A remark on Apéryʹs numbers

The Apéry numbers, introduced in Apéryʹs celebrated proof of the irrationality of ζ(3), are defined by an=∑nk=0(nk)2(n+kk)2. They have the following nice property: if p is a prime number, and n = ∑ nj pj is the base p expansion of n, then an ≡ ∏anj mod p. In a paper which appeared in this journal (64 (1995)11–19), C. Radoux asserted that the same property holds, provided p ⩾ 5, if p is replaced by p2 both for the base and for the congruence, and if p is replaced by p3 both for the base and for the congruence. We show that these two statements are not correct.