Gessel, Ira Some congruences for Apery numbers. (English) Zbl 0482.10003 J. Number Theory 14, 362-368 (1982). Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 11 ReviewsCited in 54 Documents MSC: 11A07 Congruences; primitive roots; residue systems 11B37 Recurrences Keywords:congruences for Apery numbers; arithmetical properties of algebraic hypersurfaces Citations:Zbl 0428.10008 × Cite Format Result Cite Review PDF Full Text: DOI Online Encyclopedia of Integer Sequences: Apery (Apéry) numbers: Sum_{k=0..n} (binomial(n,k)*binomial(n+k,k))^2. Numerator of 2*Sum_{k=0..n} binomial(n,k)^2*binomial(n+k,k)^2*(H(n+k)-H(n-k)) where H(n)=Sum_{k=1..n} 1/k. Denominator of 2*Sum_{k=0..n} binomial(n,k)^2*binomial(n+k,k)^2*(H(n+k)-H(n-k)) where H(n) = Sum_{k=1..n} 1/k. Number of base-p digits for which the Apéry numbers support a Lucas congruence modulo p^2, where p is the n-th prime. References: [1] Carlitz, L., The coefficients of the reciprocal of \(J_0(x)\), Arch. Mat., 6, 121-127 (1955) · Zbl 0064.06502 [2] Chowla, S.; Cowles, J.; Cowles, M., Congruence properties of Apéry numbers, J. Number Theory, 12, 188-190 (1980) · Zbl 0428.10008 [3] Cowles, J., Some congruence properties of three well-known sequences: Two notes, J. Number Theory, 12, 84-86 (1980) · Zbl 0425.10033 [4] Hardy, G. H.; Wright, E. M., (An Introduction to the Theory of Numbers (1960), Oxford Univ. Press: Oxford Univ. Press London) · Zbl 0086.25803 [5] Kazandzidis, G. S., Congruences on the binomial coefficients, Bull. Soc. Math. Grèce (N.S.), 9, 1-12 (1968) · Zbl 0179.06601 [6] Lucas, E., Sur les congruences des nombres eulériens et des coefficients différentiels des fonctions trigonométriques, suivant un module premier, Bull. Soc. Math. de France, 6, 49-54 (1878) · JFM 10.0139.04 [7] van der Poorten, A., A proof that Euler missed … Apéry’s proof of the irrationality of ζ(3). An informal report, Math. Intelligencer, 1, 195-203 (1979) · Zbl 0409.10028 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.