Draft:History of number theory

File:Ishango bone (cropped).jpg as one of the earliest arithmetic artifacts.]]

Number theory is the branch of mathematics that studies the structure and properties of integers as well as the relations and laws between them.{{multiref | {{harvnb|Bukhshtab|Nechaev|2016}} | {{harvnb|Grigorieva|2018|pp=[https://books.google.com/books?id=mEpjDwAAQBAJ&pg=PR8 viii–ix]}} | {{harvnb|Page|2003|p=[https://www.sciencedirect.com/science/article/abs/pii/B0122274105005032 15]}} }} The integers comprise a set that extends the set of natural numbers $\backslash \{1,\; 2,\; 3,\; \backslash dots\backslash \}$ to include their negations $-1,\; -2,\; -3,\; \backslash dots$ and the number $0$.{{multiref|{{harvnb|Romanowski|2008|p=304}}|{{harvnb|Nagel|2002|pp=180–181}}|{{harvnb|Hindry|2011|p=x}}|{{harvnb|Hafstrom|2013|p=[https://books.google.com/books?id=mj_DAgAAQBAJ&pg=PA95 95]}}}}

Number theory is closely related to arithmetic and some authors use the terms as synonyms.{{multiref|{{harvnb|Lozano-Robledo|2019|p=[https://books.google.com/books?id=ESiODwAAQBAJ&pg=PR13 xiii]}}|{{harvnb|Nagel|Newman|2008|p=[https://books.google.com/books?id=WgwUCgAAQBAJ&pg=PA4 4]}}}} However, the word "arithmetic" is used today to mean the study of numerical operations and extends to the real numbers.{{multiref|{{harvnb|Romanowski|2008|pp=302–303}}|{{harvnb|HC staff|2022b}}|{{harvnb|MW staff|2023}}|{{harvnb|Bukhshtab|Pechaev|2020}}}} In a more specific sense, number theory is restricted to the study of integers and focuses on their properties and relationships such as divisibility, factorization, and primality.{{multiref|{{harvnb|Wilson|2020|pp=[https://books.google.com/books?id=fcDgDwAAQBAJ&pg=PA1 1–2]}}|{{harvnb|Karatsuba|2020}}|{{harvnb|Campbell|2012|p=[https://books.google.com/books?id=yoEFp-Q2OXIC&pg=PT33 33]}}|{{harvnb|Robbins|2006|p=[https://books.google.com/books?id=TtLMrKDsDuIC&pg=PR12-IA1 1]}}}} Traditionally, it is known as higher arithmetic.{{multiref|{{harvnb|Duverney|2010|p=[https://books.google.com/books?id=sr5S9oN1xPAC&pg=PR5 v]}}|{{harvnb|Robbins|2006|p=[https://books.google.com/books?id=TtLMrKDsDuIC&pg=PR12-IA1 1]}}}}

By the early twentieth century, the term number theory or theory of numbers had been widely adopted.The term 'arithmetic' may have regained some ground, arguably due to French influence. Take, for example, {{harvnb|Serre|1996}}. In 1952, Davenport still had to specify that he meant The Higher Arithmetic. Hardy and Wright wrote in the introduction to An Introduction to the Theory of Numbers (1938): "We proposed at one time to change [the title] to An introduction to arithmetic, a more novel and in some ways a more appropriate title; but it was pointed out that this might lead to misunderstandings about the content of the book." {{harv|Hardy|Wright|2008}} The term number refers either to natural numbers or whole numbers, the latter of which means either the positive integers or all integers.{{harvnb|Weisstein|2003|loc=Number Theory, Whole Numbers}}{{harvnb|Effinger|Mullen|2022|loc=Divisibility in the Integers Z}}{{harvnb|Watkins|2014|p=xi}} There is no consensus on the precise definition of number theory as to whether it is construed to the integers or natural numbers.{{harvnb|Karatsuba|2020}}{{harvnb|Davenport|2008|loc=Introduction}} The wider scope of the integers is more convenient most purposes.{{harvnb|Ore|1948|p=28}}

The earliest forms of arithmetic are sometimes traced back to counting and tally marks used to keep track of quantities. Some historians suggest that the Lebombo bone (dated about 43,000 years ago) and the Ishango bone (dated about 22,000 to 30,000 years ago) are the oldest arithmetic artifacts but this interpretation is disputed.{{multiref|{{harvnb|Burgin|2022|pp=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA2 2–3]}}|{{harvnb|Ore|1948|pp=1, 6, 8, 10}}|{{harvnb|Thiam|Rochon|2019|p=[https://books.google.com/books?id=EWSsDwAAQBAJ&pg=PA164 164]}}}} The latter bone is claimed to list a sequence of prime numbers, though this is highly disputed. Opponents argue that the concepts of division and prime numbers evolved only around 10,000 BC in agricultural civilisations and 500 BC in ancient Greece, respectively.{{harvnb|Rudman|2007|pp=62-65}} However, a basic sense of numbers may predate these findings and might even have existed before the development of language.{{multiref|{{harvnb|Burgin|2022|p=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA3 3]}}|{{harvnb|Ponticorvo|Schmbri|Miglino|2019|p=[https://books.google.com/books?id=zSiXDwAAQBAJ&pg=PA33 33]}}}} Number theory arose out of problems related to multiplications and division of integers.{{harvnb|Karatsuba|2020}}

Based on written evidence, knowledge of numbers existed in the ancient civilisations of Mesopotamia, Egypt, China, and India.{{harvnb|Dunham|2025}} The first positional numeral system was developed by the Babylonians starting around 1800 BCE. This was a significant improvement over earlier numeral systems since it made the representation of large numbers and calculations on them more efficient.{{multiref|{{harvnb|Burgin|2022|pp=[https://books.google.com/books?id=rWF2EAAAQBAJ&pg=PA6 6–7, 9]}}|{{harvnb|Ore|1948|pp=16–18}}|{{harvnb|ITL Education Solutions Limited|2011|p=[https://books.google.com/books?id=CsNiKdmufvYC&pg=PA28 28]}}}}File:Plimpton_322.jpg

The earliest historical find of an arithmetical nature is a fragment of a Babylonian tablet. Plimpton 322, dated c. 1800 BC, contains a list of Pythagorean triples, that is, integers $(a,b,c)$ such that $a^2+b^2=c^2$. The triples are too numerous and too large to have been obtained by brute force.{{harvnb|Neugebauer|Sachs|1945|p=40}}{{harvnb|Robson|2001|p=192}}{{harvnb|Watkins|2014|loc=Number Theory Begins}} The table's layout suggests that it was constructed by means of what amounts, in modern language, to the identity

$$\backslash left(\backslash frac\{1\}\{2\}\; \backslash left(x\; -\; \backslash frac\{1\}\{x\}\backslash right)\backslash right)^2\; +\; 1\; =\; \backslash left(\backslash frac\{1\}\{2\}\; \backslash left(x\; +\; \backslash frac\{1\}\{x\}\; \backslash right)\backslash right)^2,$$

which is implicit in routine Old Babylonian 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.Neugebauer {{harv|Neugebauer|1969|pp=36–40}} discusses the table in detail and mentions in passing Euclid's method in modern notation {{harv|Neugebauer|1969|p=39}}.

