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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 problems ranged greatly in topic and precision. Some of them, like the 3rd problem, which was the first to be solved, or the 8th problem (the Riemann hypothesis), which still remains unresolved, were presented precisely enough to enable a clear affirmative or negative answer.
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 ]
Hilbert proposed that the consistency of more complicated systems, such as real analysis, could be proven in terms of simpler systems. Ultimately, the consistency of all of mathematics could be reduced to basic arithmetic. Gödel's incompleteness theorems, published in 1931, showed that Hilbert's program was unattainable for key areas of ...
Hilbert's second problem; Hilbert's third problem; Hilbert's fourth problem; Hilbert's fifth problem; No small subgroup; Hilbert's sixth problem; Hilbert's seventh problem; Hilbert's eighth problem; Hilbert's ninth problem; Hilbert's tenth problem; Hilbert's eleventh problem; Hilbert's twelfth problem; Hilbert's thirteenth problem; Hilbert's ...
In 1963-1964 he completed 10th grade at the Moscow State University physics and mathematics boarding school No. 18 named after A. N. Kolmogorov. [ 1 ] [ 2 ] In 1964, he won a gold medal at the International Mathematical Olympiad [ 3 ] and was enrolled in the Mathematics and Mechanics Department of St. Petersburg State University without exams.
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