  We consider representing of natural numbers by arithmetical expressions using
ones, addition, multiplication and parentheses. The (integer) complexity of n
-- denoted by ||n|| -- is defined as the number of ones in the shortest
expressions representing n. We arrive here very soon at the problems that are
easy to formulate, but (it seems) extremely hard to solve. In this paper we
represent our attempts to explore the field by means of experimental
mathematics. Having computed the values of ||n|| up to 10^12 we present our
observations. One of them (if true) implies that there is an infinite number of
Sophie Germain primes, and even that there is an infinite number of Cunningham
chains of length 4 (at least). We prove also some analytical results about
integer complexity.