It is not known what these applications may have been, or whether there could have been any; Babylonian astronomy, for example, truly came into its own many centuries later. It has been suggested instead that the table was a source of numerical examples for school problems.{{sfn|Friberg|1981|p=302}}{{harvnb|Robson|2001|p=201}}. This is controversial. See Plimpton 322. Robson's article is written polemically {{harv|Robson|2001|p=202}} with a view to "perhaps [...] knocking [Plimpton 322] off its pedestal" {{harv|Robson|2001|p=167}}; at the same time, it settles to the conclusion that

[...] the question "how was the tablet calculated?" does not have to have the same answer as the question "what problems does the tablet set?" The first can be answered most satisfactorily by reciprocal pairs, as first suggested half a century ago, and the second by some sort of right-triangle problems {{harv|Robson|2001|p=202}}.

The Rhind Mathematical Papyrus, from around 1550 BC, has Egyptian fraction expansions of different forms for prime and composite numbers.Bruins, Evert Marie, review in Mathematical Reviews of {{cite journal |last=Gillings |first=R.J. |year=1974 |title=The recto of the Rhind Mathematical Papyrus. How did the ancient Egyptian scribe prepare it? |journal=Archive for History of Exact Sciences |volume=12 |issue=4 |pages=291–298 |doi=10.1007/BF01307175 |mr=0497458 |s2cid=121046003}}

{{Further|Ancient Greek mathematics}}

However, the earliest surviving records of the study of prime numbers come from the ancient Greek mathematicians, who called them {{Transliteration|grc|prōtos arithmòs}} ({{lang|grc|πρῶτος ἀριθμὸς}}). Euclid's Elements (c. 300 BC) proves the infinitude of primes and the fundamental theorem of arithmetic, and shows how to construct a perfect number from a Mersenne prime.{{cite book|title=Mathematics and Its History|series=Undergraduate Texts in Mathematics|first=John|last=Stillwell|author-link=John Stillwell |edition=3rd |publisher=Springer |year=2010 |isbn=978-1-4419-6052-8 |page=40 |url=https://books.google.com/books?id=V7mxZqjs5yUC&pg=PA40}} Another Greek invention, the Sieve of Eratosthenes, is still used to construct lists of {{nowrap|primes.{{cite journal |title=The Search for Prime Numbers|first=Carl|last=Pomerance|author-link=Carl Pomerance|journal=Scientific American|volume=247|issue=6|date=December 1982|pages=136–147|jstor=24966751|doi=10.1038/scientificamerican1282-136|bibcode=1982SciAm.247f.136P}}{{cite journal | last = Mollin | first = Richard A. | doi = 10.2307/3219180 | issue = 1 | journal = Mathematics Magazine | mr = 2107288 | pages = 18–29 | title = A brief history of factoring and primality testing B. C. (before computers) | volume = 75 | year = 2002| jstor = 3219180 }}}}

Although other civilizations probably influenced Greek mathematics at the beginning,{{harvnb|van der Waerden|1961|p=87-90}} all evidence of such borrowings appear relatively late,Iamblichus, Life of Pythagoras,(trans., for example, {{harvnb|Guthrie|1987}}) cited in {{harvnb|van der Waerden|1961|p=108}}. See also Porphyry, Life of Pythagoras, paragraph 6, in {{harvnb|Guthrie|1987|para=6}}Herodotus (II. 81) and Isocrates (Busiris 28), cited in: {{harvnb|Huffman|2011}}. On Thales, see Eudemus ap. Proclus, 65.7, (for example, {{harvnb|Morrow|1992|p=52}}) cited in: {{harvnb|O'Grady|2004|p=1}}. Proclus was using a work by Eudemus of Rhodes (now lost), the Catalogue of Geometers. See also introduction, {{harvnb|Morrow|1992|p=xxx}} on Proclus's reliability. and it is likely that Greek {{tlit|grc|arithmētikḗ}} (the theoretical or philosophical study of numbers) is an indigenous tradition. Aside from a few fragments, most of what is known about Greek mathematics in the 6th to 4th centuries BC (the Archaic and Classical periods) comes through either the reports of contemporary non-mathematicians or references from mathematical works in the early Hellenistic period.{{sfn|Boyer|Merzbach|1991|p=82}} In the case of number theory, this means largely Plato, Aristotle, and Euclid.

Plato had a keen interest in mathematics, and distinguished clearly between {{tlit|grc|arithmētikḗ}} and calculation ({{tlit|grc|logistikē}}). Plato reports in his dialogue Theaetetus that Theodorus had proven that $\backslash sqrt\{3\},\; \backslash sqrt\{5\},\; \backslash dots,\; \backslash sqrt\{17\}$ are irrational. Theaetetus, a disciple of Theodorus's, worked on distinguishing different kinds of incommensurables, and was thus arguably a pioneer in the study of number systems. Aristotle further claimed that the philosophy of Plato closely followed the teachings of the Pythagoreans,Metaphysics, 1.6.1 (987a) and Cicero repeats this claim: {{lang|la|Platonem ferunt didicisse Pythagorea omnia}} ("They say Plato learned all things Pythagorean").Tusc. Disput. 1.17.39.

Euclid devoted part of his Elements (Books VII–IX) to topics that belong to elementary number theory, including prime numbers and divisibility.{{harvnb|Corry|2015|loc=Construction Problems and Numerical Problems in the Greek Mathematical Tradition}} He gave an algorithm, the Euclidean algorithm, for computing the greatest common divisor of two numbers (Prop. VII.2) and a proof implying the infinitude of primes (Prop. IX.20). There is also older material likely based on Pythagorean teachings (Prop. IX.21–34), such as "odd times even is even" and "if an odd number measures [= divides] an even number, then it also measures [= divides] half of it".{{harvnb|Becker|1936|p=533}}, cited in: {{harvnb|van der Waerden|1961|p=108}}. This is all that is needed to prove that $\backslash sqrt\{2\}$ is irrational.{{sfn|Becker|1936}} Pythagoreans apparently gave great importance to the odd and the even.{{sfn|van der Waerden|1961|p=109}} The discovery that $\backslash sqrt\{2\}$ is irrational is credited to the early Pythagoreans, sometimes assigned to Hippasus, who was expelled or split from the Pythagorean community as a result.Plato, Theaetetus, p. 147 B, (for example, {{harvnb|Jowett|1871}}), cited in {{harvnb|von Fritz|2004|p=212}}: "Theodorus was writing out for us something about roots, such as the roots of three or five, showing that they are incommensurable by the unit;..." See also Spiral of Theodorus.{{sfn|von Fritz|2004}} This forced a distinction between numbers (integers and the rationals—the subjects of arithmetic) and lengths and proportions (which may be identified with real numbers, whether rational or not).

The Pythagorean tradition also spoke of so-called polygonal or figurate numbers.{{sfn|Heath|1921|p=76}} While square numbers, cubic numbers, etc., are seen now as more natural than triangular numbers, pentagonal numbers, etc., the study of the sums of triangular and pentagonal numbers would prove fruitful in the early modern period (17th to early 19th centuries).

