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Jitse Niesen (talk | contribs) →Pell's equation: comment |
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I think it should be stated that <math>\sqrt{S}\,</math> is '''not''' a Liouville number, because it satisfies that property for ''n = 2'' but not for all ''n'' (does it fail for ''n = 3''?). [[User:Albmont|Albmont]] ([[User talk:Albmont|talk]]) 18:29, 5 August 2008 (UTC)
:Hmm, it doesn't have much to do with methods of computing square roots, so the article on [[square root]] seems a more natural notation. However, the fact that <math>\sqrt{S}</math> is an [[algebraic number]] (which is a bit hidden in that article; perhaps that could be improved) and Liouville numbers cannot be algebraic implies that square roots are not Liouville numbers, so I'm not sure it should be mentioned at that article either.
:The square root of 6 is 2.44949… so
::<math>0 < \left| \sqrt{6} - \frac52 \right| = 0.05051 < 0.0625 = \frac{1}{2^4}. </math>
:However, what is true is that there are only finitely many pairs (''p'', ''q'') such that
::<math>0< \left| \sqrt{S} - \frac{p}{q} \right| < \frac{1}{q^{2}} </math>
:(the [[irrationality measure]] is 2). On the other hand, a Liouville number has infinitely many such pairs. -- [[User:Jitse Niesen|Jitse Niesen]] ([[User talk:Jitse Niesen|talk]]) 20:17, 5 August 2008 (UTC)
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