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66-449: 60 is the 4th superior highly composite number , the 4th colossally abundant number , and is also a highly composite number , a unitary perfect number , and an abundant number . It is the smallest number divisible by the numbers 1 to 6. The smallest group that is not a solvable is the alternating group A 5 , which has 60 elements. There are 60 one-sided hexominoes , the polyominoes made from six squares. There are 60 seconds in

132-661: A 1 mod m 1 {\displaystyle n\equiv a_{1}{\bmod {m}}_{1}} , n ≡ a 2 mod m 2 {\displaystyle n\equiv a_{2}{\bmod {m}}_{2}} could be solved by a method he called kuṭṭaka , or pulveriser ; this is a procedure close to (a generalisation of) the Euclidean algorithm, which was probably discovered independently in India. Āryabhaṭa seems to have had in mind applications to astronomical calculations. Brahmagupta (628 AD) started

198-491: A Diophantine equations a polynomial equations to which rational or integer solutions are sought. While Greek astronomy probably influenced Indian learning, to the point of introducing trigonometry , it seems to be the case that Indian mathematics is otherwise an indigenous tradition; 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 n ≡

264-514: 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 , 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. While evidence of Babylonian number theory is only survived by the Plimpton 322 tablet, some authors assert that Babylonian algebra

330-455: A first foray towards both Évariste Galois 's work and algebraic number theory . Starting early in the nineteenth century, the following developments gradually took place: 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

396-604: A half-open interval I ⊂ R + {\displaystyle I\subset \mathbb {R} ^{+}} such that ∀ x ∈ I : s ( x ) = s ′ {\displaystyle \forall x\in I:s(x)=s'} . This representation implies that there exist an infinite sequence of π 1 , π 2 , … ∈ P {\displaystyle \pi _{1},\pi _{2},\ldots \in \mathbb {P} } such that for

462-405: A list of " Pythagorean triples ", that is, integers ( a , b , c ) {\displaystyle (a,b,c)} such that a 2 + b 2 = c 2 {\displaystyle a^{2}+b^{2}=c^{2}} . The triples are too many and too large to have been obtained by brute force . The heading over the first column reads: "The takiltum of

528-471: A minute, as well as 60 minutes in a degree. The first fullerene to be discovered was buckminsterfullerene C 60 , an allotrope of carbon with 60 atoms in each molecule , arranged in a truncated icosahedron . This ball is known as a buckyball, and looks like a soccer ball . The atomic number of neodymium is 60, and cobalt-60 (Co) is a radioactive isotope of cobalt . The electrical utility frequency in western Japan, South Korea, Taiwan,

594-407: A naïve sense, analogous to primes among the integers.) The initial impetus for the development of ideal numbers (by Kummer ) seems to have come from the study of higher reciprocity laws, that is, generalisations of quadratic reciprocity . 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,

660-473: 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 may be defined Some subjects generally considered to be part of analytic number theory, for example, sieve theory , are better covered by

726-712: A number "n" is not highly composite, it cannot be superior highly composite. An effective construction of the set of all superior highly composite numbers is given by the following monotonic mapping from the positive real numbers. Let e p ( x ) = ⌊ 1 p x − 1 ⌋ {\displaystyle e_{p}(x)=\left\lfloor {\frac {1}{{\sqrt[{x}]{p}}-1}}\right\rfloor } for any prime number p and positive real x . Then s ( x ) = ∏ p ∈ P p e p ( x ) {\displaystyle s(x)=\prod _{p\in \mathbb {P} }p^{e_{p}(x)}}

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792-637: 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 . Pierre de Fermat (1607–1665) never published his writings; in particular, his work on number theory

858-408: Is Dirichlet's theorem on arithmetic progressions (1837), whose proof introduced L-functions and involved some asymptotic analysis and a limiting process on a real variable. The first use of analytic ideas in number theory actually goes back to Euler (1730s), who used formal power series and non-rigorous (or implicit) limiting arguments. The use of complex analysis in number theory comes later:

924-448: Is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions . German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers ), or defined as generalizations of

990-485: 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 were set in the late nineteenth century, when ideal numbers , the theory of ideals and valuation theory were introduced; these are three complementary ways of dealing with

1056-511: Is a superior highly composite number. Note that the product need not be computed indefinitely, because if p > 2 x {\displaystyle p>2^{x}} then e p ( x ) = 0 {\displaystyle e_{p}(x)=0} , so the product to calculate s ( x ) {\displaystyle s(x)} can be terminated once p ≥ 2 x {\displaystyle p\geq 2^{x}} . Also note that in

1122-441: Is an algebraic number. Fields of algebraic numbers are also called algebraic number fields , 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

