JOURNAL OF NUMBER THEORY 14, 362-368 (1982) Some Congruences for Ap6ry Numbers IRA GESSEL* Department of Mathematics, Massachusetts Institute of Technology. Cambridge, Massachusetts 02139 Communicated by H. Zassenhaus Received September 10. 1980 Congruences for the Aptry numbers are proved which generalize the results and conjectures of Chowla, .I. Cowles, and M. Cowles. R. Apery’s proof of the irrationality of 4(3) used the numbers a,, defined by a, = 1, a, = 5, and for n > 2, n3a, - (34n3 - 51~’ + 27n - 5)a,_, + (n - 1)3a,_2 = 0. (1) These numbers have the explicit formula (2) The first few values are a, = 1, a, = 5, a2 = 73, a3 = 1445, a4 = 33,001, and a, = 8 19,005. (For an exposition of Apery’s work, seevan der Poorten [7 1.) Congruence properties of these numbers were considered by Chowla et al. [2 1, who made the following conjectures: (Cla) uZn E 1 (mod 8). (Clb) a,,,, z 5 (mod 8). (C2a) uZnE 1 (mod 3). ( (C2b) a,,,, , E 2 (mod 3). (C3) For all primes p 2 5, a, G 5 (mod p3). (C4) For odd primes p, ap = 0 (mod 5). We shall prove some congruences for the Aptry numbers which include all of these conjectures. In particular, we shall prove the following generalizations of (C3) and (C4): (C3’) For all primes p > 5, apn s a, (mod p’). A Partially supported by ONR Contract NOO014-76-C-0366, 362 0022.314X/82/030362-07%02.00/O Copyright 0 1982 by Academic Press, Inc. All rinhts of reorcduction in anv form reserved.