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Proof of a conjecture of Beukers on Apéry numbers. (English) Zbl 0634.10004

p-adic analysis, Proc. Conf., Houthalen/Belg. 1986, 189-195 (1986).
Consider the numbers \(b(n)=\sum_{k=0}^n \binom{n}{k}^2\binom{n+k}{k}\), \(c(n)=\sum_{k=0}^n \binom{n}{k}\binom{n+k}{k}\) which occur in irrationality proofs for \(\zeta(3)\) and \(\log 2\). Let \(p\) be a prime which is \(\equiv 1\bmod 4\) and write \(p=a^2+b^2\). In Theorems 1 and 2 the author proves that \[ c^2\left(\frac{p-1}{2}\right)\equiv 4a^2-2p\pmod{p^2}\quad\text{and}\quad b\left(\frac{p-1}{2}\right)\equiv 4a^2-2p\pmod{p^2}, \] thereby answering a question of the reviewer. The congruence mod \(p\) only can be derived by using formal group theory. It is not clear what the mechanism behind the author’s stronger congruence is. Here they are proved by clever ad hoc methods. In Theorems 3 and 4 the author gives some interesting congruences for \(\binom{2n}{n}\) which were treated independently by M. Coster in “Generalization of a congruence of Gauss”, J. Number Theory 29, No. 3, 300–310 (1988; Zbl 0651.10004)].
[For the entire collection see Zbl 0624.00007.]

MSC:

11A07 Congruences; primitive roots; residue systems
11B65 Binomial coefficients; factorials; \(q\)-identities