In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, for which only integer solutions are of interest. A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents.
Diophantus of Alexandria 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.
Number theory 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, or defined as generalizations of the integers.
In mathematics, a Diophantine equation is an equation of the form P(x1, ..., xj, y1, ..., yk) = 0 (usually abbreviated P(x, y) = 0) where P(x, y) is a polynomial with integer coefficients, where x1, ..., xj indicate parameters and y1, ..., yk indicate unknowns.
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.
In mathematics, Roth's theorem or Thue–Siegel–Roth theorem is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that algebraic numbers cannot have many rational approximations that are 'very good'. Over half a century, the meaning of very good here was refined by a number of mathematicians, starting with Joseph Liouville in 1844 and continuing with work of Axel Thue, Carl Ludwig Siegel, Freeman Dyson, and Klaus Roth.
Yuri Ivanovich Manin was a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics.
Arithmetica is an Ancient Greek text on mathematics written by the mathematician Diophantus in the 3rd century AD. It is a collection of 130 algebraic problems giving numerical solutions of determinate equations and indeterminate equations.
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality.
A height function is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of solutions to Diophantine equations and are typically functions from a set of points on algebraic varieties to the real numbers.
Ivan Georgiyevich Petrovsky was a Soviet mathematician working mainly in the field of partial differential equations. He greatly contributed to the solution of Hilbert's 19th and 16th problems, and discovered what are now called Petrovsky lacunas. He also worked on the theories of boundary value problems, probability, and on the topology of algebraic curves and surfaces.
In mathematics, the subspace theorem says that points of small height in projective space lie in a finite number of hyperplanes. It is a result obtained by Wolfgang M. Schmidt.
Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the theory of equations. For example, the fundamental theorem of algebra belongs to the theory of equations and is not, nowadays, considered as belonging to algebra.
The eighth problem of the second book of Arithmetica by Diophantus is to divide a square into a sum of two squares.
This is a timeline of the theory of abelian varieties in algebraic geometry, including elliptic curves.
Adolph-Andrei Pavlovich Yushkevich was a Soviet historian of mathematics, leading expert in medieval mathematics of the East and the work of Leonhard Euler. He is a winner of George Sarton Medal by the History of Science Society for a lifetime of scholarly achievement.
Tatyana Olegovna Shaposhnikova is a Russian-born Swedish mathematician. She is best known for her work in the theory of multipliers in function spaces, partial differential operators and history of mathematics, some of which was partly done jointly with Vladimir Maz'ya. She is also a translator of both scientific and literary texts.
Karl Mikhailovich Peterson was a Russian mathematician, known by an earlier formulation of the Gauss–Codazzi 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 and into English by Abe Shenitzer as Diophantus and Diophantine Equations.
Laura Toti Rigatelli (1941-2023) was an Italian historian of mathematics, founder of the Center for Medieval Mathematics at the University of Siena, biographer of Évariste Galois, and author of many books on the history of mathematics.
