskarāchārya ("Bhāskara, the teacher"), and as Bhāskara II to avoid confusion with Bhāskara I, was an Indian mathematician and astronomer. From verses, in his main work, Siddhānta Shiroman ...
'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
Henry Thomas Colebrooke FRS FRSE (15 June 1765 – 10 March 1837) was an English orientalist and mathematician. He has been described as "the first great Sanskrit scholar in Europe".
Biography
Henry Thomas Colebrooke was born on 15 Jun ...
.
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's Brāhmasphuṭasiddhānta).
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, philosopher, astronomer, mathematician and translator. He was born in Baalbek. Travelling to parts of the Byzantine Empire, he brought back Greek te ...
(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 as Alhazen (; full name ; ), was a medieval mathematician, astronomer, and physicist of the Islamic Golden Age from present-day Iraq.For the description of his main fields, see e.g. ("He is one of the prin ...
knew what would later be called
Wilson's theorem
In algebra and number theory, Wilson's theorem states that a natural number ''n'' > 1 is a prime number if and only if the product of all the positive integers less than ''n'' is one less than a multiple of ''n''. That is (using the notations of m ...
.
Western Europe in the Middle Ages
Other than a treatise on squares in arithmetic progression by Fibonacci—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 in European history marking the transition from the Middle Ages to modernity and covering the 15th and 16th centuries, characterized by an effort to revive and surpass ideas ...
, thanks to a renewed study of the works of Greek antiquity. A catalyst was the textual emendation and translation into Latin of Diophantus' ''Arithmetica''.
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 who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
(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, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number.
T ...
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 and music theorist, best known for his works ''Introduction to Arithmetic'' and ''Manual of Harmonics'' in Greek. He was born in ...
(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 (60–120 AD).
Summary
The work contains both philosophical prose and ...
". See . these topics led him to work on integer
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
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 ''p'' is a prime number, then for any integer ''a'', the number a^p - a is an integer multiple of ''p''. In the notation of modular arithmetic, this is expressed as
: a^p \equiv a \pmod p.
For example, if = ...
(1640): if ''a'' is not divisible by a prime ''p'', then Here, as usual, given two integers ''a'' and ''b'' and a non-zero integer ''m'', we write (read "''a'' is congruent to ''b'' modulo ''m''") to mean that ''m'' divides ''a'' − ''b'', or, what is the same, ''a'' and ''b'' leave the same residue when divided by ''m''. This notation is actually much later than Fermat's; it first appears in section 1 of Gauss's Disquisitiones Arithmeticae. Fermat's little theorem is a consequence of the
fact
A fact is a datum about one or more aspects of a circumstance, which, if accepted as true and proven true, allows a logical conclusion to be reached on a true–false evaluation. Standard reference works are often used to check facts. Scient ...
that the
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
of an element of a group divides the
order
Order, ORDER or Orders may refer to:
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
* Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...
of the group. The modern proof would have been within Fermat's means (and was indeed given later by Euler), even though the modern concept of a group came long after Fermat or Euler. (It helps to know that inverses exist modulo ''p'', that is, given ''a'' not divisible by a prime ''p'', there is an integer ''x'' such that ); this fact (which, in modern language, makes the residues mod ''p'' into a group, and which was already known to Āryabhaṭa; see above) was familiar to Fermat thanks to its rediscovery by Bachet . Weil 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 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 . 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 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 was not 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 has no non-trivial solutions in the integers. Fermat also mentioned to his correspondents that 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) to have shown there are no solutions to for all ; 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, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
(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 and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...
, 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 at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at the ...
, 1666) and
Russia
Russia (, , ), or the Russian Federation, is a List of transcontinental countries, transcontinental country spanning Eastern Europe and North Asia, Northern Asia. It is the List of countries and dependencies by area, largest country in the ...
(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 ''p'' is a prime number, then for any integer ''a'', the number a^p - a is an integer multiple of ''p''. In the notation of modular arithmetic, this is expressed as
: a^p \equiv a \pmod p.
For example, if = ...
(generalised by Euler to non-prime moduli); the fact that if and only if ; 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'', 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.'' In his work of sums of four squares,
partitions
Partition may refer to:
Computing Hardware
* Disk partitioning, the division of a hard disk drive
* Memory partition, a subdivision of a computer's memory, usually for use by a single job
Software
* Partition (database), the division of a ...
,
pentagonal numbers
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The ...
, and the
distribution Distribution may refer to:
Mathematics
*Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
* Probability distribution, the probability of a particular value or value range of a vari ...
