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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.
Hilbert's tenth problem does not ask whether there exists an algorithm for deciding the solvability of Diophantine equations, but rather asks for the construction of such an algorithm: "to devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers". That this ...
In 1972, at the age of 25, he defended his doctoral dissertation on the unsolvability of Hilbert's tenth problem. [7] From 1974 Matiyasevich worked in scientific positions at LOMI, first as a senior researcher, in 1980 he headed the Laboratory of Mathematical Logic.
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.
Franzén introduces Hilbert's tenth problem and the MRDP theorem (Matiyasevich-Robinson-Davis-Putnam theorem) which states that "no algorithm exists which can decide whether or not a Diophantine equation has any solution at all". MRDP uses the undecidability proof of Turing: "... the set of solvable Diophantine equations is an example of a ...
The difficulty of solving Diophantine equations is illustrated by Hilbert's tenth problem, which was set in 1900 by David Hilbert; it was to find an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. Matiyasevich's theorem implies that such an algorithm cannot exist.
Julia Hall Bowman Robinson (December 8, 1919 – July 30, 1985) was an American mathematician noted for her contributions to the fields of computability theory and computational complexity theory—most notably in decision problems. Her work on Hilbert's tenth problem (now known as Matiyasevich's theorem or the MRDP theorem) played a crucial ...
Franzén (2005) explains how Matiyasevich's solution to Hilbert's 10th problem can be used to obtain a proof to Gödel's first incompleteness theorem. [11] Matiyasevich proved that there is no algorithm that, given a multivariate polynomial p(x 1, x 2,...,x k) with integer coefficients, determines whether there is an integer solution to the ...