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In mathematics, the Lagrange numbers 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 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 φ is irrational but if we replace √5 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 √5, in fact he showed we may replace it with 2√2. Again this new bound is best possible in the new setting, but this time the number √2 is the problem. If we don't allow √2 then we can increase the number on the right hand side of the inequality from 2√2 to (√221)/5. Repeating this process we get an infinite sequence of numbers √5, 2√2, (√221)/5, ... which converge to 3.〔Cassels (1957) p.14〕 These numbers are called the Lagrange numbers,〔Conway&Guy (1996) pp.187-189〕 and are named after Joseph Louis Lagrange. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lagrange number」の詳細全文を読む スポンサード リンク
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