Liouville,s Approximation Theorem
المؤلف:
Apostol, T. M.
المصدر:
"Liouville,s Approximation Theorem." §7.3 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag
الجزء والصفحة:
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14-10-2020
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Liouville's Approximation Theorem
For any algebraic number 
of degree 
, a rational approximation 
to 
must satisfy
for sufficiently large 
. Writing 
leads to the definition of the irrationality measure of a given number. Apostol (1997) states the theorem in the slightly modified but equivalent form that there exists a positive constant 
depending only on 
such that for all integers 
and 
with 
,
REFERENCES:
Apostol, T. M. "Liouville's Approximation Theorem." §7.3 in Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 146-148, 1997.
Courant, R. and Robbins, H. "Liouville's Theorem and the Construction of Transcendental Numbers." §2.6.2 in What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 104-107, 1996.
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