On a conjecture of the Euler numbers. (English) Zbl 1197.11027
Summary: The main purpose of this paper is to prove a conjecture of the Euler numbers and its generalization by using analytic methods. That is, for any prime \(p \equiv 1\pmod 4\) and integer \(\alpha\geq 1\) we prove \(E_{\varphi(p^{\alpha})/2}\not\equiv 0\pmod {p^{\alpha}}\), where \(E_{2n}\) are the Euler numbers and \(\varphi(n)\) the Euler function.
MSC:
| 11B68 | Bernoulli and Euler numbers and polynomials |
| 11A07 | Congruences; primitive roots; residue systems |
References:
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