From numerical experiments, D. E. Knuth conjectured that $0<D_{n+4}<D_n$ for a combinatorial sequence $(D_n)$ defined as the difference $D_n = R_n-L_n$ of two definite hypergeometric sums. The conjecture implies an identity of type $L_n= \lfloor R_n \rfloor$, involving the floor function. We prove Knuth's conjecture by applying Zeilberger's algorithm as well as classical hypergeometric machinery.

Let $\sigma^0(n) = n$ and $\sigma^m(n) = \sigma(\sigma^{m-1}(n))$, where $m\ge1$ and $\sigma$ is the sum-of-divisors function. We say that $n$ is $(m,k)$-perfect if $\sigma^m(n) = kn$. We have tabulated all $(2,k)$-perfect numbers up to $10^9$ and all $(3,k)$- and $(4,k)$-perfect numbers up to $2\cdot10^8$. These tables have suggested several conjectures, some of which we prove here. We ask in particular: For any fixed $m\ge1$, are there infinitely many $(m,k)$-perfect numbers? Is every positive integer $(m,k)$-perfect, for sufficiently large $m\ge1$? In this connection, we have obtained the smallest value of $m$ such that $n$ is $(m,k)$-perfect, for $1\le n\le1000$. We also address questions concerning the limiting behaviour of $\sigma^{m+1}(n)/\sigma^m(n)$ and $(\sigma^m(n))^{1/m}$, as $m\to\infty$.

Using a Cayley program, we get all firm, residually connected geometries whose rank-two residues satisfy the intersection property, on which $M_{11}$ acts flag-transitively, and in which the stabilizer of each element is a maximal subgroup of $M_{11}$.

We show that the norm of a composition operator on the classical Hardy space $H^2$ cannot be computed using only the set of $H^2$ reproducing kernels, answering a question raised by Cowen and MacCluer.

This is an experimental study of the rank of elliptic curves over $\Q$ obtained by specializing to integer values the parameter $t$ of curves over $\Q(t)$ having rank 0 to 4.

On étudie expérimentalement le rang des courbes elliptiques sur $\Q$ obtenues par spécialisation entière du paramètre $t$ de courbes de rang allant de 0 à 4 sur $\Q(t)$.

Cameron has introduced a natural one-to-one correspondence between infinite binary sequences and sets of positive integers with the property that no two elements add up to a third. He observed that, if a sum-free set is ultimately periodic, so is the corresponding binary sequence, and asked if the converse also holds. We present here necessary and sufficient conditions for a sum-free set to be ultimately periodic, and show how these conditions can be used to test specific sets. These tests produce the first evidence of a positive nature that certain sets are, in fact, not ultimately periodic.

We address the question: How should $N$ $n$-dimensional subspaces of $m$-dimensional Euclidean space be arranged so that they are as far apart as possible? The results of extensive computations for modest values of $N, n,m$ are described, as well as a reformulation of the problem that was suggested by these computations. The reformulation gives a way to describe $n$-dimensional subspaces of $m$-space as points on a sphere in dimension $\half(m-1) (m+2)$, which provides a (usually) lower-dimensional representation than the Plücker embedding, and leads to a proof that many of the new packings are optimal. The results have applications to the graphical display of multi-dimensional data via Asimov's grand tour method.