Introduction to Diophantine Approximation. Part II
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System 1
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SummaryIn the article we present in the Mizar system [1], [2] the formalized proofs for Hurwitz’ theorem [4, 1891] and Minkowski’s theorem [5]. Both theorems are well explained as a basic result of the theory of Diophantine approximations appeared in [3], [6]. A formal proof of Dirichlet’s theorem, namely an inequation |θ−y/x| ≤ 1/x2has infinitely many integer solutions (x, y) where θ is an irrational number, was given in [8]. A finer approximation is given by Hurwitz’ theorem: |θ− y/x|≤ 1/√5x2. Minkowski’s theorem concerns an inequation of a product of non-homogeneous binary linear forms such that |a1x + b1y + c1| · |a2x + b2y + c2| ≤ ∆/4 where ∆ = |a1b2− a2b1| ≠ 0, has at least one integer solution.
1983 ◽
Vol 34
(1)
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pp. 114-122
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2013 ◽
Vol 5
(1)
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pp. 16-34
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2016 ◽
Vol 59
(2)
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pp. 349-357
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2018 ◽
Vol 154
(5)
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pp. 1014-1047
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1991 ◽
Vol 51
(2)
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pp. 324-330
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2002 ◽
Vol 132
(3)
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pp. 639-659
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2015 ◽
Vol 17
(01)
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pp. 1540003
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2013 ◽
Vol 09
(03)
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pp. 769-782
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