1188-1025: Is another superior highly composite number because it has the highest ratio of divisors to itself raised to the .4 power. 9 36 .4 ≈ 2.146 , 10 48 .4 ≈ 2.126 , 12 60 .4 ≈ 2.333 , 16 120 .4 ≈ 2.357 , 18 180 .4 ≈ 2.255 , 20 240 .4 ≈ 2.233 , 24 360 .4 ≈ 2.279 {\displaystyle {\frac {9}{36^{.4}}}\approx 2.146,{\frac {10}{48^{.4}}}\approx 2.126,{\frac {12}{60^{.4}}}\approx 2.333,{\frac {16}{120^{.4}}}\approx 2.357,{\frac {18}{180^{.4}}}\approx 2.255,{\frac {20}{240^{.4}}}\approx 2.233,{\frac {24}{360^{.4}}}\approx 2.279} The first 15 superior highly composite numbers, 2 , 6 , 12 , 60 , 120 , 360 , 2520 , 5040 , 55440, 720720, 1441440, 4324320, 21621600, 367567200, 6983776800 (sequence A002201 in

1254-526: Is conducted to felicitate this birthday. It represents a milestone in his life. There are 60 years mentioned in the historic Indian calendars. It is: Superior highly composite number In number theory , a superior highly composite number is a natural number which, in a particular rigorous sense, has many divisors . Particularly, it is defined by a ratio between the number of divisors an integer has and that integer raised to some positive power. For any possible exponent , whichever integer has

1320-433: 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: The interest of Leonhard Euler (1707–1783) in number theory was first spurred in 1729, when a friend of his, the amateur Goldbach , pointed him towards some of Fermat's work on

1386-502: Is credited to the early Pythagoreans (pre- Theodorus ). 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

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1452-406: Is even", "if an odd number measures [= divides] an even number, then it also measures [= divides] half of it"), but it is all that is needed to prove that 2 {\displaystyle {\sqrt {2}}} is irrational . Pythagorean mystics gave great importance to the odd and the even. The discovery that 2 {\displaystyle {\sqrt {2}}} is irrational

1518-403: Is through one of Plato's dialogues—namely, Theaetetus —that it is known that Theodorus had proven that 3 , 5 , … , 17 {\displaystyle {\sqrt {3}},{\sqrt {5}},\dots ,{\sqrt {17}}} are irrational. Theaetetus was, like Plato, a disciple of Theodorus's; he worked on distinguishing different kinds of incommensurables , and

1584-462: Is thus that algebraic and analytic number theory intersect. For example, one may define prime ideals (generalizations of prime numbers 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 , a key analytic object at

1650-888: The OEIS ) are also the first 15 colossally abundant numbers , which meet a similar condition based on the sum-of-divisors function rather than the number of divisors. Neither set, however, is a subset of the other. All superior highly composite numbers are highly composite . This is easy to prove: if there is some number k that has the same number of divisors as n but is less than n itself (i.e. d ( k ) = d ( n ) {\displaystyle d(k)=d(n)} , but k < n {\displaystyle k<n} ), then d ( k ) k ε > d ( n ) n ε {\displaystyle {\frac {d(k)}{k^{\varepsilon }}}>{\frac {d(n)}{n^{\varepsilon }}}} for all positive ε, so if

1716-492: The OEIS ). In other words, the quotient of two successive superior highly composite numbers is a prime number. The first few superior highly composite numbers have often been used as radices , due to their high divisibility for their size. For example: Bigger SHCNs can be used in other ways. 120 appears as the long hundred , while 360 appears as the number of degrees in a circle. Number theory Number theory (or arithmetic or higher arithmetic in older usage)

1782-842: The divisor function , denotes the number of divisors of n . The term was coined by Ramanujan (1915). For example, the number with the most divisors per square root of the number itself is 12; this can be demonstrated using some highly composites near 12. 2 2 .5 ≈ 1.414 , 3 4 .5 = 1.5 , 4 6 .5 ≈ 1.633 , 6 12 .5 ≈ 1.732 , 8 24 .5 ≈ 1.633 , 12 60 .5 ≈ 1.549 {\displaystyle {\frac {2}{2^{.5}}}\approx 1.414,{\frac {3}{4^{.5}}}=1.5,{\frac {4}{6^{.5}}}\approx 1.633,{\frac {6}{12^{.5}}}\approx 1.732,{\frac {8}{24^{.5}}}\approx 1.633,{\frac {12}{60^{.5}}}\approx 1.549} 120

1848-713: The early modern period (17th to early 19th centuries). The Chinese remainder theorem appears as an exercise in Sunzi Suanjing (3rd, 4th or 5th century CE). (There is one important step glossed over in Sunzi's solution: it is the problem that was later solved by Āryabhaṭa 's Kuṭṭaka – see below .) The result was later generalized with a complete solution called Da-yan-shu ( 大衍術 ) in Qin Jiushao 's 1247 Mathematical Treatise in Nine Sections which

