Apéry numbers

The Apéry numbers , are defined by the finite sums

for every integer . They were introduced in 1978 by R. Apéry in his highly remarkable irrationality proofs of and , respectively. In the case of , Apéry showed that there exists a sequence of rational numbers with denominator dividing such that for all . Together with the fact that , this implies the irrationality of . For a very lively and amusing account of Apéry's discovery, see [a4]. In 1979 F. Beukers [a1] gave a very short irrationality proof of , motivated by the shape of the Apéry numbers. Despite much efforts by many people there is no generalization to an irrationality proof of so far (2001).

T. Rival [a5] proved the very surprising result that for infinitely many .

It did not take long before people noticed a large number of interesting congruence properties of Apéry numbers. For example, for all positive integers , and all prime numbers . Another congruence is for all prime numbers . Here, denotes the coefficient of in the -expansion of a modular cusp form. For more details see [a2], [a3].

References

[a1]	F. Beukers, "A note on the irrationality of " Bull. London Math. Soc. , 11 (1979) pp. 268–272
[a2]	F. Beukers, "Some congruences for the Apéry numbers" J. Number Theory , 21 (1985) pp. 141–155
[a3]	F. Beukers, "Another conguence for the Apéry numbers" J. Number Theory , 25 (1987) pp. 201–210
[a4]	A.J. van der Poorten, "A proof that Euler missed Apéry's proof of the irrationality of " Math. Intelligencer , 1 (1979) pp. 195–203
[a5]	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