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About the period of Bell numbers modulo a prime. (English) Zbl 1195.11034

Let p be a prime number and let g(p)=(p p -1)/(p-1). It is well-known that the order o(r) of a root r of the irreducible polynomial x r -x-1 over 𝔽 p divides g(p). S. Wagstaff in [Math. Comput. 65, No. 213, 383–391 (1996; Zbl 0852.11008)] has conjectured that o(r)=g(p) holds for all primes p. In this paper, the authors point out some subsets S{1,...,g(p)} that do not contain o(r). Here are a couple of examples of the authors results. First it is not hard to see that if we write the positive integer d in base p as

d=d 0 +d 1 p++d p-1 p p-1 ,

with 0d i p-1 for all i=0,...,p-1, and we put

P(x)=x d 0 (x+1) d 1 (x+p-1) d p-1 -1,

then d is a multiple of o(r) if and only if P(r)=0. In particular, d=o(r) if and only if the only solution of the exponential equation P(r)=0 is obtained when d 0 =d 1 ==d p-1 . Armed with the above fact, the authors prove that under the assumption that d<o(r) the following hold:

1.

2p-1d 0 ++d p-1 p 2 -3p+1·

2. At least five of the d i ’s are positive.

The proofs involve clever manipulations of algebraic equations in 𝔽 p . We point out that o(r) is also the period of the sequence of Bell numbers modulo p, whence, the title.

MSC:
11B73Bell and Stirling numbers
11T06Polynomials over finite fields or rings
11T55Arithmetic theory of polynomial rings over finite fields