1914-415: The n -th superior highly composite number s n {\displaystyle s_{n}} holds s n = ∏ i = 1 n π i {\displaystyle s_{n}=\prod _{i=1}^{n}\pi _{i}} The first π i {\displaystyle \pi _{i}} are 2, 3, 2, 5, 2, 3, 7, ... (sequence A000705 in

1980-1255: The Babylonian system. The number system in the Mali Empire was based on 60, reflected in the counting system of the Maasina Fulfulde , a variant of the Fula language spoken in contemporary Mali . The Ekagi of Western New Guinea used base 60, and the sexagenary cycle plays a role in Chinese calendar and numerology. From Polish–Lithuanian Commonwealth in Slavic and Baltic languages 60 has its own name kopa ( Polish : kopa , Belarusian : капа́ , Lithuanian : kapa , Czech : kopa , Russian : копа , Ukrainian : копа́ ), in Germanic languages: German : Schock , Danish : skok , Dutch : schok , Swedish : Skock , Norwegian : Skokk and in Latin : sexagena refer to 60 = 5 dozen = ⁠ 1 / 2 ⁠ small gross . This quantity

2046-530: 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

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2112-645: The Philippines, the United States, and several other countries in the Americas is 60 Hz . An exbibyte (sometimes called exabyte ) is 2 bytes . The Babylonian cuneiform numerals had a base of 60, inherited from the Sumerian and Akkadian civilizations, and possibly motivated by the large number of divisors that 60 has. The sexagesimal measurement of time and of geometric angles is a legacy of

2178-674: The Quran, 60 is mentioned once: "..he should feed sixty indigent ones..", but it is mentioned many times in the Hadith, most notably Muhammad being reported to say, "..Allah, the Exalted and Glorious, created Adam in His own image with His length of sixty cubits.." In Hinduism, the 60th birthday of a man is called Sashti poorthi. A ceremony called Sashti (60) Abda (years) Poorthi (completed) in Sanskrit

2244-671: 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

2310-474: 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 {\displaystyle mX^{2}+nY^{2}} )—defining their equivalence relation, showing how to put them in reduced form, etc. Adrien-Marie Legendre (1752–1833)

2376-636: The caliph Al-Ma'mun ordered translations of many Greek mathematical works and at least one Sanskrit work (the Sindhind , which may or may not 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 – ca. 1029) builds on it to some extent. According to Rashed Roshdi, Al-Karajī's contemporary Ibn al-Haytham knew what would later be called Wilson's theorem . Other than

2442-455: The complex numbers C are an extension of the reals R , and the reals R are an extension of the rationals Q .) Classifying the possible extensions of a given number field is a difficult and partially open problem. Abelian extensions—that is, extensions L of K such that the Galois group Gal( L / K ) of L over K is an abelian group —are relatively well understood. Their classification

2508-432: The definition of e p ( x ) {\displaystyle e_{p}(x)} , 1 / x {\displaystyle 1/x} is analogous to ε {\displaystyle \varepsilon } in the implicit definition of a superior highly composite number. Moreover, for each superior highly composite number s ′ {\displaystyle s'} exists

2574-423: 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 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 {\displaystyle c/a} , presumably for actual use as

2640-503: 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 a + b d {\displaystyle a+b{\sqrt {d}}} , where a {\displaystyle a} and b {\displaystyle b} are rational numbers and d {\displaystyle d}

2706-476: The early Hellenistic period . In the case of number theory, this means, by and large, Plato and Euclid , respectively. While Asian mathematics influenced Greek and Hellenistic learning, it seems to be the case that Greek mathematics is also an indigenous tradition. Eusebius , PE X, chapter 4 mentions of Pythagoras : "In fact the said Pythagoras, while busily studying the wisdom of each nation, visited Babylon, and Egypt, and all Persia, being instructed by

60 (number) - Misplaced Pages Continue

2772-510: The greatest common divisor of two numbers (the Euclidean algorithm ; Elements , Prop. VII.2) and the first known proof of the infinitude of primes ( Elements , Prop. IX.20). In 1773, Lessing published an epigram he had found in a manuscript during his work as a librarian; it claimed to be a letter sent by Archimedes to Eratosthenes . The epigram proposed what has become known as Archimedes's cattle problem ; its solution (absent from

2838-695: The greatest ratio is a superior highly composite number. It is a stronger restriction than that of a highly composite number , which is defined as having more divisors than any smaller positive integer. The first ten superior highly composite numbers and their factorization are listed. For a superior highly composite number n there exists a positive real number ε > 0 such that for all natural numbers k > 1 we have d ( n ) n ε ≥ d ( k ) k ε {\displaystyle {\frac {d(n)}{n^{\varepsilon }}}\geq {\frac {d(k)}{k^{\varepsilon }}}} where d ( n ) ,

