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This is a linear Diophantine equation, related to Bézout's identity. + = + The smallest nontrivial solution in positive integers is 12 3 + 1 3 = 9 3 + 10 3 = 1729.It was famously given as an evident property of 1729, a taxicab number (also named Hardy–Ramanujan number) by Ramanujan to Hardy while meeting in 1917. [1]
A Diophantine equation = is called partition regular if the collection of all infinite subsets of containing a solution is partition regular. Rado's theorem characterises exactly which systems of linear Diophantine equations A x = 0 {\displaystyle \mathbf {A} \mathbf {x} =\mathbf {0} } are partition regular.
Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm that, for any given Diophantine equation (a polynomial equation with integer coefficients and a finite number of unknowns), can decide whether the equation has a solution with all unknowns taking integer values.
Then, it has been applied to solve different kinds of problems belonging to different fields, such as: 1) nonlinear functions inversion and intersection, 2) extensive sampling data generation with assigned analytical (or numerical) distribution, 3) find approximate solutions of nonlinear Diophantine equations, and 4) iso-surface identification ...
When searching for integer solutions, the equation a 2 + b 2 = c 2 is a Diophantine equation. Thus Pythagorean triples are among the oldest known solutions of a nonlinear Diophantine equation. Examples
In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study these equations. [1] Diophantine geometry is part of the broader field of arithmetic geometry.
A height function in Diophantine geometry quantifies the size of solutions to Diophantine equations. [17] Hilbertian fields A Hilbertian field K is one for which the projective spaces over K are not thin sets in the sense of Jean-Pierre Serre. This is a geometric take on Hilbert's irreducibility theorem which shows the rational numbers are ...
Diophantine approximation. Lecture Notes in Mathematics 785. Springer. (1980 [1996 with minor corrections]) (Pages 125-128 and 283–285) Wolfgang M. Schmidt. "Chapter I: Siegel's Lemma and Heights" (pages 1–33). Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, Springer Verlag 2000.