On congruence for Apéry numbers (Q1907266)

The Apéry numbers are defined by \(u(n)= \sum^n_{k=0} {n\choose k} \cdot {{n+k} \choose k}\); \(v(n)= \sum^n_{k=0} {n\choose k}^2 \cdot {{n+k} \choose k}\) where \({n \choose k}= C^k_n\) denotes a binomial coefficient. These numbers appear in certain irrationality proofs (e.g. of \(\log 2\) and \(\zeta (2)\)) by \textit{F. Beukers} [Séminaire de Théorie des Nombres, Paris 1982-83 (Birkhäuser, Boston 1983)]. The author proves the second part of a conjecture by Beukers, namely that for a prime \(p\equiv 1\pmod 4\), one has the congruence \(v((p- 1)/2)\equiv 4a^2- 2p\pmod {p^2}\), where \(p= a^2+ b^2\) and \(a\equiv 1\pmod 4\). The first part of Beukers' conjecture, which asserts that \(u({{p-1} \over 2})^2 \equiv 4a^2- 2p\pmod {p^2}\), has been proved by the author in 1992 [Southeast Asian Bull. Math. 16, 165-170 (1992; Zbl 0774.11005)].