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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.
As with the Hilbert problems, one of the prize problems (the Poincaré conjecture) was solved relatively soon after the problems were announced. The Riemann hypothesis is noteworthy for its appearance on the list of Hilbert problems, Smale's list, the list of Millennium Prize Problems, and even the Weil conjectures, in its geometric guise.
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 ...
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]". [ 4 ]
The Riemann Hypothesis. Today’s mathematicians would probably agree that the Riemann Hypothesis is the most significant open problem in all of math. It’s one of the seven Millennium Prize ...
In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane. Several existence theorems for Riemann–Hilbert problems have been produced by Mark Krein , Israel Gohberg and others.