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Zbl 0655.10008
Kimura, Noriaki
On the degree of an irreducible factor of the Bernoulli polynomials.
(English)
[J] Acta Arith. 50, No.3, 243-249 (1988). ISSN 0065-1036; ISSN 1730-6264/e

Let p be a prime number, $n=\sum\sp{h}\sb{i=0}n\sb ip\sp i$ and $A(n,p)=\sum\sp{h}\sb{i=0}n\sb i$ $(0\le n\sb i\le p-1$ for all i, $n\sb h\ne 0$, $n\in {\bbfN})$. Define by $N(n,p)$ a natural number given by the conditions $0<N(n,p)\le n$, $p \nmid \binom{n}{N(n,p)}$ and if $(p- 1)\vert t,$ $0<t<N(n,p)$, then $p \nmid \binom{n}{t}.$ The author proves that the number N(n,p) exists. Further some theorems concerning irreducible factors of Bernoulli and related polynomials are considered. In particular, it is proved that the Bernoulli polynomial $B\sb{2m}(x)$ has an irreducible factor of degree $\ge N(2m,p)$ if $A(2m,p)\ge p-1.$
[I.Sh.Slavutskij]
MSC 2000:
*11B39 Special numbers, etc.
12E05 Polynomials over general fields
11T06 Polynomials over finite fields or rings

Keywords: divisibility of binomial coefficients; irreducible factors; Bernoulli polynomial; degree

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