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 that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
, 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 + \dots
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 (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
.
*''Quadratic forms''. Following Fermat's lead, Euler did further research on the question of which primes can be expressed in the form , some of it prefiguring quadratic reciprocity.
*''Diophantine equations''. Euler worked on some Diophantine equations of genus 0 and 1. In particular, he studied Diophantus's work; he tried to systematise it, but the time was not yet ripe for such an endeavour—algebraic geometry was still in its infancy. He did notice there was a connection between Diophantine problems and elliptic integrals, whose study he had himself initiated.
Lagrange, Legendre, and Gauss
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, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to a ...
s in full generality (as opposed to )—defining their equivalence relation, showing how to put them in reduced form, etc.
Adrien-Marie Legendre (1752–1833) was the first to state the law of quadratic reciprocity. He also
conjectured what amounts to the
prime number theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...
and
Dirichlet's theorem on arithmetic progressions
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers ''a'' and ''d'', there are infinitely many primes of the form ''a'' + ''nd'', where ''n'' is als ...
. He gave a full treatment of the equation and worked on quadratic forms along the lines later developed fully by Gauss. In his old age, he was the first to prove Fermat's Last Theorem for (completing work by Peter Gustav Lejeune Dirichlet, and crediting both him and Sophie Germain).
In his ''Disquisitiones Arithmeticae'' (1798), Carl Friedrich Gauss (1777–1855) proved the law of quadratic reciprocity and developed the theory of quadratic forms (in particular, defining their composition). He also introduced some basic notation ( congruences) and devoted a section to computational matters, including primality tests. The last section of the ''Disquisitiones'' established a link between roots of unity and number theory:
The theory of the division of the circle...which is treated in sec. 7 does not belong
by itself to arithmetic, but its principles can only be drawn from higher arithmetic.
In this way, Gauss arguably made a first foray towards both
Évariste Galois
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 radicals, ...
's work and
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
.
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 that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
,
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, Galois theory—accompanied by greater rigor in analysis and abstraction in algebra.
* The rough subdivision of number theory into its modern subfields—in particular, analytic and algebraic number theory.
Algebraic number theory may be said to start with the study of reciprocity and
cyclotomy
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
, but truly came into its own with the development of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
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 theorem, states that for any two positive coprime integers ''a'' and ''d'', there are infinitely many primes of the form ''a'' + ''nd'', where ''n'' is als ...
(1837), whose proof introduced
L-functions
In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may give ri ...
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 who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rig ...
(1859) on the
zeta function
In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function
: \zeta(s) = \sum_^\infty \frac 1 .
Zeta functions include:
* Airy zeta function, related to the zeros of the Airy function
* A ...
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, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...
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'' 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 that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
. For example, the
prime number theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...
was first proven using complex analysis in 1896, but an elementary proof was found only in 1949 by
Erdős
Erdős, Erdos, or Erdoes is a Hungarian surname.
People with the surname include:
* Ágnes Erdős (born 1950), Hungarian politician
* Brad Erdos (born 1990), Canadian football player
* Éva Erdős (born 1964), Hungarian handball player
* Józse ...
and Selberg. The term is somewhat ambiguous: for example, proofs based on complex Tauberian theorems (for example, Wiener–Ikehara) are often seen as quite enlightening but not elementary, in spite of using
Fourier analysis
In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Josep ...
, 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,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 sieve In 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 congruences. It was developed by V ...
) 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 where up to half of all residue classes of numbers are removed, as opposed to small sieves such as the Selberg sieve wherein only ...
s; the study of the latter now includes ideas from
harmonic
A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', the ...
and
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Defini ...
. yet it does belong to analytic number theory.
The following are examples of problems in analytic number theory: the
prime number theorem
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...
, the
Goldbach conjecture
Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that every even natural number greater than 2 is the sum of two prime numbers.
The conjecture has been shown to hold ...
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 the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
. Some of the most important tools of analytic number theory are the
circle method
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
L-functions
In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may give ri ...
(or, rather, the study of their properties). The theory of
modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...
s (and, more generally, automorphic forms) 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 a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
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 ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
s (generalizations of
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a 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 because the only ways ...
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 functions, which are generalizations of the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
, 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
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
(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 with rational coefficients; for example, every solution of (say) is an algebraic number. Fields of algebraic numbers are also called ''
algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
s'', or shortly '' number fields''. Algebraic number theory studies algebraic number fields. Thus, analytic and algebraic number theory can and do overlap: the former is defined by its methods, the latter by its objects of study.
