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On Beukers’ conjecture. (English) Zbl 0687.10003

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

MSC:

11A07 Congruences; primitive roots; residue systems
11B37 Recurrences
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