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84-432: Al-Karaji wrote on mathematics and engineering. Some consider him to be merely reworking the ideas of others (he was influenced by Diophantus ) but most regard him as more original, in particular for the beginnings of freeing algebra from geometry. Among historians, his most widely studied work is his algebra book al-fakhri fi al-jabr wa al-muqabala , which survives from the medieval era in at least four copies. He expounded

168-435: A {\displaystyle c/a} , presumably for actual use as a "table", for example, with a view to applications. It is not known what these applications may have been, or whether there could have been any; Babylonian astronomy , 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

252-661: 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

336-491: 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 ≡

420-719: A comprehensive commentary written by the earlier Greek scholar Maximos Planudes (1260 – 1305), who produced an edition of Diophantus within the library of the Chora Monastery in Byzantine Constantinople . In addition, some portion of the Arithmetica probably survived in the Arab tradition (see above). In 1463 German mathematician Regiomontanus wrote: No one has yet translated from the Greek into Latin

504-507: A cube would be expressed, in words rather than in numbers, as a square-cube, the numerical property of adding exponents was not clear. His work on algebra and polynomials gave the rules for arithmetic operations for adding, subtracting and multiplying polynomials; though he was restricted to dividing polynomials by monomials. F. Woepcke was the first historian to realise the importance of al-Karaji's work and later historians mostly agree with his interpretation. He praised Al-Karaji for being

588-498: 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 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

672-455: 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

756-529: 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 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,

840-407: 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,

924-473: 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

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1008-424: A symbol for a general number n . Where we would write ⁠ 12 + 6 n / n − 3 ⁠ , Diophantus has to resort to constructions like: "... a sixfold number increased by twelve, which is divided by the difference by which the square of the number exceeds three". Algebra still had a long way to go before very general problems could be written down and solved succinctly. "But what we really want to know

1092-637: 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

1176-537: A value of 84 years. However, the accuracy of the information cannot be confirmed. In popular culture, this puzzle was the Puzzle No.142 in Professor Layton and Pandora's Box as one of the hardest solving puzzles in the game, which needed to be unlocked by solving other puzzles first. Arithmetica is the major work of Diophantus and the most prominent work on premodern algebra in Greek mathematics. It

1260-408: 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:

1344-572: Is a collection of problems giving numerical solutions of both determinate and indeterminate equations . Of the original thirteen books of which Arithmetica consisted only six have survived, though there are some who believe that four Arabic books discovered in 1968 are also by Diophantus. Some Diophantine problems from Arithmetica have been found in Arabic sources. It should be mentioned here that Diophantus never used general methods in his solutions. Hermann Hankel , renowned German mathematician made

1428-485: 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

1512-479: Is also known to have written on polygonal numbers , a topic of great interest to Pythagoras and Pythagoreans . Fragments of a book dealing with polygonal numbers are extant. A book called Preliminaries to the Geometric Elements has been traditionally attributed to Hero of Alexandria . It has been studied recently by Wilbur Knorr , who suggested that the attribution to Hero is incorrect, and that

1596-441: 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

1680-453: Is based on algebra. How much he affected India is a matter of debate. Diophantus has been considered "the father of algebra" because of his contributions to number theory, mathematical notations and the earliest known use of syncopated notation in his book series Arithmetica . However this is usually debated, because Al-Khwarizmi was also given the title as "the father of algebra", nevertheless both mathematicians were responsible for paving

1764-459: Is believed that Fermat did not actually have the proof he claimed to have. Although the original copy in which Fermat wrote this is lost today, Fermat's son edited the next edition of Diophantus, published in 1670. Even though the text is otherwise inferior to the 1621 edition, Fermat's annotations—including the "Last Theorem"—were printed in this version. Fermat was not the first mathematician so moved to write in his own marginal notes to Diophantus;

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1848-589: 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 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

1932-433: 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

2016-412: Is entirely lost. Although The Porisms is lost, we know three lemmas contained there, since Diophantus refers to them in the Arithmetica . One lemma states that the difference of the cubes of two rational numbers is equal to the sum of the cubes of two other rational numbers, i.e. given any a and b , with a > b , there exist c and d , all positive and rational, such that Diophantus

2100-470: Is irrational 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

2184-653: Is only survived by the Plimpton 322 tablet, some authors assert that Babylonian algebra 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

2268-498: Is that he did not have any notion for zero and he avoided negative coefficients by considering the given numbers a , b , c to all be positive in each of the three cases above. Diophantus was always satisfied with a rational solution and did not require a whole number which means he accepted fractions as solutions to his problems. Diophantus considered negative or irrational square root solutions "useless", "meaningless", and even "absurd". To give one specific example, he calls

2352-571: Is the earliest extant proof of the sum formula for integral cubes . Diophantus Diophantus of Alexandria (born c.  AD 200  – c.  214 ; died c.  AD 284  – c.  298 ) was a Greek mathematician , who was the author of two main works: On Polygonal Numbers , which survives incomplete, and the Arithmetica in thirteen books, most of it extant, made up of arithmetical problems that are solved through algebraic equations . His Arithmetica influenced

