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Liouville approximation theorem
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(Theorem)
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Given , a real algebraic number of degree , there is a constant
such that for all rational numbers
, the inequality
holds.
Many mathematicians have worked at strengthening this theorem:
- Thue: If
is an algebraic number of degree , then there is a constant
such that for all rational numbers , the inequality
holds.
- Siegel: If
is an algebraic number of degree , then there is a constant
such that for all rational numbers , the inequality
holds.
- Dyson: If
is an algebraic number of degree , then there is a constant
such that for all rational numbers with , the inequality
holds.
- Roth: If
is an irrational algebraic number and
, then there is a constant
such that for all rational numbers , the inequality
holds.
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"Liouville approximation theorem" is owned by KimJ.
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Cross-references: irrational, inequality, rational numbers, degree, algebraic number, real
There are 4 references to this entry.
This is version 8 of Liouville approximation theorem, born on 2001-10-15, modified 2003-02-01.
Object id is 211, canonical name is LiouvillesTheorem.
Accessed 3628 times total.
Classification:
| AMS MSC: | 11J68 (Number theory :: Diophantine approximation, transcendental number theory :: Approximation to algebraic numbers) |
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Pending Errata and Addenda
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