An epigram published by Lessing in 1773 appears to be a letter sent by Archimedes to Eratosthenes.{{sfn|Vardi|1998|pp=305–319}} 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 it is known, such equations were first successfully treated by Indian mathematicians. It is not known whether Archimedes himself had a method of solution.

File:Diophantus-cover.png}}, translated into Latin by Bachet (1621)]]

Aside from the elementary work of Neopythagoreans such as Nicomachus and Theon of Smyrna, the foremost authority in {{tlit|grc|arithmētikḗ}} in Late Antiquity was Diophantus of Alexandria, who probably lived in the 3rd century AD, approximately five hundred years after Euclid. Little is known about his life, but he wrote two works that are extant: On Polygonal Numbers, a short treatise written in the Euclidean manner on the subject, and the Arithmetica, a work on pre-modern algebra (namely, the use of algebra to solve numerical problems). Six out of the thirteen books of Diophantus's Arithmetica survive in the original Greek and four more survive in an Arabic translation. The {{lang|la|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 $f(x,y)=z^2$ or $f(x,y,z)=w^2$. In modern parlance, Diophantine equations are polynomial equations to which rational or integer solutions are sought.

The Chinese remainder theorem appears as an exerciseSunzi Suanjing, Chapter 3, Problem 26. This can be found in {{harvnb|Lam|Ang|2004|pp=219–220}}, which contains a full translation of the Suan Ching (based on {{harvnb|Qian|1963}}). See also the discussion in {{harvnb|Lam|Ang|2004|pp=138–140}}. in Sunzi Suanjing (between the third and fifth centuries).The date of the text has been narrowed down to 220–420 AD (Yan Dunjie) or 280–473 AD (Wang Ling) through internal evidence (= taxation systems assumed in the text). See {{harvnb|Lam|Ang|2004|pp=27–28}}. (There is one important step glossed over in Sunzi's solution:Sunzi Suanjing, Ch. 3, Problem 26, in {{harvnb|Lam|Ang|2004|pp=219–220}}:

[26] Now there are an unknown number of things. If we count by threes, there is a remainder 2; if we count by fives, there is a remainder 3; if we count by sevens, there is a remainder 2. Find the number of things. Answer: 23.

Method: If we count by threes and there is a remainder 2, put down 140. If we count by fives and there is a remainder 3, put down 63. If we count by sevens and there is a remainder 2, put down 30. Add them to obtain 233 and subtract 210 to get the answer. If we count by threes and there is a remainder 1, put down 70. If we count by fives and there is a remainder 1, put down 21. If we count by sevens and there is a remainder 1, put down 15. When [a number] exceeds 106, the result is obtained by subtracting 105.

While Greek astronomy probably influenced Indian learning, to the point of introducing trigonometry,{{sfn|Plofker|2008|p=119}} it seems to be the case that Indian mathematics is otherwise an autochthonous tradition;Any early contact between Babylonian and Indian mathematics remains conjectural {{harv|Plofker|2008|p=42}}. in particular, there is no evidence that Euclid's Elements reached India before the eighteenth century.{{sfn|Mumford|2010|p=387}} Āryabhaṭa (476–550 AD) showed that pairs of simultaneous congruences $n\backslash equiv\; a\_1\; \backslash bmod\; m\_1$, $n\backslash equiv\; a\_2\; \backslash bmod\; m\_2$ could be solved by a method he called kuṭṭaka, or pulveriser;Āryabhaṭa, Āryabhatīya, Chapter 2, verses 32–33, cited in: {{harvnb|Plofker|2008|pp=134–140}}. See also {{harvnb|Aryabhata|1930|pp=42–50}}. A slightly more explicit description of the kuṭṭaka was later given in Brahmagupta, Brāhmasphuṭasiddhānta, XVIII, 3–5 (in {{harvnb|Colebrooke|1817|p=325}}, cited in {{harvnb|Aryabhata|1930|p=42}}). this is a procedure close to (a generalization of) the Euclidean algorithm, which was probably discovered independently in India.{{sfn|Mumford|2010|p=388}} Āryabhaṭa seems to have had in mind applications to astronomical calculations.{{sfn|Plofker|2008|p=119}}

Brahmagupta (628 AD) started the systematic study of indefinite quadratic equations—in particular, the misnamed Pell equation, in which Archimedes may have first been interested, and which did not start to be solved in the West until the time of Fermat and Euler. Later Sanskrit authors would follow, using Brahmagupta's technical terminology. A general procedure (the chakravala, or "cyclic method") 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's Bīja-gaṇita (twelfth century).{{sfn|Plofker|2008|p=194}}

Indian mathematics remained largely unknown in Europe until the late eighteenth century;{{sfn|Plofker|2008|p=283}} Brahmagupta and Bhāskara's work was translated into English in 1817 by Henry Colebrooke.{{sfn|Colebrooke|1817}}

{{Further|Mathematics in medieval Islam|Islamic Golden Age}}

File:Hevelius_Selenographia_frontispiece.png as seen by the West: on the frontispiece of Selenographia Alhasen{{sic}} represents knowledge through reason and Galileo knowledge through the senses.]]

In the early ninth century, the caliph al-Ma'mun ordered translations of many Greek mathematical works and at least one Sanskrit work (the Sindhind, which may{{harvnb|Colebrooke|1817|p=lxv}}, cited in {{harvnb|Hopkins|1990|p=302}}. See also the preface in {{harvnb|Sachau|Bīrūni|1888}} cited in {{harvnb|Smith|1958|pp=168}} or may not{{harvnb|Pingree|1968|pp=97–125}}, and {{harvnb|Pingree|1970|pp=103–123}}, cited in {{harvnb|Plofker|2008|p=256}}. be Brahmagupta's Brāhmasphuṭasiddhānta). Diophantus's main work, the Arithmetica, was translated into Arabic by Qusta ibn Luqa (820–912). Part of the treatise al-Fakhri (by al-Karajī, 953 – c. 1029) builds on it to some extent. According to Rashed Roshdi, Al-Karajī's contemporary Ibn al-Haytham knew{{sfn|Rashed|1980|pp=305–321}} what would later be called Wilson's theorem.

Around 1000 AD, the Islamic mathematician Ibn al-Haytham (Alhazen) found Wilson's theorem, characterizing the prime numbers as the numbers {{tmath|n}} that evenly divide {{tmath|(n-1)!+1}}. He also conjectured that all even perfect numbers come from Euclid's construction using Mersenne primes, but was unable to prove it.{{MacTutor Biography|id=Al-Haytham|title=Abu Ali al-Hasan ibn al-Haytham|mode=cs1}} Another Islamic mathematician, Ibn al-Banna' al-Marrakushi, observed that the sieve of Eratosthenes can be sped up by considering only the prime divisors up to the square root of the upper limit. Fibonacci took the innovations from Islamic mathematics to Europe. His book Liber Abaci (1202) was the first to describe trial division for testing primality, again using divisors only up to the square root. a

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, thanks to a renewed study of the works of Greek antiquity. A catalyst was the textual emendation and translation into Latin of Diophantus' Arithmetica.Bachet, 1621, following a first attempt by Xylander, 1575

File:Pierre_de_Fermat.png]]

Pierre de Fermat (1607–1665) never published his writings but communicated through correspondence instead. Accordingly, his work on number theory is contained almost entirely in letters to mathematicians and in private marginal notes.{{sfn|Weil|1984|pp=45–46}} Although he drew inspiration from classical sources, in his notes and letters Fermat scarcely wrote any proofs—he had no models in the area.{{harvnb|Weil|1984|p=118}}. This was more so in number theory than in other areas ({{harvnb|Mahoney|1994|p=|pp=283-289}}). Bachet's own proofs were "ludicrously clumsy" {{harv|Weil|1984|p=33}}.

