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Hilbert's ninth problem. Hilbert's ninth problem, from the list of 23 Hilbert's problems (1900), asked to find the most general reciprocity law for the norm residues of k -th order in a general algebraic number field, where k is a power of a prime .
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 ...
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.
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 ...
In mathematics, Hilbert's program, formulated by German mathematician David Hilbert in the early 1920s, [1] was a proposed solution to the foundational crisis of mathematics, when early attempts to clarify the foundations of mathematics were found to suffer from paradoxes and inconsistencies. As a solution, Hilbert proposed to ground all ...
Hilbert's 2nd and 10th problems introduced the "Entscheidungsproblem" (the "decision problem"). In his 2nd problem he asked for a proof that "arithmetic" is "consistent". Kurt Gödel would prove in 1931 that, within what he called "P" (nowadays called Peano Arithmetic), "there exist undecidable sentences [propositions]".
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 ...
Hilary Whitehall Putnam (/ ˈpʌtnəm /; July 31, 1926 – March 13, 2016) was an American philosopher, mathematician, computer scientist, and figure in analytic philosophy in the second half of the 20th century. He contributed to the studies of philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science. [5]