2436-478: 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 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

2520-462: 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

2604-510: Is to what extent the Alexandrian mathematicians of the period from the first to the fifth centuries C.E. were Greek. Certainly, all of them wrote in Greek and were part of the Greek intellectual community of Alexandria. And most modern studies conclude that the Greek community coexisted [...] So should we assume that Ptolemy and Diophantus, Pappus and Hypatia were ethnically Greek, that their ancestors had come from Greece at some point in

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2688-438: Is unreasonable to portray them with purely European features when no physical descriptions exist." "Diophantos was most likely a Hellenized Babylonian." Number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions . German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics

2772-453: 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 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

2856-425: Is very simple material ("odd times even 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}}}

2940-510: The Byzantine scholar John Chortasmenos (1370–1437) had written "Thy soul, Diophantus, be with Satan because of the difficulty of your other theorems and particularly of the present theorem" next to the same problem. Diophantus wrote several other books besides Arithmetica , but only a few of them have survived. Diophantus himself refers to a work which consists of a collection of lemmas called The Porisms (or Porismata ), but this book

3024-401: The basic principles of hydrology and this book reveals his profound knowledge of this science and has been described as the oldest extant text in this field. He systematically studied the algebra of exponents, and was the first to define the rules for monomials like x,x²,x³ and their reciprocals in the cases of multiplication and division. However, since for example the product of a square and

3108-474: 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)

3192-577: 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

3276-455: 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

3360-399: The development of algebra by Arabs, and his equations influenced modern work in both abstract algebra and computer science . The first five books of his work are purely algebraic. Furthermore, recent studies of Diophantus's work have revealed that the method of solution taught in his Arithmetica matches later medieval Arabic algebra in its concepts and overall procedure. Diophantus was

3444-503: 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}

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3528-522: The earliest, the Arithmetica has the best-known use of algebraic notation to solve arithmetical problems coming from Greek antiquity, and some of its problems served as inspiration for later mathematicians working in analysis and number theory . In modern use, Diophantine equations are algebraic equations with integer coefficients for which integer solutions are sought. Diophantine geometry and Diophantine approximations are two other subareas of number theory that are named after him. Diophantus

3612-650: The equation 4 = 4 x + 20 'absurd' because it would lead to a negative value for x . One solution was all he looked for in a quadratic equation. There is no evidence that suggests Diophantus even realized that there could be two solutions to a quadratic equation. He also considered simultaneous quadratic equations. Diophantus made important advances in mathematical notation, becoming the first person known to use algebraic notation and symbolism. Before him everyone wrote out equations completely. Diophantus introduced an algebraic symbolism that used an abridged notation for frequently occurring operations, and an abbreviation for

3696-471: The first Greek mathematician who recognized positive rational numbers as numbers, by allowing fractions for coefficients and solutions. He coined the term παρισότης ( parisotēs ) to refer to an approximate equality. This term was rendered as adaequalitas in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves. Although not

3780-416: The first Latin edition that was widely available. Pierre de Fermat owned a copy, studied it and made notes in the margins. A later 1895 Latin translation by Paul Tannery was said to be an improvement by Thomas L. Heath , who used it in the 1910 second edition of his English translation. The 1621 edition of Arithmetica by Bachet gained fame after Pierre de Fermat wrote his famous " Last Theorem " in

3864-394: 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 which is implicit in routine Old Babylonian exercises. If some other method was used, the triples were first constructed and then reordered by c /

3948-493: The first who introduced the theory of algebraic calculus. Al-Karaji gave the first formulation of the binomial coefficients and the first description of Pascal's triangle . He is also credited with the discovery of the binomial theorem. In a now lost work known only from subsequent quotation by al-Samaw'al , Al-Karaji introduced the idea of argument by mathematical induction . As Katz says Another important idea introduced by al-Karaji and continued by al-Samaw'al and others

4032-451: The following remark regarding Diophantus: Our author (Diophantos) not the slightest trace of a general, comprehensive method is discernible; each problem calls for some special method which refuses to work even for the most closely related problems. For this reason it is difficult for the modern scholar to solve the 101st problem even after having studied 100 of Diophantos's solutions. Like many other Greek mathematical treatises, Diophantus

4116-580: The founding of Alexandria, small numbers of Egyptians were admitted to the privileged classes in the city to fulfill numerous civic roles. Of course, it was essential in such cases for the Egyptians to become "Hellenized," to adopt Greek habits and the Greek language. Given that the Alexandrian mathematicians mentioned here were active several hundred years after the founding of the city, it would seem at least equally possible that they were ethnically Egyptian as that they remained ethnically Greek. In any case, it

4200-510: 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

4284-873: 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

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4368-405: 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

4452-509: 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

4536-455: The margins of his copy: If an integer n is greater than 2, then a + b = c has no solutions in non-zero integers a , b , and c . I have a truly marvelous proof of this proposition which this margin is too narrow to contain. Fermat's proof was never found, and the problem of finding a proof for the theorem went unsolved for centuries. A proof was finally found in 1994 by Andrew Wiles after working on it for seven years. It

