Let be a prime number and let . It is well-known that the order of a root of the irreducible polynomial over divides . S. Wagstaff in [Math. Comput. 65, No. 213, 383–391 (1996; Zbl 0852.11008)] has conjectured that holds for all primes . In this paper, the authors point out some subsets that do not contain . Here are a couple of examples of the authors results. First it is not hard to see that if we write the positive integer in base as
with for all , and we put
then is a multiple of if and only if . In particular, if and only if the only solution of the exponential equation is obtained when . Armed with the above fact, the authors prove that under the assumption that the following hold:
2. At least five of the ’s are positive.
The proofs involve clever manipulations of algebraic equations in . We point out that is also the period of the sequence of Bell numbers modulo , whence, the title.