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Euler tangent numbers modulo 720 and Genocchi numbers modulo 45. (English) Zbl 1510.11077

Summary: We establish congruences for higher order Euler polynomials modulo 720. We apply this result for constructing analogues of Stern congruences for Euler secant numbers \(E_{4n}\equiv 5\pmod{60}, E_{4n+2}\equiv -1\pmod{60}\) to Euler tangent numbers and Genocchi numbers. We prove that Euler tangent numbers satisfy the following congruences \(E_{4n+1}\equiv 16\pmod{720}\), and \(E_{4n+3}\equiv -272\pmod{720}\). We establish 12-periodic property of Genocchi numbers modulo 45.

MSC:

11B68 Bernoulli and Euler numbers and polynomials
11A07 Congruences; primitive roots; residue systems
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References:

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