A remark on Apéry's numbers

✍ Scribed by Jean-Paul Allouche

• Book ID: 104338445
• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 134 KB
• Volume: 83
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/s0377-0427(97)00074-5

✦ Synopsis

The Aprry numbers, introduced in Aprry's celebrated proof of the irrationality of ~(3), are defined by n (n~ 2 (n+k)2 a, = ~k=0 ~k/ ~ k j • They have the following nice property: if p is a prime number, and n = ~ niP j is the base p expansion of n, then a, = I-la,j rood 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.