View source for Apéry numbers

The Apéry numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a1302601.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a1302602.png" /> are defined by the finite sums

<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/a/a130/a130260/a1302603.png" /></td> </tr></table>

for every integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a1302604.png" />. They were introduced in 1978 by R. Apéry in his highly remarkable irrationality proofs of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a1302605.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a1302606.png" />, respectively. In the case of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a1302607.png" />, Apéry showed that there exists a sequence of rational numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a1302608.png" /> with denominator dividing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a1302609.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026010.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026011.png" />. Together with the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026012.png" />, this implies the irrationality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026014.png" />. For a very lively and amusing account of Apéry's discovery, see [[#References|[a4]]]. In 1979 F. Beukers [[#References|[a1]]] gave a very short irrationality proof of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026015.png" />, motivated by the shape of the Apéry numbers. Despite much efforts by many people there is no generalization to an irrationality proof of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026016.png" /> so far (2001).

T. Rival [[#References|[a5]]] proved the very surprising result that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026017.png" /> for infinitely many <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026018.png" />.

It did not take long before people noticed a large number of interesting congruence properties of Apéry numbers. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026019.png" /> for all positive integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026021.png" /> and all prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026022.png" />. Another congruence is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026023.png" /> for all prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026024.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026025.png" /> denotes the coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026026.png" /> in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026027.png" />-expansion of a modular cusp form. For more details see [[#References|[a2]]], [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F. Beukers,   "A note on the irrationality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026028.png" />"  ''Bull. London Math. Soc.'' , '''11'''  (1979)  pp. 268–272</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  F. Beukers,   "Some congruences for the Apéry numbers"  ''J. Number Theory'' , '''21'''  (1985)  pp. 141–155</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  F. Beukers,   "Another conguence for the Apéry numbers"  ''J. Number Theory'' , '''25'''  (1987)  pp. 201–210</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A.J. van der Poorten,   "A proof that Euler missed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026029.png" /> Apéry's proof of the irrationality of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a130/a130260/a13026030.png" />"  ''Math. Intelligencer'' , '''1'''  (1979)  pp. 195–203</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  T. Rival,   "La fonction zêta de Riemann pren une infinité de valeurs irrationnelles aux entiers impairs"  ''C.R. Acad. Sci. Paris'' , '''331'''  (2000)  pp. 267–270</TD></TR></table>