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On congruences of Euler numbers modulo powers of two. (English) Zbl 1204.11051
The author establishes some identities involving the Euler numbers, the Euler numbers of order 2 and the central factorial numbers, and gives a new proof of a classical result due to M. A. Stern [J. Reine Angew. Math. 79, 67–98 (1874; JFM 06.0103.01)]. Stern gave a brief sketch of the following congruence \[ E_{2n}\equiv E_{2m}\pmod{2^k}\quad\text{if and only if}\quad 2n\equiv 2m\pmod{2^k}.\tag{1} \] Then F. G. Frobenius [Sitzungsber. Königl. Preuss. Akad. Wiss. Berlin, 809–847 (1910), also in: Ges. Abh. III, 440–478 (1968; Zbl 0169.28901)] amplified Stern’s sketch, and later R. Ernvall [Ann. Univ. Turku., Ser. A I 178, 72 p. (1979; Zbl 0403.12010)] gave a proof of (1) using umbral calculus. In 2000 S. S. Wagstaff jun. [Number theory for the millennium III. Proceedings of the millennial conference on number theory, Urbana-Champaign, IL, USA, 2000. Natick, MA: A K Peters, 357–374 (2002; Zbl 1050.11021)] used an induction proof, and in 2005 Z.-W. Sun [J. Number Theory 115, No. 2, 371–380 (2005; Zbl 1090.11016)] obtained an explicit congruence for Euler numbers modulo powers of two to give a new proof of (1).

MSC:
11B68 Bernoulli and Euler numbers and polynomials
11A07 Congruences; primitive roots; residue systems
11B65 Binomial coefficients; factorials; \(q\)-identities
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