The corpus of his works consists of letters sent in private correspondence and marginal notes, in which he often formulated conjectures.

In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler).{{harvnb|Sandifer|2007}}, [https://books.google.com/books?id=sohHs7ExOsYC&pg=PA45 8. Fermat's Little Theorem (November 2003), p. 45] Fermat also investigated the primality of the Fermat numbers {{tmath|2^{2^n}+1}},{{cite book|title=How Euler Did Even More|first=C. Edward|last=Sandifer|publisher=Mathematical Association of America|year=2014|isbn=978-0-88385-584-3|page=42|url=https://books.google.com/books?id=3c6iBQAAQBAJ&pg=PA42}} and Marin Mersenne studied the Mersenne primes, prime numbers of the form $2^p-1$ with {{tmath|p}} itself a prime.{{cite book|title=Elementary Number Theory with Applications|first=Thomas|last=Koshy|publisher=Academic Press|year=2002|isbn=978-0-12-421171-1|page=369|url=https://books.google.com/books?id=-9pg-4Pa19IC&pg=PA369}}

Over his lifetime, Fermat made the following contributions to the field:

• One of Fermat's first interests was perfect numbers (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 earlier times—whether in the West or the Arab-speaking world—due in part to the importance given to them by the Neopythagorean Nicomachus (c. 100 AD), who wrote a very elementary but influential book entitled Introduction to Arithmetic. See {{harvnb|van der Waerden|1961|loc=Ch. IV}}. these topics led him to work on integer divisors, which were from the beginning among the subjects of the correspondence (1636 onwards) that put him in touch with the mathematical community of the day.{{harvnb|Mahoney|1994|pp=48, 53–54}}. The initial subjects of Fermat's correspondence included divisors ("aliquot parts") and many subjects outside number theory; see the list in the letter from Fermat to Roberval, 22.IX.1636, {{harvnb|Tannery|Henry|1891|loc=Vol. II, pp. 72, 74}}, cited in {{harvnb|Mahoney|1994|p=54}}.
• In 1638, Fermat claimed, without proof, that all whole numbers can be expressed as the sum of four squares or fewer.{{Cite encyclopedia |title=Numbers and Measurements |encyclopedia=Encyclopaedia Britannica |url=https://books.google.com/books?id=5tFFDwAAQBAJ |access-date=2019-08-06 |date=2017 |isbn=978-1-5383-0042-8 |last2=Hosch |first2=William L. |last1=Faulkner |first1=Nicholas}}
• Fermat's little theorem (1640):{{harvnb|Tannery|Henry|1891|loc=Vol. II, p. 209}}, Letter XLVI from Fermat to Frenicle, 1640, cited in {{harvnb|Weil|1984|p=56}} if a is not divisible by a prime p, then $a^\{p-1\}\; \backslash equiv\; 1\; \backslash bmod\; p.$
• If a and b are coprime, then $a^2\; +\; b^2$ is not divisible by any prime congruent to −1 modulo 4;{{harvnb|Tannery|Henry|1891|loc=Vol. II, p. 204}}, cited in {{harvnb|Weil|1984|p=63}}. All of the following citations from Fermat's Varia Opera are taken from {{harvnb|Weil|1984|loc=Chap. II}}. The standard Tannery & Henry work includes a revision of Fermat's posthumous Varia Opera Mathematica originally prepared by his son {{harv|Fermat|1679}}. and every prime congruent to 1 modulo 4 can be written in the form $a^2\; +\; b^2$.{{sfn|Tannery|Henry|1891|loc=Vol. II, p. 213}} 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.{{sfn|Tannery|Henry|1891|loc=Vol. II, p. 423}}
• 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.{{sfn|Weil|1984|p=92}} 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){{sfn|Tannery|Henry|1891|loc=Vol. I, pp. 340–341}} that $x^\{4\}\; +\; y^\{4\}\; =\; z^\{4\}$ 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.{{sfn|Weil|1984|p=115}} The first known proof is due to Euler (1753; indeed by infinite descent).{{sfn|Weil|1984|pp=115–116}}
• Fermat claimed (Fermat's Last Theorem) to have shown there are no solutions to $x^n\; +\; y^n\; =\; z^n$ for all $n\backslash geq\; 3$; this claim appears in his annotations in the margins of his copy of Diophantus.

File:Leonhard_Euler.jpg

The interest of Leonhard Euler (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 {{harv|Weil|1984|pp=159, 161}}. (There were already some recognisable features of professional practice, viz., seeking correspondents, visiting foreign colleagues, building private libraries {{harv|Weil|1984|pp=160–161}}. Matters started to shift in the late seventeenth century {{harv|Weil|1984|p=161}}; scientific academies were founded in England (the Royal Society, 1662) and France (the Académie des sciences, 1666) and Russia (1724). Euler was offered a position at this last one in 1726; he accepted, arriving in St. Petersburg in 1727 ({{harvnb|Weil|1984|p=163}} and {{harvnb|Varadarajan|2006|p=7}}). In this context, the term amateur usually applied to Goldbach is well-defined and makes some sense: he has been described as a man of letters who earned a living as a spy {{harv|Truesdell|1984|p=xv}}; cited in {{harvnb|Varadarajan|2006|p=9}}). 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.{{sfn|Weil|1984|pp=2, 172}} This has been called the "rebirth" of modern number theory,{{sfn|Weil|1984|pp=1–2}} after Fermat's relative lack of success in getting his contemporaries' attention for the subject.{{harvnb|Weil|1984|p=2}} and {{harvnb|Varadarajan|2006|p=37}} Euler's work on number theory includes the following:{{harvnb|Varadarajan|2006|p=39}} and {{harvnb|Weil|1984|pp=176–189}}

Christian Goldbach formulated Goldbach's conjecture, that every even number is the sum of two primes, in a 1742 letter to Euler.{{cite book|title=Goldbach Conjecture|edition=2nd|volume=4|series=Series In Pure Mathematics|first=Wang|last=Yuan|publisher=World Scientific|year=2002|isbn=978-981-4487-52-8|page=21|url=https://books.google.com/books?id=g4jVCgAAQBAJ&pg=PA21}} Euler proved Alhazen's conjecture (now the Euclid–Euler theorem) that all even perfect numbers can be constructed from Mersenne primes. He introduced methods from mathematical analysis to this area in his proofs of the infinitude of the primes and the divergence of the sum of the reciprocals of the primes {{tmath|\tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{5}+\tfrac{1}{7}+\tfrac{1}{11}+\cdots}}.{{cite book|title=The Development of Prime Number Theory: From Euclid to Hardy and Littlewood|series=Springer Monographs in Mathematics|first=Wladyslaw|last=Narkiewicz|publisher=Springer|year=2000|isbn=978-3-540-66289-1|page=11|contribution=1.2 Sum of Reciprocals of Primes|contribution-url=https://books.google.com/books?id=VVr3EuiHU0YC&pg=PA11}}