2904-516: 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 ) that encode properties of the integers, primes or other number-theoretic objects in some fashion ( analytic number theory ). One may also study real numbers in relation to rational numbers; for example, as approximated by

2970-873: The lack of unique factorisation in algebraic number fields. (For example, in the field generated by the rationals and − 5 {\displaystyle {\sqrt {-5}}} , the number 6 {\displaystyle 6} can be factorised both as 6 = 2 ⋅ 3 {\displaystyle 6=2\cdot 3} and 6 = ( 1 + − 5 ) ( 1 − − 5 ) {\displaystyle 6=(1+{\sqrt {-5}})(1-{\sqrt {-5}})} ; all of 2 {\displaystyle 2} , 3 {\displaystyle 3} , 1 + − 5 {\displaystyle 1+{\sqrt {-5}}} and 1 − − 5 {\displaystyle 1-{\sqrt {-5}}} are irreducible, and thus, in

3036-473: The latter ( Diophantine approximation ). The older term for number theory is arithmetic . By the early twentieth century, it had been superseded by number theory . (The word arithmetic is used by the general public to mean " elementary calculations "; it has also acquired other meanings in mathematical logic , as in Peano arithmetic , and computer science , as in floating-point arithmetic .) The use of

3102-405: The lines later developed fully by Gauss. In his old age, he was the first to prove Fermat's Last Theorem for n = 5 {\displaystyle n=5} (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

3168-509: 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 the Indian school . It is not known whether Archimedes himself had a method of solution. 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

3234-426: The most important tools of analytic number theory are the circle method , sieve methods and L-functions (or, rather, the study of their properties). The theory of modular forms (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 numbers , and use analytic means to answer such questions; it

3300-694: The most interesting questions in each area remain open and are being actively worked on. The term elementary generally denotes a method that does not use complex analysis . For example, the prime number theorem was first proven using complex analysis in 1896, but an elementary proof was found only in 1949 by Erdős 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 , rather than complex analysis as such. Here as elsewhere, an elementary proof may be longer and more difficult for most readers than

3366-436: The one hand, and lengths and proportions (which may be identified 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 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

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3432-755: The roots of the subject. This is an example of a general procedure in analytic number theory: deriving information about the distribution of a sequence (here, prime ideals or prime numbers) from the analytic behavior of an appropriately constructed complex-valued function. An algebraic number is any complex number that is a solution to some polynomial equation f ( x ) = 0 {\displaystyle f(x)=0} with rational coefficients; for example, every solution x {\displaystyle x} of x 5 + ( 11 / 2 ) x 3 − 7 x 2 + 9 = 0 {\displaystyle x^{5}+(11/2)x^{3}-7x^{2}+9=0} (say)

3498-568: The second rather than the first definition: some of sieve theory, for instance, uses little analysis, yet it does belong to analytic number theory. The following are examples of problems in analytic number theory: the prime number theorem , the Goldbach conjecture (or the twin prime conjecture , or the Hardy–Littlewood conjectures ), the Waring problem and the Riemann hypothesis . Some of

3564-399: 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: 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

3630-456: 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

3696-402: The term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is commonly preferred as an adjective to number-theoretic . 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

3762-518: 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

3828-540: The thirteen books of Diophantus's Arithmetica survive in the original Greek and four more survive in an Arabic translation. The Arithmetica is a collection of worked-out problems where the task is invariably to find rational solutions to a system of polynomial equations, usually of the form f ( x , y ) = z 2 {\displaystyle f(x,y)=z^{2}} or f ( x , y , z ) = w 2 {\displaystyle f(x,y,z)=w^{2}} . Thus, nowadays,

3894-423: The work of Bernhard Riemann (1859) on the zeta function 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 forms ). 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

3960-547: Was exceptionally well developed and included the foundations of modern elementary algebra . Late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt . In book nine of Euclid's Elements , propositions 21–34 are very probably influenced by Pythagorean teachings ; it is very simple material ("odd times even

4026-452: 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). 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 . In the early ninth century,

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4092-402: 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 {\displaystyle ax^{2}+by^{2}+cz^{2}=0} and worked on quadratic forms along

4158-524: Was the object of the programme of class field theory , 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. The central problem of Diophantine geometry

4224-400: Was thus arguably a pioneer in the study of number systems . (Book X of Euclid's Elements 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

4290-458: Was translated into English in early 19th century by British missionary Alexander Wylie . There is also some numerical mysticism in Chinese mathematics, but, unlike that of the Pythagoreans, it seems to have led nowhere. 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

4356-467: Was used in international medieval treaties e.g. for ransom of captured Teutonic Knights . 60 occurs several times in the Bible ; for example, as the age of Isaac when Jacob and Esau were born, and the number of warriors escorting King Solomon . In the laws of kashrut of Judaism , 60 is the proportion (60:1) of kosher to non-kosher ingredients that can render an admixture kosher post-facto. In

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