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Diophantus And Diophantine Equations
''Diophantus and Diophantine Equations'' is a book in the history of mathematics, on the history of Diophantine equations and their solution by Diophantus of Alexandria. It was originally written in Russian by Isabella Bashmakova, and published by Nauka in 1972 under the title ''Диофант и диофантовы уравнения''. It was translated into German by Ludwig Boll as ''Diophant und diophantische Gleichungen'' (Birkhäuser, 1974) and into English by Abe Shenitzer as ''Diophantus and Diophantine Equations'' (Dolciani Mathematical Expositions 20, Mathematical Association of America, 1997). Topics In the sense considered in the book, a Diophantine equation is an equation written using polynomials whose coefficients are rational numbers. These equations are to be solved by finding rational-number values for the variables that, when plugged into the equation, make it become true. Although there is also a well-developed theory of integer (rather than rational) solution ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
History Of Mathematics
The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the patterns in nature, the field of astronomy and to record time and formulate calendars. The earliest mathematical texts available are from Mesopotamia and Egypt – '' Plimpton 322'' ( Babylonian c. 2000 – 1900 BC), the ''Rhind Mathematical Papyrus'' ( Egyptian c. 1800 BC) and the '' Moscow Mathematical Papyrus'' (Egyptian c. 1890 BC). All of these texts mention the so-called Pythagorean triples, so, by inference, the Pythagorean theorem seems to be the most anci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Genus Of An Algebraic Curve
In mathematics, genus (plural genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a Surface (topology), surface. A sphere has genus 0, while a torus has genus 1. Topology Orientable surfaces The genus of a Connected space, connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting Curve#Topological_curve, closed simple curves without rendering the resultant manifold disconnected. It is equal to the number of Handle (mathematics), handles on it. Alternatively, it can be defined in terms of the Euler characteristic ''χ'', via the relationship ''χ'' = 2 − 2''g'' for Surface_(topology)#Closed_surfaces, closed surfaces, where ''g'' is the genus. For surfaces with ''b'' Boundary (topology), boundary components, the equation reads ''χ'' = 2 − 2''g'' − ''b''. In layman's terms, it's the number of "holes" an object has ("holes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Reviews
''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also publishes an associated online bibliographic database called MathSciNet which contains an electronic version of ''Mathematical Reviews'' and additionally contains citation information for over 3.5 million items as of 2018. Reviews Mathematical Reviews was founded by Otto E. Neugebauer in 1940 as an alternative to the German journal ''Zentralblatt für Mathematik'', which Neugebauer had also founded a decade earlier, but which under the Nazis had begun censoring reviews by and of Jewish mathematicians. The goal of the new journal was to give reviews of every mathematical research publication. As of November 2007, the ''Mathematical Reviews'' database contained information on over 2.2 million articles. The authors of reviews are volunteers, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of 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 the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been known since antiquity to have infinitely many solutions.Singh, pp. 18–20. The proposition was first stated as a theorem by Pierre de Fermat around 1637 in the margin of a copy of '' Arithmetica''. Fermat added that he had a proof that was too large to fit in the margin. Although other statements claimed by Fermat without proof were subsequently proven by others and credited as theorems of Fermat (for example, Fermat's theorem on sums of two squares), Fermat's Last Theorem resisted proof, leading to doubt that Fermat ever had a correct proof. Consequently the proposition became known as a conjecture rather than a theorem. After 358 years of effort by mathematicians, the first successful proof was released in 1994 by Andrew Wiles and form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph H
Joseph is a common male given name, derived from the Hebrew Yosef (יוֹסֵף). "Joseph" is used, along with "Josef", mostly in English, French and partially German languages. This spelling is also found as a variant in the languages of the modern-day Nordic countries. In Portuguese and Spanish, the name is "José". In Arabic, including in the Quran, the name is spelled '' Yūsuf''. In Persian, the name is "Yousef". The name has enjoyed significant popularity in its many forms in numerous countries, and ''Joseph'' was one of the two names, along with ''Robert'', to have remained in the top 10 boys' names list in the US from 1925 to 1972. It is especially common in contemporary Israel, as either "Yossi" or "Yossef", and in Italy, where the name "Giuseppe" was the most common male name in the 20th century. In the first century CE, Joseph was the second most popular male name for Palestine Jews. In the Book of Genesis Joseph is Jacob's eleventh son and Rachel's first son, and k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypatia
Hypatia, Koine pronunciation (born 350–370; died 415 AD) was a neoplatonist philosopher, astronomer, and mathematician, who lived in Alexandria, Egypt, then part of the Eastern Roman Empire. She was a prominent thinker in Alexandria where she taught philosophy and astronomy. Although preceded by Pandrosion, another Alexandrine female mathematician, she is the first female mathematician whose life is reasonably well recorded. Hypatia was renowned in her own lifetime as a great teacher and a wise counselor. She wrote a commentary on Diophantus's thirteen-volume '' Arithmetica'', which may survive in part, having been interpolated into Diophantus's original text, and another commentary on Apollonius of Perga's treatise on conic sections, which has not survived. Many modern scholars also believe that Hypatia may have edited the surviving text of Ptolemy's ''Almagest'', based on the title of her father Theon's commentary on Book III of the ''Almagest''. Hypatia constructed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermat's Theorem On Sums Of Two Squares
In additive number theory, Fermat's theorem on sums of two squares states that an odd prime ''p'' can be expressed as: :p = x^2 + y^2, with ''x'' and ''y'' integers, if and only if :p \equiv 1 \pmod. The prime numbers for which this is true are called Pythagorean primes. For example, the primes 5, 13, 17, 29, 37 and 41 are all congruent to 1 modulo 4, and they can be expressed as sums of two squares in the following ways: :5 = 1^2 + 2^2, \quad 13 = 2^2 + 3^2, \quad 17 = 1^2 + 4^2, \quad 29 = 2^2 + 5^2, \quad 37 = 1^2 + 6^2, \quad 41 = 4^2 + 5^2. On the other hand, the primes 3, 7, 11, 19, 23 and 31 are all congruent to 3 modulo 4, and none of them can be expressed as the sum of two squares. This is the easier part of the theorem, and follows immediately from the observation that all squares are congruent to 0 or 1 modulo 4. Since the Diophantus identity implies that the product of two integers each of which can be written as the sum of two squares is itself expressible as t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free in .) It is always understood that the curve is really sitting in the projective plane, with the point being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free in .) It is always understood that the curve is really sitting in the projective plane, with the point being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Plane Curve
In mathematics, a cubic plane curve is a plane algebraic curve defined by a cubic equation : applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an equation. Here is a non-zero linear combination of the third-degree monomials : These are ten in number; therefore the cubic curves form a projective space of dimension 9, over any given field . Each point imposes a single linear condition on , if we ask that pass through . Therefore, we can find some cubic curve through any nine given points, which may be degenerate, and may not be unique, but will be unique and non-degenerate if the points are in general position; compare to two points determining a line and how five points determine a conic. If two cubics pass through a given set of nine points, then in fact a pencil of cubics does, and the points satisfy additional properties; see Cayley–Bacharach theorem. A cubic curve may ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Secant Line
Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to: * a secant line, in geometry * the secant variety, in algebraic geometry * secant (trigonometry) (Latin: secans), the multiplicative inverse (or reciprocal) trigonometric function of the cosine * the secant method, a root-finding algorithm in numerical analysis, based on secant lines to graphs of functions * a secant ogive Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to: * a secant line, in geometry * the secant variety, in algebraic geometry * secant (trigonometry) (Latin: secans), the multiplicative inverse (or reciproc ... in nose cone design {{mathdab sr:Секанс ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