It could be argued that the simplest kind of number fields (viz., quadratic fields) were already studied by Gauss, as the discussion of quadratic forms in ''Disquisitiones arithmeticae'' can be restated in terms of ideals and
norms in quadratic fields. (A ''quadratic field'' consists of all
numbers of the form , where
and are rational numbers and
is a fixed rational number whose square root is not rational.)
For that matter, the 11th-century chakravala method amounts—in modern terms—to an algorithm for finding the units of a real quadratic number field. However, neither Bhāskara nor Gauss knew of number fields as such.
The grounds of the subject as we know it were set in the late nineteenth century, when ''ideal numbers'', the ''theory of ideals'' and ''valuation theory'' were developed; these are three complementary ways of dealing with the lack of unique factorisation in algebraic number fields. (For example, in the field generated by the rationals
and , the number can be factorised both as and
; all of , , and
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 Kummer is a German surname. Notable people with the surname include:
*Bernhard Kummer (1897–1962), German Germanist
*Clare Kummer (1873—1958), American composer, lyricist and playwright
*Clarence Kummer (1899–1930), American jockey
* Christo ...
) seems to have come from the study of higher reciprocity laws, that is, generalisations of quadratic reciprocity.
Number fields are often studied as extensions of smaller number fields: a field ''L'' is said to be an ''extension'' of a field ''K'' if ''L'' contains ''K''.
(For example, the complex numbers ''C'' are an extension of the reals ''R'', and the reals ''R'' are an extension of the rationals ''Q''.)
Classifying the possible extensions of a given number field is a difficult and partially open problem. Abelian extensions—that is, extensions ''L'' of ''K'' such that the Galois groupThe Galois group of an extension ''L/K'' consists of the operations (
isomorphisms
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
) 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 radicals, ...
; 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, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commut ...
—are relatively well understood.
Their classification was the object of the programme of
class field theory
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field.
Hilbert is credit ...
, which was initiated in the late 19th century (partly by Kronecker and Eisenstein) and carried out largely in 1900–1950.
An example of an active area of research in algebraic number theory is Iwasawa theory. The Langlands program, one of the main current large-scale research plans in mathematics, is sometimes described as an attempt to generalise class field theory to non-abelian extensions of number fields.
Diophantine geometry
The central problem of ''Diophantine geometry'' is to determine when a
Diophantine equation
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, such that the only solutions of interest are the integer ones. A linear Diophantine equation equates to a c ...
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 (ge ...
, a surface or some other such object in ''n''-dimensional space. In Diophantine geometry, one asks whether there are any ''rational points'' (points all of whose coordinates are rationals) or
''integral points'' (points all of whose coordinates are integers) on the curve or surface. If there are any such points, the next step is to ask how many there are and how they are distributed. A basic question in this direction is if there are finitely
or infinitely many rational points on a given curve (or surface).
In the
Pythagorean equation
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
we would like to study its rational solutions, that is, its solutions
such that
''x'' and ''y'' are both rational. This is the same as asking for all integer solutions
to ; any solution to the latter equation gives
us a solution , to the former. It is also the
same as asking for all points with rational coordinates on the curve
described by . (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 , where 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 . We allow ''x'' and ''y'' to be complex numbers: . 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 or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
, loosely speaking, the surface of a doughnut (somewhat
stretched). A doughnut has one hole; hence the genus is 1. allow the variables in to be complex numbers; then defines a 2-dimensional surface in (projective) 4-dimensional space (since two complex variables can be decomposed into four real variables, that is, four dimensions). If we count the number of (doughnut) holes in the surface; we call this number the ''genus'' of . Other geometrical notions turn out to be just as crucial.
There is also the closely linked area of Diophantine approximations: given a number , 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 (with ) a good approximation to if , where is large.) This question is of special interest if is an algebraic number. If cannot be well approximated, then some equations do not have integer or rational solutions. Moreover, several concepts (especially that of
height
Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is).
For example, "The height of that building is 50 m" or "The height of an airplane in-flight is abou ...
) 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: if a number can be better approximated than any algebraic number, then it is a
transcendental number
In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and .
Though only a few classes ...
. It is by this argument that and e have been shown to be transcendental.
Diophantine geometry should not be confused with the
geometry of numbers Geometry 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 in \mathbb R^n, and the study of these lattices provides fundamental information ...
, 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 Louis Mordell that a curve of Genus (mathematics), genus greater than 1 over the field Q of rational numbers has only finitely many rational points. In 1983 it was proved by ...
