View source for Liouville number

A real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059650/l0596501.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059650/l0596502.png" /> the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059650/l0596503.png" /></td> </tr></table>

has infinitely many integer solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059650/l0596504.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059650/l0596505.png" /> satisfying the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059650/l0596506.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059650/l0596507.png" />. The fact that a Liouville number is transcendental (cf. [[Transcendental number|Transcendental number]]) follows from the Liouville theorem (cf. [[Liouville theorems|Liouville theorems]]). These numbers were studied by J. Liouville [[#References|[1]]].

Examples of Liouville numbers are:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059650/l0596508.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059650/l0596509.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059650/l05965010.png" /></td> </tr></table>

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Liouville,   "Sur des classes très étendues de quantités dont la valeur n'est ni algébrique, ni même réductible à des irrationelles algébriques"  ''C.R. Acad. Sci. Paris'' , '''18'''  (1844)  pp. 883–885</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.O. Gel'fond,   "Transcendental and algebraic numbers" , Dover, reprint  (1960)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  O. Perron,   "Die Lehre von den Kettenbrüchen" , '''1''' , Teubner  (1977)  pp. Sect. 35</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  O. Perron,   "Irrationalzahlen" , Chelsea, reprint  (1948)</TD></TR></table>