{{Short description|Type of number related to Diophantine approximation}} In mathematics, the '''Lagrange numbers''' ([https://oeis.org/A382098 A382098] and [https://oeis.org/A382099 A382099] in the OEIS) are a sequence of numbers that appear in bounds relating to the approximation of irrational numbers by rational numbers. They are linked to Hurwitz's theorem.

==Definition==

Hurwitz improved Peter Gustav Lejeune Dirichlet's criterion on irrationality to the statement that a real number <math>\alpha</math> is irrational if and only if there are infinitely many rational numbers {{nobr|<math display="inline">\frac pq</math>,}} written in simplest terms, such that

:{{nobr|<math>\left|\alpha - \frac{p}{q}\right| < \frac{1}{\sqrt{5}q^2}</math>.}}

This was an improvement on Dirichlet's result which had <math display="inline">\frac1{q^2}</math> on the right-hand side. The above result is best possible, since the golden ratio <math>\varphi</math> is irrational. If we replace <math>\sqrt5</math> with any larger number in the above expression, we will only be able to find finitely many rational numbers that satisfy the inequality for {{nobr|<math>\alpha=\varphi</math>.}}

Hurwitz also showed that if we omit <math>\varphi</math> (and numbers derived therefrom), we can increase the <math>\sqrt5</math> to {{nobr|<math>2\sqrt2</math>.}} Again this new bound is best possible, this time with <math>\sqrt2</math> being the problem. If we omit{{nobr|<math>\sqrt2</math>,}} we can further increase the <math>2\sqrt2</math> to {{nobr|<math display="inline">\frac\sqrt{221}5</math>.}} Repeating this process we get the infinite series <math display="inline">\sqrt5,\;2\sqrt2,\;\frac\sqrt{221}5,\;\frac\sqrt{1517}{13},\;\ldots</math> which converges to 3.<ref>Cassels (1957) p.14</ref> These are the '''Lagrange numbers''',<ref>Conway&Guy (1996) pp.187-189</ref> named after Joseph Louis Lagrange.{{why|date=January 2026}}

==Relation to Markov numbers==

The <math>n</math>th Lagrange number <math>L_n</math> is given by{{why|date=January 2026}}

:<math>L_n=\sqrt{9-\frac4{M_n^2}}</math>

where <math>M_n</math> is the <math>n</math>th Markov number<ref>Cassels (1957) p.41</ref>—the <math>n</math>th-smallest integer <math>m</math> such that the equation

:<math>m^2+x^2+y^2=3mxy</math>

has a solution in positive integers <math>x</math> and {{nobr|<math>y</math>.}}

==References== {{reflist}} * {{cite book | first=J.W.S. | last=Cassels | author-link=J. W. S. Cassels | title=An introduction to Diophantine approximation | series=Cambridge Tracts in Mathematics and Mathematical Physics | volume=45 | publisher=Cambridge University Press | year=1957 | zbl=0077.04801 }} * {{cite book | first1=J.H. | last1=Conway | author1-link=John Horton Conway | first2=R.K. | last2=Guy | author2-link=Richard K. Guy | title=The Book of Numbers | location=New York | publisher=Springer-Verlag | year=1996 | isbn=0-387-97993-X | url-access=registration | url=https://archive.org/details/bookofnumbers0000conw }}

==External links== *[http://mathworld.wolfram.com/LagrangeNumber.html Lagrange number]. From MathWorld at Wolfram Research. *[http://www.math.jussieu.fr/~miw/articles/pdf/IntroductionDiophantineMethods.pdf 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, Lagrange Numbers on pp.&nbsp;24–26.

Category:Diophantine approximation