Apéry's numbers are defined by

			(1)
			(2)
			(3)

where  is a binomial coefficient. The first few for , 1, 2, ... are 1, 5, 73, 1445, 33001, 819005, ... (OEIS A005259).

The first few prime Apéry numbers are 5, 73, 12073365010564729, 10258527782126040976126514552283001, ... (OEIS A092826), which have indices , 2, 12, 24, ... (OEIS A092825).

The  case of Schmidt's problem expresses these numbers in the form

(4)

(Strehl 1993, 1994; Koepf 1998, p. 55).

They are also given by the recurrence equation

(5)

with  and  (Beukers 1987).

There is also an associated set of numbers

			(6)
			(7)

(Beukers 1987), where  is a generalized hypergeometric function. The values for , 1, ... are 1, 3, 19, 147, 1251, 11253, 104959, ... (OEIS A005258). The first few prime -numbers are 5, 73, 12073365010564729, 10258527782126040976126514552283001, ... (OEIS A092827), which have indices , 2, 6, 8, ... (OEIS A092828), with no others for  (Weisstein, Mar. 8, 2004).

The  numbers are also given by the recurrence equation

(8)

with  and .

Both  and  arose in Apéry's irrationality proof of  and  (van der Poorten 1979, Beukers 1987). They satisfy some surprising congruence properties,

(9)

(10)

for  a prime  and  (Beukers 1985, 1987), as well as

{4a^2-2p (mod p) if p=a^2+b^2, a odd; 0 (mod p) if p=3 (mod 4) " src="https://mathworld.wolfram.com/images/equations/AperyNumber/NumberedEquation6.gif" style="height:42px; width:306px" />

(11)

(Stienstra and Beukers 1985, Beukers 1987). Defining  from the generating function

			(12)
			(13)

where  is a q-Pochhammer symbol, gives  of 1, , , 24, , , ... (OEIS A030211; Koike 1984) for , 3, 5, ..., and

(14)

for  an odd prime (Beukers 1987). Furthermore, for  an odd prime and ,

(15)

(Beukers 1987).

The Apéry numbers are given by the diagonal elements  in the identity

			(16)
			(17)
			(18)

(Koepf 1998, p. 119).

REFERENCES:

Apéry, R. "Irrationalité de  et ." Astérisque 61, 11-13, 1979.

Apéry, R. "Interpolation de fractions continues et irrationalité de certaines constantes." Mathématiques, Ministère universités (France), Comité travaux historiques et scientifiques. Bull. Section Sciences 3, 243-246, 1981.

Beukers, F. "Some Congruences for the Apéry Numbers." J. Number Th. 21, 141-155, 1985.

Beukers, F. "Another Congruence for the Apéry Numbers." J. Number Th. 25, 201-210, 1987.

Chowla, S.; Cowles, J.; and Cowles, M. "Congruence Properties of Apéry Numbers." J. Number Th. 12, 188-190, 1980.

Gessel, I. "Some Congruences for the Apéry Numbers." J. Number Th. 14, 362-368, 1982.

Koepf, W. "Hypergeometric Identities." Ch. 2 in Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, pp. 29 and 119, 1998.

Koike, M. "On McKay's Conjecture." Nagoya Math. J. 95, 85-89, 1984.

Schmidt, A. L. "Legendre Transforms and Apéry's Sequences." J. Austral. Math. Soc. Ser. A 58, 358-375, 1995.

Sloane, N. J. A. Sequences A005258/M3057, A005259/M4020, A030211, A092825, A092826, A092827, and A092828 in "The On-Line Encyclopedia of Integer Sequences."

Stienstra, J. and Beukers, F. "On the Picard-Fuchs Equation and the Formal Brauer Group of Certain Elliptic  Surfaces." Math. Ann. 271, 269-304, 1985.

Strehl, V. "Binomial Sums and Identities." Maple Technical Newsletter 10, 37-49, 1993.

Strehl, V. "Binomial Identities--Combinatorial and Algorithmic Aspects." Discrete Math. 136, 309-346, 1994.

van der Poorten, A. "A Proof that Euler Missed... Apéry's Proof of the Irrationality of ." Math. Intel. 1, 196-203, 1979.