In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Lagrange numbers are a sequence of numbers that appear in bounds relating to the approximation of
irrational numbers
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 ...
by
rational numbers
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rationa ...
. They are linked to
Hurwitz's theorem.
Definition
Hurwitz improved
Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
's criterion on irrationality to the statement that a real number α is irrational if and only if there are infinitely many rational numbers ''p''/''q'', written in lowest terms, such that
:
This was an improvement on Dirichlet's result which had 1/''q''
2 on the right hand side. The above result is best possible since 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 ( ...
φ is irrational but if we replace by any larger number in the above expression then we will only be able to find finitely many rational numbers that satisfy the inequality for α = φ.
However, Hurwitz also showed that if we omit the number φ, and numbers derived from it, then we ''can'' increase the number . In fact he showed we may replace it with 2. Again this new bound is best possible in the new setting, but this time the number is the problem. If we don't allow then we can increase the number on the right hand side of the inequality from 2 to /5. Repeating this process we get an infinite sequence of numbers , 2, /5, ... which converge to 3. These numbers are called the Lagrange numbers, and are named after
Joseph Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia[Markov number
A Markov number or Markoff number is a positive integer ''x'', ''y'' or ''z'' that is part of a solution to the Markov Diophantine equation
:x^2 + y^2 + z^2 = 3xyz,\,
studied by .
The first few Markov numbers are
: 1, 2, 5, 13, 29, 34, 89 ...](_blank)
,
[Cassels (1957) p.41] that is the ''n''th smallest integer ''m'' such that the equation
:
has a solution in positive integers ''x'' and ''y''.
References
*
*
External links
Lagrange number From
MathWorld
''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Dig ...
at
Wolfram Research
Wolfram Research, Inc. ( ) is an American multinational company that creates computational technology. Wolfram's flagship product is the technical computing program Wolfram Mathematica, first released on June 23, 1988. Other products include Wo ...
.
Introduction to Diophantine methods irrationality and transcendence{{Webarchive, url=https://web.archive.org/web/20120209111526/http://www.math.jussieu.fr/~miw/articles/pdf/IntroductionDiophantineMethods.pdf , date=2012-02-09 - Online lecture notes by
Michel Waldschmidt
Michel Waldschmidt (born June 17, 1946 at Nancy, France) is a French mathematician, specializing in number theory, especially transcendental numbers.
Biography
Waldschmidt was educated at Lycée Henri Poincaré and the University of Nancy unti ...
, Lagrange Numbers on pp. 24–26.
Diophantine approximation