4620-490: The mathematics of Classical Greece is known to us either through the reports of contemporary non-mathematicians or through mathematical works from the early Hellenistic period . In the case of number theory, this means, by and large, Plato and Euclid , respectively. While Asian mathematics influenced Greek and Hellenistic learning, it seems to be the case that Greek mathematics is also an indigenous tradition. Eusebius , PE X, chapter 4 mentions of Pythagoras : "In fact

4704-426: 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

4788-694: 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

4872-406: The necessary notation to express more general methods. This caused his work to be more concerned with particular problems rather than general situations. Some of the limitations of Diophantus' notation are that he only had notation for one unknown and, when problems involved more than a single unknown, Diophantus was reduced to expressing "first unknown", "second unknown", etc. in words. He also lacked

4956-436: The past but had remained effectively isolated from the Egyptians? It is, of course, impossible to answer this question definitively. But research in papyri dating from the early centuries of the common era demonstrates that a significant amount of intermarriage took place between the Greek and Egyptian communities [...] And it is known that Greek marriage contracts increasingly came to resemble Egyptian ones. In addition, even from

5040-399: The rationals—the subjects of arithmetic), on 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.,

5124-755: 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)

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5208-529: 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

5292-568: 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

5376-503: The stone tells how old: 'God gave him his boyhood one-sixth of his life, One twelfth more as youth while whiskers grew rife; And then yet one-seventh ere marriage begun; In five years there came a bouncing new son. Alas, the dear child of master and sage After attaining half the measure of his father's life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.' This puzzle implies that Diophantus' age x can be expressed as which gives x

5460-493: 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 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

5544-523: The study of the sums of triangular and pentagonal numbers would prove fruitful in 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

5628-399: 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

5712-456: 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

5796-518: 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

5880-540: 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,

5964-442: The thirteen books of Diophantus, in which the very flower of the whole of arithmetic lies hidden. Arithmetica was first translated from Greek into Latin by Bombelli in 1570, but the translation was never published. However, Bombelli borrowed many of the problems for his own book Algebra . The editio princeps of Arithmetica was published in 1575 by Xylander . The Latin translation of Arithmetica by Bachet in 1621 became

6048-456: The true author is Diophantus. Diophantus' work has had a large influence in history. Editions of Arithmetica exerted a profound influence on the development of algebra in Europe in the late sixteenth and through the 17th and 18th centuries. Diophantus and his works also influenced Arab mathematics and were of great fame among Arab mathematicians. Diophantus' work created a foundation for work on algebra and in fact much of advanced mathematics

6132-418: The two basic components of a modern argument by induction, namely the truth of the statement for n = 1 (1 = 1) and the deriving of the truth for n = k from that of n = k - 1. Of course, this second component is not explicit since, in some sense, al-Karaji's argument is in reverse; this is, he starts from n = 10 and goes down to 1 rather than proceeding upward. Nevertheless, his argument in al-Fakhri

6216-481: The unknown and for the powers of the unknown. Mathematical historian Kurt Vogel states: The symbolism that Diophantus introduced for the first time, and undoubtedly devised himself, provided a short and readily comprehensible means of expressing an equation... Since an abbreviation is also employed for the word 'equals', Diophantus took a fundamental step from verbal algebra towards symbolic algebra. Although Diophantus made important advances in symbolism, he still lacked

6300-642: The way for algebra today. Today, Diophantine analysis is the area of study where integer (whole-number) solutions are sought for equations, and Diophantine equations are polynomial equations with integer coefficients to which only integer solutions are sought. It is usually rather difficult to tell whether a given Diophantine equation is solvable. Most of the problems in Arithmetica lead to quadratic equations . Diophantus looked at 3 different types of quadratic equations: ax + bx = c , ax = bx + c , and ax + c = bx . The reason why there were three cases to Diophantus, while today we have only one case,

6384-582: 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

6468-423: 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

6552-531: Was born into a Greek family and is known to have lived in Alexandria , Egypt , during the Roman era , between AD 200 and 214 to 284 or 298. Much of our knowledge of the life of Diophantus is derived from a 5th-century Greek anthology of number games and puzzles created by Metrodorus . One of the problems (sometimes called his epitaph) states: Here lies Diophantus, the wonder behold. Through art algebraic,

6636-452: 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,

6720-698: Was forgotten in Western Europe during the Dark Ages , since the study of ancient Greek, and literacy in general, had greatly declined. The portion of the Greek Arithmetica that survived, however, was, like all ancient Greek texts transmitted to the early modern world, copied by, and thus known to, medieval Byzantine scholars. Scholia on Diophantus by the Byzantine Greek scholar John Chortasmenos (1370–1437) are preserved together with

6804-583: 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 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,

6888-462: Was that of an inductive argument for dealing with certain arithmetic sequences. Thus al-Karaji used such an argument to prove the result on the sums of integral cubes already known to Aryabhata [...] Al-Karaji did not, however, state a general result for arbitrary n . He stated his theorem for the particular integer 10 [...] His proof, nevertheless, was clearly designed to be extendable to any other integer. [...] Al-Karaji's argument includes in essence

6972-402: 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

7056-524: 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

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