Hurwitz's Theorem (number Theory)
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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 \left , \xi-\frac\right , < \frac. The condition that ''ξ'' is irrational cannot be omitted. Moreover, the constant \sqrt is the best possible; if we replace \sqrt by any number A > \sqrt and we let \xi = (1+\sqrt)/2 (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 \sqrt.


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