When does one Fibonacci number divide another? Let $F_0 = 0$, $F_1 =
1$, and for $n\\geq 2$, $F_n = F_{n-1} + F_{n-2}$. It is well known
that for $F_m > 1$

This last result was used in Yuri Matijasevi\\u{c}\'s solution of
Hilbert\'s 10th problem.

Using simple combinatorial arguments, we derive previuosly unknown
necessary and sufficient conditions for the following question: For
any $L \\geq 1$

When does $F_m^L$ divide $F_n$?

Our method allows us to answer this same question for any Lucas
sequence of the first kind, defined by $U_0 = 0$, $U_1 = 1$, and for
$n\\geq 2$, $U_n = aU_{n-1} + bU_{n-2}$. This talk is based on joint
work with Harvey Mudd College undergraduate Jeremy Rouse, while
attending the 10th International Conference on Applications of
Fibonacci Numbers.

Let X be a fixed random variable and U,Z,W three others, mutually
independent. We ask if the convolution equation X = U(Z + W) can be solved
in U for given Z,W or in W for given U,Z. We also look at certain
generalizations. A particular version arises in a model for energy
redistribution among stars. We present some general theorems, and some
special examples.

The old subject of congruences between elliptic modular forms has lately become relevant to contemporary arithmetic issues, notably by its role in Wiles\' arguments related to the Taniyama-Shimura conjecture. Some classical results on such congruences will be discussed and an approach based on the modular representation theory of reductive groups will be proposed.

The reductions modulo primes of a fixed elliptic curve defined over the
rational numbers provide an interesting and useful generalization of the
finite field Z/pZ. The study of Z/pZ as p varies has been a fertile
source of problems in classical analytic number theory. Similar problems
about elliptic curves are providing new challenges for modern analytic
number theory.

In this talk I will first review some of the analogies and especially the
insights of Serre concerning these reductions. In order to understand them
it is helpful to identify the Frobenius in explicit terms. This has some
nice side applications for "non-abelian reciprocity" . It also leads to a
set of problems where the classical sieve techniques break down and new
ones must be found. These problems are often so difficult that the
generalized Riemann hypothesis must be assumed in order to prove realistic
results.