• Proofs for Fermat's statements. This includes Fermat's little theorem (generalised by Euler to non-prime moduli); the fact that $p\; =\; x^2\; +\; y^2$ if and only if $p\backslash equiv\; 1\; \backslash bmod\; 4$; initial work towards a proof that every integer is the sum of four squares (the first complete proof is by Joseph-Louis Lagrange (1770), soon improved by Euler himself{{sfn|Weil|1984|pp=178–179}}); the lack of non-zero integer solutions to $x^4\; +\; y^4\; =\; z^2$ (implying the case n=4 of Fermat's last theorem, the case n=3 of which Euler also proved by a related method).
• Pell's equation, first misnamed by Euler.{{harvnb|Weil|1984|p=174}}. Euler was generous in giving credit to others {{harv|Varadarajan|2006|p=14}}, not always correctly. He wrote on the link between continued fractions and Pell's equation.{{sfn|Weil|1984|p=183}}
• First steps towards analytic number theory. In his work of sums of four squares, partitions, pentagonal numbers, and the distribution of prime numbers, Euler pioneered the use of what can be seen as analysis (in particular, infinite series) in number theory. Since he lived before the development of complex analysis, most of his work is restricted to the formal manipulation of power series. He did, however, do some very notable (though not fully rigorous) early work on what would later be called the Riemann zeta function.{{harvnb|Varadarajan|2006|pp=45–55}}; see also chapter III.
• 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.{{sfn|Varadarajan|2006|pp=44–47}}
• Diophantine equations. Euler worked on some Diophantine equations of genus 0 and 1.{{sfn|Varadarajan|2006|pp=55–56}} 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.{{sfn|Weil|1984|p=181}} He did notice there was a connection between Diophantine problems and elliptic integrals,{{sfn|Weil|1984|p=181}} whose study he had himself initiated.

File:Carl_Friedrich_Gauss_1840_by_Jensen.jpg

At the start of the 19th century, Legendre and Gauss conjectured that as {{tmath|x}} tends to infinity, the number of primes up to {{tmath|x}} is asymptotic to {{tmath| x/\log x }}, where $\backslash log\; x$ is the natural logarithm of {{tmath|x}}. A weaker consequence of this high density of primes was Bertrand's postulate, that for every $n\; >\; 1$ there is a prime between {{tmath|n}} and {{tmath|2n}}, proved in 1852 by Pafnuty Chebyshev.{{cite journal|first=P. |last=Tchebychev |author-link=Pafnuty Chebyshev |title=Mémoire sur les nombres premiers. |journal=Journal de mathématiques pures et appliquées |series=Série 1 |year=1852 |pages=366–390 |url=http://sites.mathdoc.fr/JMPA/PDF/JMPA_1852_1_17_A19_0.pdf |language=fr}}. (Proof of the postulate: 371–382). Also see Mémoires de l'Académie Impériale des Sciences de St. Pétersbourg, vol. 7, pp. 15–33, 1854

Joseph-Louis Lagrange (1736–1813) was the first to give full proofs of some of Fermat's and Euler's work and observations; for instance, the four-square theorem and the basic theory of the misnamed "Pell's equation" (for which an algorithmic solution was found by Fermat and his contemporaries, and also by Jayadeva and Bhaskara II before them.) He also studied quadratic forms in full generality (as opposed to $m\; X^2\; +\; n\; Y^2$), including defining their equivalence relation, showing how to put them in reduced form, etc.

Adrien-Marie Legendre (1752–1833) was the first to state the law of quadratic reciprocity. He also conjectured what amounts to the prime number theorem and Dirichlet's theorem on arithmetic progressions. He gave a full treatment of the equation $a\; x^2\; +\; b\; y^2\; +\; c\; z^2\; =\; 0${{sfn|Weil|1984|pp=327–328}} and worked on quadratic forms along the lines later developed fully by Gauss.{{sfn|Weil|1984|pp=332–334}} In his old age, he was the first to prove Fermat's Last Theorem for $n=5$ (completing work by Peter Gustav Lejeune Dirichlet, and crediting both him and Sophie Germain).{{sfn|Weil|1984|pp=337–338}}

Carl Friedrich Gauss (1777–1855) worked in a wide variety of fields in both mathematics and physics including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. The Disquisitiones Arithmeticae (1801), which he wrote three years earlier when he was 21, had an immense influence in the area of number theory and set its agenda for much of the 19th century. Gauss proved in this work 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.{{sfn|Goldstein|Schappacher|2007|p=14}} 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.From the preface of Disquisitiones Arithmeticae; the translation is taken from {{harvnb|Goldstein|Schappacher|2007|p=16}}

File:Peter_Gustav_Lejeune_Dirichlet.jpg]]

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.See the discussion in section 5 of {{harvnb|Goldstein|Schappacher|2007}}. Early signs of self-consciousness are present already in letters by Fermat: thus his remarks on what number theory is, and how "Diophantus's work [...] does not really belong to [it]" (quoted in {{harvnb|Weil|1984|p=25}}).
• The development of much of modern mathematics necessary for basic modern number theory: complex analysis, group theory, Galois theory—accompanied by greater rigor in analysis and abstraction in algebra.
• The rough subdivision of number theory into its modern subfields—in particular, analytic and algebraic number theory.

Ideas of Bernhard Riemann in his 1859 paper on the zeta-function sketched an outline for proving the conjecture of Legendre and Gauss. Although the closely related Riemann hypothesis remains unproven, Riemann's outline was completed in 1896 by Hadamard and de la Vallée Poussin, and the result is now known as the prime number theorem.{{cite book | last = Apostol | first = Tom M. | author-link = Tom M. Apostol | editor1-last = Bambah | editor1-first = R.P. | editor2-last = Dumir | editor2-first = V.C. | editor3-last = Hans-Gill | editor3-first = R.J. | contribution = A centennial history of the prime number theorem | location = Basel | mr = 1764793 | pages = 1–14 | publisher = Birkhäuser | series = Trends in Mathematics | title = Number Theory | contribution-url = https://books.google.com/books?id=aiDyBwAAQBAJ&pg=PA1 | year = 2000}} Another important 19th century result was Dirichlet's theorem on arithmetic progressions, that certain arithmetic progressions contain infinitely many primes.{{cite book | last = Apostol | first = Tom M. | author-link = Tom M. Apostol | contribution = 7. Dirichlet's Theorem on Primes in Arithmetical Progressions | contribution-url = https://books.google.com/books?id=3yoBCAAAQBAJ&pg=PA146 | location = New York; Heidelberg | mr = 0434929 | pages = 146–156 | publisher = Springer-Verlag | title = Introduction to Analytic Number Theory | year = 1976 }}

Many mathematicians have worked on primality tests for numbers larger than those where trial division is practicably applicable. Methods that are restricted to specific number forms include Pépin's test for Fermat numbers (1877),{{cite book|title=A History of Algorithms: From the Pebble to the Microchip|first=Jean-Luc|last=Chabert|publisher=Springer|year=2012|isbn=978-3-642-18192-4|page=261|url=https://books.google.com/books?id=XcDqCAAAQBAJ&pg=PA261}} Proth's theorem (c. 1878),{{cite book|title=Elementary Number Theory and Its Applications|first=Kenneth H.|last=Rosen|edition=4th|publisher=Addison-Wesley|year=2000|isbn=978-0-201-87073-2|contribution=Theorem 9.20. Proth's Primality Test|page=342}} the Lucas–Lehmer primality test (originated 1856), and the generalized Lucas primality test.

