# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
About the period of Bell numbers modulo a prime. (English) Zbl 1195.11034

Let $p$ be a prime number and let $g\left(p\right)=\left({p}^{p}-1\right)/\left(p-1\right)$. It is well-known that the order $o\left(r\right)$ of a root $r$ of the irreducible polynomial ${x}^{r}-x-1$ over ${𝔽}_{p}$ divides $g\left(p\right)$. S. Wagstaff in [Math. Comput. 65, No. 213, 383–391 (1996; Zbl 0852.11008)] has conjectured that $o\left(r\right)=g\left(p\right)$ holds for all primes $p$. In this paper, the authors point out some subsets $S\subset \left\{1,...,g\left(p\right)\right\}$ that do not contain $o\left(r\right)$. 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

$d={d}_{0}+{d}_{1}p+\cdots +{d}_{p-1}{p}^{p-1},$

with $0\le {d}_{i}\le p-1$ for all $i=0,...,p-1$, and we put

$P\left(x\right)={x}^{{d}_{0}}{\left(x+1\right)}^{{d}_{1}}\cdots {\left(x+p-1\right)}^{{d}_{p-1}}-1,$

then $d$ is a multiple of $o\left(r\right)$ if and only if $P\left(r\right)=0$. In particular, $d=o\left(r\right)$ if and only if the only solution of the exponential equation $P\left(r\right)=0$ 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 the following hold:

1.

$2p-1\le {d}_{0}+\cdots +{d}_{p-1}\le {p}^{2}-3p+1·$

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 $o\left(r\right)$ 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 or rings 11T55 Arithmetic theory of polynomial rings over finite fields