Ishikawa, Tsuneo On Beukers’ conjecture. (English) Zbl 0687.10003 Kobe J. Math. 6, No. 1, 49-51 (1989). Let a(n) be the Apéry number given by \(\sum^{n}_{k=0}\left( \begin{matrix} n\\ k\end{matrix} \right)^ 2\left( \begin{matrix} n+k\\ k\end{matrix} \right)^ 2\). Let \(\gamma\) (n) be the coefficient of \(q^ n\) in the Taylor expansion of \(q\prod^{\infty}_{n=1}(1-q^{2n})^ 4(1-q^{4n})^ 4.\) Let p be any odd prime. By a very simple combination of a result of the reviewer [J. Number Theory 25, 201-210 (1987; Zbl 0614.10011)] and of I. Gessel [J. Number Theory 14, 362-368 (1982; Zbl 0482.10003)] the author proves that if p does not divide a(p), then a(p)\(\equiv \gamma (p)\) (mod \(p^ 2)\). Moreover, \(p\nmid a(p)\) for all primes \(p<80000\) except \(p=11,3137\). Reviewer: F.Beukers Cited in 8 Documents MSC: 11A07 Congruences; primitive roots; residue systems 11B37 Recurrences Keywords:congruence properties; Apéry number; Taylor expansion Citations:Zbl 0614.10011; Zbl 0482.10003 PDFBibTeX XMLCite \textit{T. Ishikawa}, Kobe J. Math. 6, No. 1, 49--51 (1989; Zbl 0687.10003)