Algebraic number theory may be said to start with the study of reciprocity and cyclotomy, but truly came into its own with the development of abstract algebra and early ideal theory and valuation theory; see below. A conventional starting point for analytic number theory is Dirichlet's theorem on arithmetic progressions (1837),{{sfn|Apostol|1976|p=7}} whose proof introduced L-functions and involved some asymptotic analysis and a limiting process on a real variable.See the proof in {{harvnb|Davenport|Montgomery|2000|loc=section 1}} The first use of analytic ideas in number theory actually goes back to Euler (1730s),{{sfn|Iwaniec|Kowalski|2004|p=1}} who used formal power series and non-rigorous (or implicit) limiting arguments. The use of complex analysis in number theory comes later: the work of Bernhard Riemann (1859) on the zeta function is the canonical starting point;{{sfn|Granville|2008|pp=322–348}} 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 forms).See the comment on the importance of modularity in {{harvnb|Iwaniec|Kowalski|2004|p=1}}

Pafnuty Chebyshev (1821-1894) provided substantiated cases for the PNT such as the existence of primes within specific intervals.

Since 1951 all the largest known primes have been found using these tests on computers.{{efn|A 44-digit prime number found in 1951 by Aimé Ferrier with a mechanical calculator remains the largest prime not to have been found with the aid of electronic computers.{{cite book|title=The Once and Future Turing|first1=S. Barry|last1=Cooper|first2=Andrew|last2=Hodges|publisher=Cambridge University Press|year=2016|isbn=978-1-107-01083-3|pages=37–38|url=https://books.google.com/books?id=h12cCwAAQBAJ&pg=PA37}}}} The search for ever larger primes has generated interest outside mathematical circles, through the Great Internet Mersenne Prime Search and other distributed computing projects.{{harvnb|Rosen|2000}}, p. 245.

The American Mathematical Society awards the Cole Prize in Number Theory. Moreover, number theory is one of the three mathematical subdisciplines rewarded by the Fermat Prize.

The idea that prime numbers had few applications outside of pure mathematics{{efn|name="pure"|For instance, Beiler writes that number theorist Ernst Kummer loved his ideal numbers, closely related to the primes, "because they had not soiled themselves with any practical applications",{{cite book|title=Recreations in the Theory of Numbers: The Queen of Mathematics Entertains|first=Albert H.|last=Beiler|year=1999|publisher=Dover|orig-year=1966|isbn=978-0-486-21096-4|page=2|url=https://books.google.com/books?id=NbbbL9gMJ88C&pg=PA2|oclc=444171535}} and Katz writes that Edmund Landau, known for his work on the distribution of primes, "loathed practical applications of mathematics", and for this reason avoided subjects such as geometry that had already shown themselves to be useful.{{cite journal | last = Katz | first = Shaul | doi = 10.1017/S0269889704000092 | issue = 1–2 | journal = Science in Context | mr = 2089305 | pages = 199–234 | title = Berlin roots – Zionist incarnation: the ethos of pure mathematics and the beginnings of the Einstein Institute of Mathematics at the Hebrew University of Jerusalem | volume = 17 | year = 2004| s2cid = 145575536 }}}} was shattered in the 1970s when public-key cryptography and the RSA cryptosystem were invented, using prime numbers as their basis.{{cite book|title=Elementary Number Theory|series=Textbooks in mathematics|first1=James S.|last1=Kraft|first2=Lawrence C.|last2=Washington|publisher=CRC Press|year=2014|isbn=978-1-4987-0269-0|page=7|url=https://books.google.com/books?id=4NAqBgAAQBAJ&pg=PA7}}

Number theory was once regarded as the canonical example of pure mathematics with no applications outside the field. In 1970s, it became known that prime numbers would be used as the basis for the creation of public-key cryptography algorithms. Schemes such as RSA are based on the difficulty of factoring large composite numbers into their prime factors. Elementary number theory is taught in discrete mathematics courses for computer scientists.

The increased practical importance of computerized primality testing and factorization led to the development of improved methods capable of handling large numbers of unrestricted form.{{cite book|title=Secret History: The Story of Cryptology|series=Discrete Mathematics and Its Applications|first=Craig P.|last=Bauer|publisher=CRC Press|year=2013|isbn=978-1-4665-6186-1|page=468|url=https://books.google.com/books?id=EBkEGAOlCDsC&pg=PA468}}{{cite book|title=Old and New Unsolved Problems in Plane Geometry and Number Theory|volume=11|series=Dolciani mathematical expositions|first1=Victor|last1=Klee|author1-link=Victor Klee|first2=Stan|last2=Wagon|author2-link=Stan Wagon|publisher=Cambridge University Press|year=1991|isbn=978-0-88385-315-3|page=224|url=https://books.google.com/books?id=tRdoIhHh3moC&pg=PA224}} The mathematical theory of prime numbers also moved forward with the Green–Tao theorem (2004) that there are arbitrarily long arithmetic progressions of prime numbers, and Yitang Zhang's 2013 proof that there exist infinitely many prime gaps of bounded size.{{harvnb|Neale|2017}}, pp. 18, 47.

One quirk of number theory is that it deals with statements that are simple to understand but require a high degree of sophistication to solve. Open questions remain, such as the aforementioned Riemann hypothesis that is among the seven Millennium Prize Problems, Goldbach's conjecture on the representation of even numbers as the sum of two primes, and the existence of odd perfect numbers.

