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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 \({\mathbb F}_p\) 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 \(S\subset \{1,\ldots,g(p)\}\) 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 \[ d=d_0+d_1p+\cdots+d_{p-1} p^{p-1}, \] with \(0\leq d_i\leq p-1\) for all \(i=0,\ldots,p-1\), and we put \[ P(x)=x^{d_0}(x+1)^{d_1}\cdots (x+p-1)^{d_{p-1}}-1, \] 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 \(d_0=d_1=\cdots=d_{p-1}\). Armed with the above fact, the authors prove that under the assumption that \(d<o(r)\) the following hold:
1. \[ 2p-1\leq d_0+\cdots+d_{p-1}\leq p^2-3p+1. \] 2. At least five of the \(d_i\)’s are positive.
The proofs involve clever manipulations of algebraic equations in \({\mathbb F}_p\). We point out that \(o(r)\) is also the period of the sequence of Bell numbers modulo \(p\), whence, the title.

MSC:

11B73 Bell and Stirling numbers
11T06 Polynomials over finite fields
11T55 Arithmetic theory of polynomial rings over finite fields

Citations:

Zbl 0852.11008
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