Ad
related to: nonlinear diophantine equations
Search results
Results From The WOW.Com Content Network
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.
solution of linear Diophantine systems, i.e. equation systems where the coefficient matrix and the right hand side are integer valued and an integer solution is sought; this is a special but important case of Hilbert's tenth problem, the only one in practice soluble; solution of nonlinear algebraic equations;
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.
A Diophantine equation, in general, is one where the solutions are restricted to some algebraic system, typically integers. (In another usage ) Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made initial studies of integer Diophantine equations.
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 ...
Diophantine approximations and transcendental number theory are very close areas that share many theorems and methods. Diophantine approximations also have important applications in the study of Diophantine equations. The 2022 Fields Medal was awarded to James Maynard, in part for his work on Diophantine approximation.
Matiyasevich's theorem, also called the Matiyasevich–Robinson–Davis–Putnam or MRDP theorem, says: . Every computably enumerable set is Diophantine, and the converse.. A set S of integers is computably enumerable if there is an algorithm such that: For each integer input n, if n is a member of S, then the algorithm eventually halts; otherwise it runs forever.