Liouville Number
المؤلف:
Apostol, T. M.
المصدر:
Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag
الجزء والصفحة:
...
2-2-2021
2282
Liouville Number
A Liouville number is a transcendental number which has very close rational number approximations. An irrational number 
is called a Liouville number if, for each 
, there exist integers 
and 
such that
Note that the first inequality is true by definition, since it follows immediately from the fact that 
is irrational and hence cannot be equal to 
for any values of 
and 
.
Liouville's constant is an example of a Liouville number and is sometimes called "the" Liouville number or "Liouville's number" (Wells 1986, p. 26). Mahler (1953) proved that 
is not a Liouville number.
REFERENCES:
Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, p. 147, 1997.
Mahler, K. "On the Approximation of 
." Nederl. Akad. Wetensch. Proc. Ser. A. 56/Indagationes Math. 15, 30-42, 1953.
Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986.
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