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 ![]() |
[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 ![]() ![]() |
[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 |
Apéry numbers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ap%C3%A9ry_numbers&oldid=13350