A remark on Apery's numbers

[摘要] The Apery numbers, introduced in Apery's celebrated proof of the irrationality of xi(3), are defined by a(n) = Sigma(k=0)(n) ((n)(k))(2)((n+k)(k))(2). They have the following nice property: if p is a prime number, and n = Sigma n(j)p(j) is the base p expansion of n, then a(n) = Pi a(nj) mod p. In a paper which appeared in this journal (64 (1995)11-19), C. Radoux asserted that the same property holds, provided p greater than or equal to 5, if p is replaced by p(2) both for the base and for the congruence, and if p is replaced by p(3) both for the base and for the congruence. We show that these two statements are not correct.

[发布日期] 1997-09-30

[关键词] Apery's numbers;Lucas property