## Hilbert's ProblemsMadison, Bailey, Darien, Ryan

David Hilbert was a German mathematician who reduced geometry to a series of statements that substantially contributed to the foundation of mathematics.

Hilbert established a list of 23 problems with math.

Problem 1 - Cantor's problem of the cardinal number of the continuum. Has been proven impossible to prove or disprove.

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Problem 2 - The compatibility of the arithmetic axioms. Can not be proven.

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Problem 3 - The equality of two volumes of two tetrahedra of equal bases and equal altitudes. Has been disproven using the Dean invariants.

Problem 4 - Problem of the straight line as the shortest distance between two points. There is not enough detail to be resolved.

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Problem 5 - Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group. (i.e., are continuous groups automatically differential groups?). Resolved by Sndrew Gleason based on interpretation.

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Problem 6 - Mathematical treatment of the axioms of physics. Partially resolved based on interpretation.

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Problem 7 - Irrationality and transcendence of certain numbers.

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Problem 8 - Problems (with the distribution) of prime numbers. Proven by Gelfonds theorem.

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Problem 9 - Proof of the most general law of reciprocity in any number field. Unresolved.

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Problem 10 - Determination of the solvability of a diophantine equation. Proven impossible by Matiyasevichs theorem.

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Problem 11 - Quadratic forms with any algebraic numerical coefficients. Partially resolved.

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Problem 12 - Extension of Kronecker's theorem on abelian fields. Unresolved.

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Problem 13 - Impossibility of the solution of the general equation of the 7th degree. Partially solved by Vladimir Arnold.

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Problem 14 - Proof of the finiteness of certain complete systems of functions. Counter example was constructed by Masayoshi Nagata.

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Problem 15 - Rigorous foundation of Schubert's calculus. Partially resolved.

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Problem 16 - Problem of the topology of algebraic curves and surfaces. Unresolved.

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Problem 17 - Expression of definite forms by squares. Resolved by Emil Artin who established an upper limit.

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Problem 18 - Building space from congruent polyhedra. Resolved by Karl Reinhardt.

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Problem 19 - Are the solutions of regular problems in the calculus of variations always necessarily analytic? Proven by Ennio de Giorgi.

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Problem 20 - The general problem of boundary curves. Resolved.

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Problem 21 - Proof of the existence of linear differential equations having a prescribed monodromic group. Resolved depending on one or more exact statements.

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Problem 22 - Uniformization of analytic relations by means of automorphic functions. Resolved.

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Problem 23 - Further development of the methods of the calculus of variations. Too vague to be proven or disproven.