In
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, Hurwitz's theorem, named after
Adolf Hurwitz
Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, mathematical analysis, analysis, geometry and number theory.
Early life
He was born in Hildesheim, then part of the Kingdom of Hanover, to a ...
, gives a
bound on a
Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.
The first problem was to know how well a real number can be approximated ...
. The theorem states that for every
irrational number
In mathematics, the irrational numbers are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, ...
''ξ'' there are infinitely many
relatively prime
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...
integers ''m'', ''n'' such that
The condition that ''ξ'' is irrational cannot be omitted. Moreover, the constant
is the best possible; if we replace
by any number
and we let
(the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if
\fr ...
) then there exist only ''finitely'' many relatively prime integers ''m'', ''n'' such that the formula above holds.
The theorem is equivalent to the claim that the
Markov constant
In number theory, specifically in Diophantine approximation theory, the Markov constant M(\alpha) of an irrational number \alpha is the factor for which Dirichlet's approximation theorem can be improved for \alpha.
History and motivation
Cert ...
of every number is larger than
.
See also
*
Dirichlet's approximation theorem
In number theory, Dirichlet's theorem on Diophantine approximation, also called Dirichlet's approximation theorem, states that for any real numbers \alpha and N , with 1 \leq N , there exist integers p and q such that 1 \leq q \leq N and
...
*
Lagrange number
References
*
*
*
* {{cite book
, author=
Ivan Niven
, title=Diophantine Approximations
, publisher=Courier Corporation
, year=2013
, isbn=978-0486462677
Diophantine approximation
Theorems in number theory