{{reflist|group=note}}

{{reflist}}

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• {{cite book |last=Heath |first=Thomas L. |year=1921 |author-link=Thomas Little Heath |title=A History of Greek Mathematics, Volume 1: From Thales to Euclid |location=Oxford |publisher=Clarendon Press |url=https://archive.org/details/historyofgreekma01heat |access-date=2016-02-28}}

• {{cite book |last1=Hindry |first1=Marc |title=Arithmetics |series=Universitext |date=2011 |publisher=Springer |doi=10.1007/978-1-4471-2131-2 |isbn=978-1-4471-2130-5 |url=https://link.springer.com/book/10.1007/978-1-4471-2131-2}}
• {{cite book |last=Hopkins |first=J. F. P. |editor1-last=Young |editor1-first=M. J. L. |editor2-last=Latham |editor2-first=J. D. |editor3-last=Serjeant |editor3-first=R. B. |year=1990 |chapter=Geographical and Navigational Literature |title=Religion, Learning and Science in the 'Abbasid Period |series=The Cambridge history of Arabic literature |publisher=Cambridge University Press |isbn=978-0-521-32763-3}}
• {{cite encyclopedia |last=Huffman |first=Carl A. |editor1-last=Zalta |editor1-first=Edward N. |date=8 August 2011 |title=Pythagoras |encyclopedia=Stanford Encyclopaedia of Philosophy |edition=Fall 2011 |url=http://plato.stanford.edu/archives/fall2011/entries/pythagoras/ |access-date=7 February 2012 |archive-date=2 December 2013 |archive-url=https://web.archive.org/web/20131202071830/http://plato.stanford.edu/archives/fall2011/entries/pythagoras/ |url-status=live}}
• {{cite book |author=ITL Education Solutions Limited |title=Introduction to Computer Science |date=2011 |publisher=Pearson Education India |isbn=978-81-317-6030-7 |url=https://books.google.com/books?id=CsNiKdmufvYC&pg=PA28 |language=en}}
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• {{cite book |ref={{harvid|Jowett|1871}} |translator-last=Jowett |translator-first=Benjamin |translator-link=Benjamin Jowett |year=1871 |last1=Plato |author1-link=Plato |title=Theaetetus |url=http://classics.mit.edu/Plato/theatu.html |access-date=2012-04-10 |archive-date=2011-07-09 |archive-url=https://web.archive.org/web/20110709194524/http://classics.mit.edu/Plato/theatu.html |url-status=live}}
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• {{citation |first=Ulrich |last=Libbrecht |title=Chinese Mathematics in the Thirteenth Century: the "Shu-shu Chiu-chang" of Ch'in Chiu-shao |publisher=Dover Publications Inc |isbn=978-0-486-44619-6 |date=1973}}
• {{ cite book |last=Long |first=Calvin T. |year=1972 |title=Elementary Introduction to Number Theory |edition=2nd |publisher=D.C. Heath and Company |location=Lexington, VA |lccn=77171950}}
• {{cite book |last1=Lozano-Robledo |first1=Álvaro |title=Number Theory and Geometry: An Introduction to Arithmetic Geometry |date=2019 |publisher=American Mathematical Soc. |isbn=978-1-4704-5016-8 |url=https://books.google.com/books?id=ESiODwAAQBAJ&pg=PR13 |language=en}}
• {{cite book |last=Mahoney |first=M. S. |year=1994 |title=The Mathematical Career of Pierre de Fermat, 1601–1665 |edition=Reprint, 2nd |publisher=Princeton University Press |isbn=978-0-691-03666-3 |url=https://books.google.com/books?id=My19IcewAnoC |access-date=2016-02-28}}
• {{cite web

