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On the degree of an irreducible factor of the Bernoulli polynomials. (English) Zbl 0655.10008
Let p be a prime number, n= i=0 h n i p i and A(n,p)= i=0 h n i (0n i p-1 for all i, n h 0, n). Define by N(n,p) a natural number given by the conditions 0<N(n,p)n, pn N(n,p) and if (p-1)|t, 0<t<N(n,p), then pn 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 2m (x) has an irreducible factor of degree N(2m,p) if A(2m,p)p-1·
Reviewer: I.Sh.Slavutskij
MSC:
11B39Fibonacci and Lucas numbers, etc.
12E05Polynomials over general fields
11T06Polynomials over finite fields or rings