) 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 within mathematical logic and the theory of computation within computer science. The computability of a problem is close ...
dates only from the 1930s and 1940s, and
computational complexity theory
In theoretical computer science and mathematics, 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 ...
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
* Independ ...
. 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
The probabilistic method is a nonconstructive method, primarily used in combinatorics and pioneered by Paul Erdős, for proving the existence of a prescribed kind of mathematical object. It works by showing that if one randomly chooses objects ...
uses the fact that whatever happens with probability greater than must happen sometimes; one may say with equal justice that many applications of probabilistic number theory hinge on the fact that whatever is unusual must be rare. If certain algebraic objects (say, rational or integer solutions to certain equations) can be shown to be in the tail of certain sensibly defined distributions, it follows that there must be few of them; this is a very concrete non-probabilistic statement following from a probabilistic one.
At times, a non-rigorous, probabilistic approach leads to a number of
heuristic
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 optimal, perfect, or rational, but is nevertheless sufficient for reaching an immediate, ...
If we begin from a fairly "thick" infinite set , does it contain many elements in arithmetic progression: ,
, say? Should it be possible to write large integers as sums of elements of ?
These questions are characteristic of ''arithmetic combinatorics''. This is a presently coalescing field; it subsumes '' additive number theory'' (which concerns itself with certain very specific sets of arithmetic significance, such as the primes or the squares) and, arguably, some of the ''geometry of numbers'',
together with some rapidly developing new material. Its focus on issues of growth and distribution accounts in part for its developing links with
ergodic theory
Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...
model theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the s ...
, and other fields. The term ''additive combinatorics'' is also used; however, the sets being studied need not be sets of integers, but rather subsets of non-commutative
groups
A group is a number of persons 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 iden ...
, for which the multiplication symbol, not the addition symbol, is traditionally used; they can also be subsets of rings, in which case the growth of and · may be
compared.
Computational number theory
While the word ''algorithm'' goes back only to certain readers of
al-Khwārizmī
Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persian polymath from Khwarazm, who produced vastly influential works in mathematics, astronomy ...
, 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 its basic form (namely, as an algorithm for computing the greatest common divisor) it appears as Proposition 2 of Book VII in ''Elements'', together with a proof of correctness. However, in the form that is often used in number theory (namely, as an algorithm for finding integer solutions to an equation ,
or, what is the same, for finding the quantities whose existence is assured by the
Chinese remainder theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
) it first appears in the works of
Āryabhaṭa
Aryabhata (ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy. He flourished in the Gupta Era and produced works such as the '' Aryabhatiya'' (whic ...
(5th–6th century CE) as an algorithm called
''kuṭṭaka'' ("pulveriser"), without a proof of correctness.
There are two main questions: "Can we compute this?" and "Can we compute it rapidly?" Anyone can test whether a number is prime or, if it is not, split it into prime factors; doing so rapidly is another matter. We now know fast algorithms for testing primality, but, in spite of much work (both theoretical and practical), no truly fast algorithm for factoring.
The difficulty of a computation can be useful: modern protocols for encrypting messages (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
Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm which, for any given Diophantine equation (a polynomial equati ...
, that there is no
Turing machine
A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...
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 (1874–1954) said "Thank God that number theory is unsullied by any application". Such a view is no longer applicable to number theory. In 1974, Donald Knuth said "...virtually every theorem in elementary number theory arises in a natural, motivated way in connection with the problem of making computers do high-speed numerical calculations".
Elementary number theory is taught in
discrete mathematics
Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous f ...
courses for
computer scientist
A computer scientist is a person who is trained in the academic study of computer science.
Computer scientists typically work on the theoretical side of computation, as opposed to the hardware side on which computer engineers mainly focus (al ...
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). It is the study of ...
. As well as the well-known applications to
cryptography
Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adver ...
, 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 mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...
Algebraic function field
In mathematics, an algebraic function field (often abbreviated as function field) of ''n'' variables over a field ''k'' is a finitely generated field extension ''K''/''k'' which has transcendence degree ''n'' over ''k''. Equivalently, an algebraic ...
*
Finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
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* 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:
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Further reading
Two of the most popular introductions to the subject are:
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Hardy and Wright's book is a comprehensive classic, though its clarity sometimes suffers due to the authors' insistence on elementary methods ( Apostol n.d.).
Vinogradov's main attraction consists in its set of problems, which quickly lead to Vinogradov's own research interests; the text itself is very basic and close to minimal. Other popular first introductions are:
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Popular choices for a second textbook include:
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