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• {{cite book |last1=Montgomery |first1=Hugh L. |author-link=Hugh Montgomery (mathematician) |year=2007 |last2=Vaughan |first2=Robert C. |author2-link=Bob Vaughan |title=Multiplicative Number Theory: I, Classical Theory |publisher=Cambridge University Press |isbn=978-0-521-84903-6 |url=https://books.google.com/books?id=nGb1NADRWgcC |access-date=2016-02-28}}
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• {{cite web |author1=MW staff |title=Definition of Arithmetic |url=https://www.merriam-webster.com/dictionary/arithmetic |website=www.merriam-webster.com |access-date=19 October 2023 |language=en |date=2023 }}
• {{cite book |last1=Nagel |first1=Rob |title=U-X-L Encyclopedia of Science |date=2002 |publisher=U-X-L |isbn=978-0-7876-5440-5 |url=https://www.encyclopedia.com/science-and-technology/mathematics/mathematics/arithmetic |language=en}}
• {{cite book |last1=Nagel |first1=Ernest |last2=Newman |first2=James Roy |title=Godel's Proof |date=2008 |publisher=NYU Press |isbn=978-0-8147-5837-3 |url=https://books.google.com/books?id=WgwUCgAAQBAJ&pg=PA4 |language=en}}
• {{cite book |last=Neugebauer |first=Otto E. |year=1969 |author-link=Otto E. Neugebauer |title=The Exact Sciences in Antiquity |volume=9 |location=New York |publisher=Dover Publications |isbn=978-0-486-22332-2}}
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• {{cite book |last1=Ore |first1=Øystein |author-link=Øystein Ore |title=Number Theory and Its History |date=1948 |publisher=McGraw-Hill |url=https://archive.org/details/numbertheoryitsh00ore/ |url-access=limited }} Dover reprint, 1988, {{isbn|978-0-486-65620-5}}.
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• {{cite web |last=O'Grady |first=Patricia |author-link=Patricia O'Grady |date=September 2004 |title=Thales of Miletus |url=http://www.iep.utm.edu/thales/ |publisher=The Internet Encyclopaedia of Philosophy |access-date=7 February 2012 |archive-date=6 January 2016 |archive-url=https://web.archive.org/web/20160106182825/http://www.iep.utm.edu/thales/ |url-status=live}}
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• {{cite book |last=Plofker |first=Kim |author-link=Kim Plofker |year=2008 |title=Mathematics in India |title-link=Mathematics in India (book) |publisher=Princeton University Press |isbn=978-0-691-12067-6}}
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• {{cite book |editor1-last=Qian |editor1-first=Baocong |year=1963 |language=zh |title=Suanjing shi shu (Ten Mathematical Classics) |location=Beijing |publisher=Zhonghua shuju |url=https://www.scribd.com/doc/53797787/Jigu-Suanjing%E3%80%80%E7%B7%9D%E5%8F%A4%E7%AE%97%E7%B6%93-Qian-Baocong-%E9%8C%A2%E5%AF%B6%E7%90%AE |access-date=2016-02-28 |archive-date=2013-11-02 |archive-url=https://web.archive.org/web/20131102154812/http://www.scribd.com/doc/53797787/Jigu-Suanjing%E3%80%80%E7%B7%9D%E5%8F%A4%E7%AE%97%E7%B6%93-Qian-Baocong-%E9%8C%A2%E5%AF%B6%E7%90%AE |url-status=live}}
• {{cite journal |last=Rashed |first=Roshdi |year=1980 |title=Ibn al-Haytham et le théorème de Wilson |journal=Archive for History of Exact Sciences |volume=22 |issue=4 |pages=305–321 |doi=10.1007/BF00717654 |s2cid=120885025}}
• {{cite book |last1=Robbins |first1=Neville |title=Beginning Number Theory |date=2006 |publisher=Jones & Bartlett Learning |isbn=978-0-7637-3768-9 |url=https://books.google.com/books?id=TtLMrKDsDuIC&pg=PR12-IA1 |language=en}}
• {{cite journal |last=Robson |first=Eleanor |author-link=Eleanor Robson |year=2001 |title=Neither Sherlock Holmes nor Babylon: a Reassessment of Plimpton 322 |volume=28 |journal=Historia Mathematica |issue=3 |pages=167–206 |doi=10.1006/hmat.2001.2317 |url=http://www.hps.cam.ac.uk/people/robson/neither-sherlock.pdf |archive-url=https://web.archive.org/web/20141021070742/http://www.hps.cam.ac.uk/people/robson/neither-sherlock.pdf |archive-date=2014-10-21}}
• {{cite book |last1=Romanowski |first1=Perry |editor1-last=Lerner |editor1-first=Brenda Wilmoth |editor2-last=Lerner |editor2-first=K. Lee |title=The Gale Encyclopedia of Science |date=2008 |publisher=Thompson Gale |isbn=978-1-4144-2877-2 |edition=4th |chapter=Arithmetic |url=https://www.encyclopedia.com/science-and-technology/mathematics/mathematics/arithmetic}}
• {{cite book |last1=Rudman |first1=Peter S. |title=How Mathematics Happened: The First 50,000 Years |date=2007 |publisher=Prometheus Books |location=New York |isbn=978-1-59102-477-4}}
• {{cite book |last1=Sachau |first1=Eduard |author-link=Eduard Sachau |last2=Bīrūni |first2=̄Muḥammad ibn Aḥmad |author2-link=Abū Rayḥān al-Bīrūnī |year=1888 |title=Alberuni's India: An Account of the Religion, Philosophy, Literature, Geography, Chronology, Astronomy and Astrology of India, Vol. 1 |location=London |publisher=Kegan, Paul, Trench, Trübner & Co. |url=http://onlinebooks.library.upenn.edu/webbin/book/lookupname?key=Sachau%2C%20Eduard%2C%201845-1930 |access-date=2016-02-28 |archive-date=2016-03-03 |archive-url=https://web.archive.org/web/20160303235035/http://onlinebooks.library.upenn.edu/webbin/book/lookupname?key=Sachau,%20Eduard,%201845-1930 |url-status=live}}
• {{cite book |last=Serre |first=Jean-Pierre |year=1996 |orig-year=1973 |author-link=Jean-Pierre Serre |title=A Course in Arithmetic |series=Graduate Texts in Mathematics |volume=7 |publisher=Springer |isbn=978-0-387-90040-7 |url=https://archive.org/details/courseinarithmet00serr}}
• {{cite book |last=Smith |first=D. E. |year=1958 |title=History of Mathematics, Vol I |location=New York |publisher=Dover}}
• {{cite book |ref={{harvid|Tannery|Henry|1891}} |last1=Tannery |first1=Paul |author1-link=Paul Tannery |editor1=Charles Henry |editor1-link=Charles Henry (librarian) |year=1891 |last2=Fermat |first2=Pierre de |author2-link=Pierre de Fermat |language=fr, la |title=Oeuvres de Fermat |series=(4 Vols.) |location=Paris |publisher=Imprimerie Gauthier-Villars et Fils |url=https://archive.org/details/oeuvresdefermat01ferm}} [https://archive.org/details/oeuvresdefermat01ferm Volume 1] [https://archive.org/details/oeuvresdefermat02ferm Volume 2] [https://archive.org/details/oeuvresdefermat03ferm Volume 3] [https://archive.org/details/oeuvresdefermat04ferm Volume 4 (1912)]
• {{Cite book |last=Tanton |first=James |title=Encyclopedia of Mathematics |publisher=Facts On File |year=2005 |isbn=0-8160-5124-0 |location=New York |language=en |chapter=Number theory}}
• {{cite book |ref={{harvid|Taylor|1818}} |translator-last=Taylor |translator-first=Thomas |translator-link=Thomas Taylor (neoplatonist) |year=1818 |author1=Iamblichus |author1-link=Iamblichus |title=Life of Pythagoras or, Pythagoric Life |location=London |publisher=J. M. Watkins}} For other editions, see Iamblichus#List of editions and translations
• {{cite book |last1=Thiam |first1=Thierno |last2=Rochon |first2=Gilbert |title=Sustainability, Emerging Technologies, and Pan-Africanism |date=2019 |publisher=Springer Nature |isbn=978-3-030-22180-5 |url=https://books.google.com/books?id=EWSsDwAAQBAJ&pg=PA164 |language=en}}
• {{cite book |last=Truesdell |first=C. A. |author-link=Clifford Truesdell |year=1984 |translator-last=Hewlett |translator-first=John |chapter=Leonard Euler, Supreme Geometer |title=Leonard Euler, Elements of Algebra |edition=reprint of 1840 5th |location=New York |publisher=Springer-Verlag |isbn=978-0-387-96014-2 |chapter-url=https://books.google.com/books?id=mkOhy6v7kIsC}} This Google books preview of Elements of algebra lacks Truesdell's intro, which is reprinted (slightly abridged) in the following book:
• {{cite book |last=Truesdell |first=C. A. |author-link=Clifford Truesdell |year=2007 |editor1-last=Dunham |editor1-first=William |chapter=Leonard Euler, Supreme Geometer |title=The Genius of Euler: reflections on his life and work |series=Volume 2 of MAA tercentenary Euler celebration |location=New York |publisher=Mathematical Association of America |isbn=978-0-88385-558-4 |chapter-url=https://books.google.com/books?id=M4-zUnrSxNoC |access-date=2016-02-28}}
• {{cite book |last=Varadarajan |first=V. S. |year=2006 |title=Euler Through Time: A New Look at Old Themes |publisher=American Mathematical Society |isbn=978-0-8218-3580-7 |url=https://books.google.com/books?id=CYyKTREGYd0C |access-date=2016-02-28}}
• {{cite journal |last=Vardi |first=Ilan |title=Archimedes' Cattle Problem |date=April 1998 |journal=American Mathematical Monthly |volume=105 |issue=4 |pages=305–319 |doi=10.2307/2589706 |url=https://www.cs.drexel.edu/~crorres/Archimedes/Cattle/cattle_vardi.pdf |jstor=2589706 |citeseerx=10.1.1.383.545 |access-date=2012-04-08 |archive-date=2012-07-15 |archive-url=https://web.archive.org/web/20120715031904/https://www.cs.drexel.edu/~crorres/Archimedes/Cattle/cattle_vardi.pdf |url-status=live}}
• {{cite book |last1=van der Waerden |first1=Bartel L. |translator-last=Dresden |translator-first=Arnold |year=1961 |author-link=Bartel Leendert van der Waerden |title=Science Awakening |volume=1 or 2 |location=New York |publisher=Oxford University Press}}

• {{cite book |last1=Watkins |first1=John J. |title=Number Theory: A Historical Approach |date=2014 |publisher=Princeton University Press |isbn=978-0-691-15940-9}}
• {{cite book |last=Weil |first=André |year=1984 |author-link=André Weil |title=Number Theory: an Approach Through History – from Hammurapi to Legendre |location=Boston |publisher=Birkhäuser |isbn=978-0-8176-3141-3 |url=https://books.google.com/books?id=XSV0hDFj3loC |access-date=2016-02-28}}
• {{Cite book |last=Weisstein |first=Eric W. |title=CRC Concise Encyclopedia of Mathematics |publisher=Chapman & Hall/CRC |year=2003 |isbn=1-58488-347-2 |edition=2nd}}
• {{cite book |last1=Wilson |first1=Robin |title=Number Theory: A Very Short Introduction |date=2020 |publisher=Oxford University Press |isbn=978-0-19-879809-5 |url=https://books.google.com/books?id=fcDgDwAAQBAJ&pg=PA1 |language=en}}

{{refend}}

• {{Citizendium}}