In
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, Hurwitz's theorem, named after
Adolf Hurwitz
Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, analysis, geometry and number theory.
Early life
He was born in Hildesheim, then part of the Kingdom of Hanover, to a Jewish family and died ...
, gives a
bound
Bound or bounds may refer to:
Mathematics
* Bound variable
* Upper and lower bounds, observed limits of mathematical functions
Physics
* Bound state, a particle that has a tendency to remain localized in one or more regions of space
Geography
*B ...
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 by r ...
. The theorem states that for every
irrational number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...
''ξ'' there are infinitely many
relatively prime
In mathematics, 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 equivale ...
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 sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
) 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
C ...
of every number is larger than
.
References
*
*
*
* {{cite book
, author=
Ivan Niven
Ivan Morton Niven (October 25, 1915 May 9, 1999) was a Canadian-American mathematician, specializing in number theory and known for his work on Waring's problem. He worked for many years as a professor at the University of Oregon, and was preside ...
, title=Diophantine Approximations
, publisher=Courier Corporation
, year=2013
, isbn=978-0486462677
Diophantine approximation
Theorems in number theory