Hurwitz,s Irrational Number Theorem
المؤلف:
Apostol, T. M.
المصدر:
Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag
الجزء والصفحة:
...
14-10-2020
1649
Hurwitz's Irrational Number Theorem
As Lagrange showed, any irrational number 
has an infinity of rational approximations 
which satisfy
 |
(1)
|
Furthermore, if there are no integers 
with 
and 
(corresponding to values of 
associated with the golden ratio 
through their continued fractions), then
 |
(2)
|
and if values of 
associated with the silver ratio 
are also excluded, then
 |
(3)
|
In general, even tighter bounds of the form
 |
(4)
|
can be obtained for the best rational approximation possible for an arbitrary irrational number 
, where the 
are called Lagrange numbers and get steadily larger for each "bad" set of irrational numbers which is excluded.
REFERENCES:
Apostol, T. M. Modular Functions and Dirichlet Series in Number Theory, 2nd ed. New York: Springer-Verlag, p. 145, 1997.
Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, p. 40, 1987.
Chandrasekharan, K. An Introduction to Analytic Number Theory. Berlin: Springer-Verlag, p. 23, 1968.
Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 187-189, 1996.
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